Catholic Russia Converted Ideas.

The Goal is to simplify and to show, in an intuitive way, some important concepts of Physics... not everything is explained, of course, but you may have already a 1^st idea of the reasoning, when this kind of subject appears at school or in the books of Physics...

Physics For Kids: Physics for everybody, for kids also, for the children of God also, of course... University and Secondary education not needed to understand this...

Understanding the Idea Behind the Dynamos, the Dams and the Cars...

Young people don't generally feel difficulties with this kind of concepts anymore, but the older generations may feel it, and, in lands, where the education is not yet correctly restructured, some students may, possibly, still have difficulties with this kind of concepts...

The idea behind the Dynamo, the Electromagnetism, the Production of Electricity is very simple:

A Magnet, in Movement, Produces Electricity!

The Opposite Is Also True:

Electricity (Electrons in Movement) May Move Magnets Also!

This Is the Basic Idea Behind the Production of Household Electrical Appliances, as Wash Machines, etc.

Concerning the Bike and the Dynamo, everything becomes very clear:

The Head of the Dynamo moves (rotates), rotating the magnet of the Dynamo, producing movement of Electrons, which are sent, through an electric wire, to the lamp of the Bike, producing, thus, in this way, light...

We can go a bit further in this reasoning, with the help of God: the magnet has two poles, a positive and a negative one. The negative pole has negative electric charge: electrons concentrate there. This is often the strongest negative component in an electric electric circuit.

In this way, when the dynamo rotates, the electrons of the electric wire, flee the negative component of the magnet, in direction of the lamp of the bike.

But the dynamo keeps rotating, and, immediately after, the positive pole of the magnet is near the electric wire, and, as such, the electrons of the electric wire, react, and are attracted by the positive pole of the magnet.

These movements proceed, while the bike keeps in movement, producing thus, in this way, Alternating electric current, enough to produce some light...

Now that you understand the basic idea behind the production of electricity, you may read other related literature, generally more complicated, for further information, e.g., the Wikipedia articles over 'dynamos' and 'Dynamo Systems for Bicycle Lighting'...

The idea behind the Dam, is exactly the same, as for the dynamo, for production of electricity and electromagnetism:

With a barrier inside a river, the water accumulates and falls. It makes then move / rotate a magnet, connected to electric wires. The rotation / movement of the magnets, produces electricity (electrons in movement). It's exactly the same principle of the dynamo, but applied to the water of a river and not to a bike anymore...

Dams are used since the ancient times, for electricity production also, says God. Because of severe punishments, when knowledge advances too much (Great Flood, fall of Continents), this production of electricity was limited and available only for the wealthy elite, as the beautiful platinum blond Aseneth of ancient Egypt...

The idea behind the (slightly polluting) car, is not exactly the same anymore, but it is of the same style:

The combustion of the fuels is required to activate the procedure of production of electricity by the battery of the car.

Once electricity is produced, everything becomes very spontaneous: that electricity is conducted by electric wires to the proximity of materials with magnetic properties (magnets or electromagnets), which consequently move / rotate...

These movements / rotations set then the car in movement...

Section Maths...

Section School Maths...

Some Mathematics follow... Concerns: all levels, but also Post Secondary Maths, levels: 1^st to 3^rd year, after secondary Maths (higher education or post secondary equivalent).

Maths for Kids: Multiple Integrals, Etc., for Everybody...

The Goal is to simplify and to show, in an intuitive way, some important concepts of Mathematics... not everything is explained, of course, but you may have already a 1^st idea of the reasoning, when this kind of subject appears at school or in the books of Maths...

Maths For Kids: Maths for everybody, for kids also, for the children of God also, of course... University and Secondary education not needed to understand this...

Understanding Fractions, i.e., Numbers of the Type '¾'.

A Fraction is simply a Composed Number, another way to represent a division.

It is composed of two important parts: an Upper Number, called Numerator, and a Lower number (i.e., the number under the fraction dash), called Denominator... The following figure shows this:

Left: Our Lady, Nordic Look. Right: Those who honor Jesus, with full trust, through this picture, shall be saved*

Negative Numbers and Negative Areas, for Kids...

In order to understand the concept of Negative Numbers, it's enough to see a Thermometer, in further detail:

The temperatures may be positive and negative. When the temperatures become negative, it's cold and the water freezes, becoming ice / snow... What happens to the temperatures, near 0 °C?

If we have 3 °C, and the temperature falls, we have, thus, this kind of evolution:

3 °C, 2 °C, 1 °C, 0, -1 °C, -2 °C, -3 °C, ...

Inversely, If we have -3 °C, and the temperature rises, we have, thus, this kind of evolution, respecting the conventions for the official Cartesian Coordinate Systems and for the graduated straight line:

-∞ , ... ,-3 °C, -2 °C, -1 °C, 0, 1 °C, 2 °C, 3 °C, ..., +∞

(where "∞" means infinite...)

Thus, negative numbers do exist: it's enough to analyze a reality in the correct context...

Still two other examples:

debts / loans may be considered as "negative money", while salaries may be considered as "positive money". In this case, a bank account, with a balance available of 1000 $, means that his / her owner possesses + 1000 $,
while a bank account,
with a balance available of - 100 $, means that his / her owner has spent all his / her money, and has already a debt of 100 $, towards the credit institution (bank).
Altitudes / Depths (Measuring Vertical Relief Distances): When going up, the Altitude increases, e.g., the highest point of the Everest is near + 8900 m high,
while going down,
under the sea levels, we may have a depth of 1000 m deep, in the deep waters, which means, in practice, -1000 m...

What About Negative Areas?

The idea is the same, but for areas. As we know, the easiest formulas to compute the Area of a figure, are those of the Rectangle and of the Square...

Area of the Rectangle = Length × Width

Area_Square = Side × Side = Side²

Normally these surfaces / Areas are positive... but we may have negative Areas, also... But How?

If we have Negative Numbers, one of the Dimensions of the Square / Rectangle may be positive and the other negative, in order to produce a negative area...

Please note that the length of a side of a Rectangle / Square, with a negative value is precisely the same length of the same side, but positive... that side, is simply located in the negative side of the Coordinate System, that's all!

In this way, a Square of side 2 cm, gives an area of 4 cm²...

This may interpreted as: in two minutes, I sold 2 ice creams, 2$ / ice cream... thus, I earned 4$, in total...

This can be represented by a square with Area 4, and side 2...

Indeed:

Area_{this Square} = Side × Side = 2 × 2 = 4...

On the other hand, 2 clients bought these same 2 ice creams, in two minutes time... each client "lost" 2$: each client had less 2$, -2$, after the transaction... Together, they "lost" 4$, -4$...

Indeed:

Area_{this Square / Rectangle} = Length × Width

Area_{this Square / Rectangle} = 2 Minutes × (-2)$= 2 × (-2) = -4...

This is also the idea behind the integral calculus, for negative integrals / areas... More information over integral calculus, in the sections "Integrals for Kids"...

The following figure shows this...

Examples of Negative Numbers and Negative Areas

We may, thus, also have negative Volumes... the idea is exactly the same, but applied to the 3D space: a negative Volume may appear, when 1 of the dimensions is negative, and the other two positive, or, either, alternatively, having all three dimensions with negative values... e.g., when computing the Volume of a parallelipiped. Indeed:

Volume_{Parallelipiped / Cube} = Length × Width × Height

Volume_{Parallelipiped / Cube} = 1 dm (Length) × 1 dm (Width) × -1 dm (Depth)

Volume_{Parallelipiped / Cube} = -1 dm³ (of water), from the surface of the swimming pool...

This figure shows this:

Example of a Negative Volume: - 1 dm<sup>3</sup>
of water of a Swimming Pool...

Example of a Negative Volume: - 1 dm³ of water of a Swimming Pool...

Basic Equations for Kids.

Assume we have a problem like this: I want to invite two friends to drink a glass of cola. I have only a glass, right now, but we need 3 glasses: one per person. How many extra glasses do I need, in order to have three glasses, in total?

Indeed, we may set the problem in this way: 1 plus "something" (the value we need to calculate) equals 3. We may represent this "something" by a box (a square), in this way:

1 + ⬜ = 3

Alternatively, since this is the value we want to calculate, we may call it the variable 'x', in this way:

1 + x = 3

We call these kind of equalities, where there's an unknown value (a variable) we want to compute, an Equation.

And How do we calculate the value of 'x', the number of additional euros I need, in order to buy the required ice cream?

We need to define a methodology for that. In this case, we can still solve it spontaneously, but in other cases, more complicated, without a methodology, we're won't be able to solve it, in an easier, quicker way...

That Methodology Does Exist: It's the Methodology of the Traditional Mechanical Balance, with 2 Plates, in Equilibrium.

This Figure tries to Show this:

Solving the Equation 1 + x = 3, with the help of a Traditional Balance, with 2 Plates, in Equilibrium

Matrices for Teenagers.

A basic equation, with an unknown variable named 'x', is a Mathematical expression of the type:

2x + 1 = 3

We can simplify a bit this basic equation, removing the '1' from the left side of the equation, following the general methodology to solve simple equations, (analog to a balance of plates in equilibrium):

Removing one unit (the digit '1'), from the both sides of the equation / equality / balance, doesn't affect the equilibrium / equality (both sides remain with the same number of units):

2x + 1 - 1 = 3 - 1

Thus:

2x = 2

This kind of equations are simple equations, with one unknown variable, here named 'x'. These equations are also called equations of first degree in 'x', because they contain simple factors in 'x' and not expressions of the type 'x²' nor 'x³' nor '√x'.

Equations could include also several different variables, which we could name 'x' and 'y', or, either, alternatively, x₁, x₂...

For this analysis, we discard, expressions of the type 'xy', 'xy²', 'x²', 'x³', '√x', 'y²', 'y³', '√y'...

We could have then an equation with 2 unknowns (variables) named here 'x' and 'y', which may define a function, y = f(x), or a straight line, in the 2D Cartesian Coordinate System. E.g.:

x + 3y = 5

This kind of equality has infinite solutions; it defines a straight line in the 2D Plan. If we add a 2^nd simple equation, of the same style, we may, in many cases, define the expression for another straignt line, different from the first one. In the 2D Cartesian Coordinate System, these 2 equations may define 2 different straight lines, that intersect in a given point... We could have, in many cases, thus, 1 solution, for both variables 'x' and 'y', the intersection point: the coordinates in x (horizontal axis) and y (vertical axis), in the plan...

These 2 equations could be of this style:

1x + 3y = 5 <=> x + 3y = 5

2x + 4y = 8

These 2 equations could express a problem of this kind:

We name the (unknown) price of an ice cream, as the variable 'x' and the (unknown) price of a cake, as the variable 'y'.

The mother buys, in a coffee shop, 1 ice cream and 3 cakes, for her children. She spends, for that, 5 €...

Thus:

x + 3y = 5 <=> 1x + 3y = 5

Later in time, the mother buys, in the coffee shop, 2 ice creams and 4 cakes, for her children. She spends, for that, 8 €......

Thus:

2x + 4y = 8

We're not going to solve these equations, but we may observe that "x = 2", with the meaning "an ice cream costs 2 €, and "y = 1", with the meaning "a cake costs 1 €", are solutions for this system of equations, replacing "x" by "2" and "y" by "1", in the previous system of equations:

With "x = 2" and "y = 1", we have, thus:

1x + 3y = 5 <=> (1).(2) + (3).(1) = 5 <=> 2 + 3 = 5

2x + 4y = 8 <=> (2).(2) + (4).(1) = 8 <=> 4 + 4 = 8

Note: "." designs here "multiplication", in this example...

In the 2 dimensional Cartesian Coordinate System, the intersection of the 2 straight lines, gives, thus, the solution, the intersection point P = (x,y) = (2,1), when solving, either this system of equations or either representing it graphically, in the 2D plan, because each of these equations represents also a straight line, in the 2D plan...

For this kind of simple (called linear), equations, containing several (unknown) variables, there's an alternative notation, called 'Matricial Notation'.

The Following Figure shows this:

Introducing the Matrix Notation, based in a System of 2 simple (linear) Equations, in the unknown variables 'x' and 'y'...

Going Further With Matrix Notation...

Matrices are simply tables, with coefficients of each variable, set in a specific organized way. Matrices may, thus, be interpreted as "coefficient" matrices... they're just another way to express sets of equations, with several unknown variables...

We could still go further, with matrix notation for linear equation systems, with 3 equations and 3 unknown variables: this case, if the equations are adequate, gives a unique solution for the variables, 'x', 'y' and 'z' (which we could, alternatively, rename as 'x₁', 'x₂' and 'x₃').

In many situations, such a linear system, with 3 equations and 3 variables, may give the intersection point P = (x, y, z) of the 3 planes in the 3D space, visible in a 3D Cartesian Coordinate System, but more difficult to implement...

Each equation with 3 variables, may, thus, in many cases, implement a plane in the 3D space (in the Cartesian Coordinate System of ℝ³, thus) ...

The generalization can still go further, with matrix notation for linear equation systems, with several equations and several unknown variables: variable names 'x', 'y', 'z' become then less adequate, e.g., if we have 5 equations with 5 unknown variables... variable names as 'x₁', 'x₂', 'x₃', 'x₄' and 'x₅', become then more adequate...

The coefficients, inside the matrix table, when described, in a generic way (replacing, thus, the coefficient numbers by "letters") are, often, represented in this way:

a_ij or c_ij,

Where:

the letters 'a' or 'c' mean 'coefficient'
'i' gives the number of the row in the given matrix
'j' gives the number of the column in the given matrix

E.g, c₄₃ means that the coefficient of the matrix, in position 'row 4' and 'column 3', corresponds to the coefficient of the variable x₃ (x₃ is sometimes called 'z'), in the 4^th equation of the linear system with several equations (at least 4) and several variables (at least 3)...

The following picture goes, thus, a bit further, in this analysis over matrix notation and tries to show some of this:

Going further with Matrix Notation...
References: Bibliography from IST Lisbon, VUB Brussels, Francophone Secondary Schools from Brussels

Square Roots for Kids: Understanding Square Roots, i.e., Numbers of the Type i = '√-1'.

In order to understand the concept of square root of number 'y', '√y' we shall begin to review the formula for the Area of a square:

Area_square = Length_square × Width_square

In Practice:

Area_square = Side_square × Side_square

Thus, if we have a square with 3 cm of side, the Area of this Square is:

Area_square = 3 cm × 3 cm = 9 cm²

This means also that:

'Three squared' ('the square of three'), '3²' equals:

3²= 3 × 3 = 9

Per analogy, a square with side 5 m, has an Area of 25 m²

This means:

Area_{second square} = 5 m × 5 m = 25 m²

This means also that:

'Five squared' ('the square of five'), '5²' equals:

5²= 5 × 5 = 25

With the terms 'Five squared', (named also 'the square of five'), and represented by '5²', we make a simple Introduction to an Important Subject in Mathematics, the 'Exponentiation', the 'Exponent of a Number'...

E.g., in the case of 5² = 25, the base number is the number '5' and the exponent (power) is the number '2'...

For further details, you may, e.g., read the Wikipedia article over this subject, 'Exponentiation'...

What About the Square Roots, the √???

Once you understand good the Concept of a 'Square of a Number', It's Not Difficult Anymore to Understand the Concept of a 'Square Root of a Number "y"', '√y'...

Indeed:

If the Square of 5, 5² equals 25,

This Means also that the inverse operation may also exist...

Thus, if a square with side 5 has an Area of 25,

If Area_square = 5 × 5 = 5² = 25

Then, if we have a square with Area equaling 25, and we want to know the length of its Sides, we have to use the inverse operation for that, √25

√25 = 5,

because the side of the corresponding square, with area 25, has to measure 5...

Per analogy:

√9 = 3,

because the side of the corresponding square, with area 9, has to measure 3...

Watch out: For numbers like, 0, 1, 2, 4, 9, 16, 25, 36, 49, ..., it's relatively easy to compute the Square root of these numbers, using only the knowledge of the tables of Multiplication of the Primary Education... because:

0 = 0 × 0 = 0², and, thus, √0 = 0

1 = 1 × 1 = 1², and, thus, √1 = 1

4 = 2 × 2 = 2², and, thus, √4 = 2

9 = 3 × 3 = 3², and, thus, √9 = 3

...

49 = 7 × 7 = 7², and, thus, √49 = 7

...

But for other areas, the respective squares don't have sides, with such perfect measures!

E.g., if a square has Area of 2, What Shall its Sides Measure?

The side of this square measures, thus, √2, but this number is not a perfect integer anymore...

There are ways to compute this value, but it's more complicated to compute it... e.g., you may approximate values, use the calculator, use an algorithm (method), which exists, for that...

The perfect square areas to approximate √2 are 1, because 1 = 1 × 1 and 4, because 4 = 2 × 2...

This means that a square with area 2, has a side with length between 1 and 2, because those squares have area between 1 = 1 × 1 and 4 = 2 × 2...

Indeed,

√2 = 1.4142...

because:

√2 × √2 = 1.4142 × 1.4142 = 1.999 ~ 2...

... but it's not very easy to compute it anymore... you may use the calculator to see a better approximation for √2...

And What About Square Roots of Squares with Negative Areas, e.g., √-1?

That subject is targeted by Maths, also, in the subject 'Complex Analysis', which studies the Algebra and Geometry of the Sides of Squares with Negative Areas, e.g., sides measuring √-1, etc.

An introduction to this subject appears here, in this Chapter, in the section 'Introduction to the Theory of Supernatural Numbers of ℂ, Complex Analysis of 2^nd Order: Another graphical representation for i = √-1', but, Watch out, level Maths of the Secondary Education recommended, in order to fully understand it, and, preferably, to read also first the section 'Integrals for Kids', if you don't understand this subject good yet...

Integrals for Kids: Simple and Multiple Integrals...

Simple Integrals for Kids...

What is a Simple Integral?

An integral, from the Integral Calculus, is simply a middle to compute the area of a figure...

Some integrals may be solved analytically, with formulas, sometimes a bit complicated...

Other integrals are hard to be solved analytically... It's possible, but, sometimes, it becomes so difficult, that it's only waste of time... it becomes a parasite activity, confirms God: those demonstrations of 700 pages A4, completely full of calculus, non stop, may be valid for a Doctorate thesis, but only for that... it's waste of time...

Indeed, all Integrals may be solved numerically, but they compute Areas of Figures, in the case of the simple Integrals, or Volumes of Objects, in the case of the double Integrals...

And How Can We Compute an Integral, In an Easy Way?

The Simplest way to Compute an Area, is to use the Formula for the Rectangle:

Area_Rectangle = Length_Rectangle × Width_Rectangle

The Problem: That formula is valid for Rectangles (and for squares), but most of Surfaces / figures have other forms...

That's true. But we can approach all other formulas, with rectangles: I mean, we can approach the area of a given figure by a sum of rectangles / squares equivalents, or either decompose that figure in rectangles.

Those rectangles don't need to cover completely the figure (which we want to compute the area), they only need to cover it, with an adequate approach / rigor.

This is similar to the number of digits, after the comma, for a decimal number given by the fraction ⅓...

Indeed,

1/3 = 0.3333333333(3)

The digit 3, in order to represent 1/3 correctly, should be repeated, up to the infinite...

In practice, we don't do that, and we approach 1/3 to ~0.333... That' enough to give an idea of the decimal value of 1/3...

With the numeric integrals, it's the same: We approach the figure, which we want to compute the Area, by a near equivalent figure, similar, but not the same anymore (in most cases), composed only by rectangles...

The following pictures show this:

Integrals For Integrals: Understanding Integrals

Computing the Numeric Integral, for a Half Circle. Auxiliar Explanations.

Please note that we approached the half circle (superior part) with three rectangles... we can use a grid for that, in order to try to find the dimensions of the desired bounding rectangles...

When x = 1/2 = 0.5, the respective y, in the circumference, measures already 0.87 (upper half circle)...

The bounding rectangle of the middle, named here R2, with -0.5 ≤ x ≤ 0.5, has a width of 1 (difference between 0.5 and -0.5 equals 1): we can notice it, observing the right side of the previous figure (example 3b).

This bounding rectangle of the middle, may still have a height of 1. We can see in the figure that the error is not yet big.

We can notice also from the figure, that, when x = 0, the respective y equals 1...

The remaining parts of the circle are approached by 2 smaller rectangles, equal in dimensions and symmetrical...

The value of the y coordinate of the circumference, when x=1/2 or x=-1/2, equals:

f(0.5)=y(1/2)=0.87

We computed these values in the previous picture, when observing some auxiliary data for the lower half circle, with negative area, but similar to this one...

From the image of the lower half circle, we may observe that the bounding square with side 0.5, e.g., when 0.5 ≤ x ≤ 1 is too big (too high) and introduces a huge amount of error, if it's used to approach the remainder surfaces of the half circle...

A higher height is, thus, a better approach, for the dimensions of R1 and R3...

Watching the half circle, and using a grid (or a ruler...), we could try to find an adequate height (here, in the y axis), for the remaining bounding rectangles R1 and R3...

Observing the figure, we may conclude that the Y coordinate in the circumference, when x equals 3/4=3:4= 0.75, could give a relative good average height, for the Rectangles R1 and R3....

We can try to get this value directly from the graphic, or we may compute it, using the function f(x), which gives the values on the circumference, over the upper half circle...

Taking the grid, the value of this Height seems to be ~0.65, while the Width = 0.5, when x varies between 0.5 and 1, for R3, or, either, Width = 0.5, when -1 ≤ x ≤ -1/2...

Computing f(0.75), to confirm it, we get:

f(3/4) = y(0.75) = Square Root of (1-(3/4)²) = √(1-(9/16)) = √(7/16) = √(0.44) = 0.66

Thus, f(0.75) seems to be more or less OK and this height equals 0.66 (the calculator used for this...)

This confirms the value read from the graphic of 0.65 (almost identical...), which was used to compute the areas of the smaller Rectangles, R1 and R3...

The Simple Numeric Integral, As a Sum of Products of 2 Factors.

Please note that a Numeric Integral, is nothing else than a Sum of the Areas of the Concerned Rectangles!

Once we have approached the function / figure (we want to compute the Area), by adequate (smaller) rectangles, we compute the area (of each adequate rectangle), simply performing products of 2 factors (terms): a term 'Length' and a term 'Width'...

The Numeric Integral, the Area of the desired Figure, is, thus, given by:

Area_{desired Figure} = Area_{Rectangle Auxiliary₁} + Area_{Rectangle Auxiliary₂} + ... + Area_{Rectangle Auxiliary_n}

Because the Area_{Rectangle Auxiliary₂}= Length × Width, the previous Sum may be given by:

Area_{desired Figure} = Length_Rectangle₁ × Width_Rectangle₁ + Length_Rectangle₂ × Width_Rectangle₂ + ... + Length_{Rectangle_m} × Width_{Rectangle_m}

We can observe this, in the previous figure, when we computed the numeric integral (Area) of the upper half circle, with ray 1, centered in the origin of the Cartesian Coordinate System...

We approached the half circle by 3 (smaller) Rectangles: R1, R2 and R3...

The Area of the Half Circle was, thus, approached by:

Area_{Half Circle} = Area_{Rectangle R1} + Area_{Rectangle R2} + ... + Area_{Rectangle R3}

Thus, Area_{Half Circle}, is, thus, approached by:

Area_{Half Circle} = Length_R1 × Width_R1 + Length_R2 × Width_R2 + Length_R3 × Width_R3

We replaced this, by the corresponding dimensions, getting, thus:

Area_{Half Circle} = 0.65 * 0.5 + 1 * 1 + 0.65 * 0.5

WHAT THIS MEANS IS THAT THE SIMPLE NUMERICAL INTEGRAL, THE APPROACHED AREA OF THE GIVEN FIGURE / FUNCTION, MAY BE ALWAYS COMPUTED AS A SUM OF SIMPLE MULTIPLICATIONS (each multiplication with only 2 terms / factors).

INVERSELY, THIS MEANS ALSO, THAT, WHEN YOU SEE AN ALGEBRAIC EXPRESSION, as:

Result = 1.5 × 2 + 2 × 3 + 4 × 2 + 0.5 × 2

THIS ALGEBRAIC EXPRESSION MAY HIDE THE CALCULUS OF AN AREA / of a Numeric, Simple Integral...

Thus, in this last example, that sum expressed also the calculus of the Area of a hidden figure / function, approached by (m = ) 4 different Rectangles... the values may also be negative, of course (negative areas / integrals)...

Double Integrals for Kids: The Double Numeric Integral, As a Sum of Products of 3 Factors.

Once we have understood the ideas, behind the calculus of the universal, simple, numeric integral and the form of its algebraic expression, which computes a given simple numeric integral / the area of a figure (function), it's relatively easy to generalize it to Volumes (3 dimensions) and to upper dimensions...

To Compute Volumes / the Double Numeric Integral, the idea is similar to that for the Simple Numeric Integrals, adequate to compute Areas of figures / functions...

Indeed, every object (with volume), may be approached by Parallelipipeds.

The Volume of an irregular object may be difficult to compute, but the Volume of a Parallelipiped is very easy to compute:

Volume_{Parallelipiped} = Length × Width × Height

A Volume of a Parallelipiped expresses, thus, a Product of 3 Factors...

It May Be Negative also: it suffices to place parts of it in negative parts of the 3D Cartesian Coordinate System...

Thus, an irregular object may be approached by adequate, smaller, (m) Parallelipipeds...

The error may be greater or smaller, according to the quality of the approach: with more (smaller) adequate Parallelipipeds, the error is generally smaller, but it requires more time to compute it, and, with less adequate Parallelipipeds, the error is generally greater, for irregular objects...

Thus, the Double Numeric Integral, the Volume of the Desired Object (Function of 2 variables, in ℝ² → ℝ or isomorph), may be simply given by a sum of products of 3 factors (terms).

Thus:

Volume_Object = Volume_{Adequate Parallelipiped₁} + Volume_{Adequate Parallelipiped₂} + ... + Volume_{Adequate Parallelipiped_n}

Thus, the Volume_Object may be approached by:

Volume_Object = Length_{Adequate Parallelipiped₁} × Width_{Adequate Parallelipiped₁} × Height_{Adequate Parallelipiped₁}

Length_{Adequate Parallelipiped₂} × Width_{Adequate Parallelipiped₂} × Height_{Adequate Parallelipiped₂}

...

Length_{Adequate Parallelipiped_m} × Width_{Adequate Parallelipiped_m} × Height_{Adequate Parallelipiped_m}

INVERSELY, THIS MEANS ALSO, THAT, WHEN YOU SEE AN ALGEBRAIC EXPRESSION, as:

Result = 1.5 × 2 × 3 + 2 × 3 × 4 + 4 × 2 × (-1) + 0.5 × 2 × (-2)

THIS ALGEBRAIC EXPRESSION MAY HIDE THE CALCULUS OF A VOLUME OF AN OBJECT / THE CALCULUS of a NUMERIC, Double Integral...

Thus, in this last example, that sum expressed also the calculus of the Volume of a hidden Object / function, approached by (m = ) 4 different Parallelipipeds... the values may also be negative, of course (negative Volumes / double integrals)...

And What Are the Symbols for the Double Integrals?

The Nomenclature used for the analytical, complicated, double integral is:

∫∫f(x,y)dydx

Example: ∫∫f(x,y)dydx = ∫∫(x² + 2xy)dydx,

where f(x,y) = x² + 2xy

The Principle is similar to that of the simple integral. More information, over the simple integral in the section 'Simple Integrals for Kids'.

Above and below the symbols of integrals, '∫', we have, generally, the upper and lower indexes (not visible here).

In the case of:

∫∫f(x,y)dydx = ∫∫(x² + 2xy)dydx

We have a 3D function of 2 variables, f(x,y) = x² + 2xy, where the 2 variables are denominated 'x' and 'y'.

'dydx' means that the double integral makes vary the variables 'x' and 'y', being 'x' (here, in this case) linked with the 1^st, outer integral and 'y' (here, in this case) linked with the inner, 2 ^nd integral.

Why Is 'y' Linked with the 2^nd, Inner Integral and 'x' linked with the 1^st, Outer Integral?

The notation 'dydx' defines the order of the variables, for the multiple integral, double integral, in this case. In order to have the opposite order, with the same variables, 'x' and 'y', we have to write 'dxdy' instead, which gives this kind of notation:

∫∫f(x,y)dxdy = ∫∫(x² + 2xy)dxdy

with f(x, y) being given, in this case, again, by:

f(x,y) = x² + 2xy

Multiple Integrals for Kids: The Multiple Numeric Integral, As a Sum of Products of n Factors.

Once we have understood the ideas, behind the calculus of the universal, simple and double numeric integrals and its algebraic expression, it's relatively easy to generalize it to n Dimensional Volumes (n dimensions)...

To Compute n Dimensional Volumes / Multiple Numeric Integrals, the idea is similar to that for the Simple and Double Numeric Integrals, adequate to compute the Areas and Volumes / functions...

Indeed, every n dimensional object (with n dimensional volume), may be approached by n dimensional Parallelipipeds.

The Volume of an irregular n dimensional object may be difficult to compute, but the Volume of a n dimensional Parallelipiped is very easy to compute:

Volume_{n dimensional Parallelipiped} = Measure Side 1 × Measure Side 1 × ... × Measure Side n

A Volume of a n dimensional Parallelipiped expresses, thus, a Product of n Factors...

It May Be Negative also: it suffices to place parts of it in negative parts of the n dimensional Cartesian Coordinate System...

Thus, an irregular object may be approached by m (how many? Answer: m) adequate, smaller, n dimensional, Parallelipipeds...

The error may be greater or smaller, according to the quality of the approach: with more (smaller) adequate n dimensional Parallelipipeds, the error is generally smaller, but it requires more time to compute it, and, with less adequate n dimensional Parallelipipeds, the error is generally greater, for irregular n dimensional objects...

Thus, the Triple Numeric Integral, the 4 dimensional Volume of the Desired Object (Function of 3 variables, in ℝ³ → ℝ or isomorph), may be simply given by a sum of products of 4 factors (terms).

For the 4 dimensional Volume of the Desired Object, we may consider the following dimensions:

Dimension 1, in the x axis: Length of the Parallelipiped (measure of Side 1)

Dimension 2, in the y axis: Width of the Parallelipiped (measure of Side 2)

Dimension 3, in the z axis: Height of the Parallelipiped (measure of Side 3)

Dimension 4, in the w axis: Timeline of the Parallelipiped (measure of Side 4)

Thus:

4D Volume_Object = 4D Volume_{Adequate 4D Parallelipiped₁} + 4D Volume_{Adequate 4D Parallelipiped₂} + ... + 4D Volume_{Adequate 4D Parallelipiped_m}

Thus, the (4D =) 4 Dimensional Volume_Object may be approached by:

4D Volume_Object = Length_{Adequate 4D Parallelipiped₁} × Width_{Adequate 4D Parallelipiped₁} × Height_{Adequate 4D Parallelipiped₁} × Time_{Adequate 4D Parallelipiped₁}

Length_{Adequate 4D Parallelipiped₂} × Width_{Adequate 4D Parallelipiped₂} × Height_{Adequate 4D Parallelipiped₂} × Time_{Adequate 4D Parallelipiped₂}

...

Length_{Adequate 4D Parallelipiped_m} × Width_{Adequate 4D Parallelipiped_m} × Height_{Adequate 4D Parallelipiped_m} × Time_{Adequate 4D Parallelipiped_m}

INVERSELY, THIS MEANS ALSO, THAT, WHEN YOU SEE AN ALGEBRAIC EXPRESSION, as:

Result = 1.5 × 2 × 3 × 5 + 2 × 3 × 4 × 1 + 4 × 2 × (-1) × 1.5 + 0.5 × 2 × (-2) × 1/3 + √2 × 4 × (-1) × 0.2

Which means:

Result = Product₁ + Product₂ + Product₃ + Product₄ + Product₅

Which means (with 4D meaning 4 Dimensional):

Result = 4D Volume 4D Parallelipiped₁ + 4D Volume 4D Parallelipiped₂ + 4D Volume 4D Parallelipiped₃ + 4D Volume 4D Parallelipiped₄ + 4D Volume 4D Parallelipiped₅

THIS ALGEBRAIC EXPRESSION MAY HIDE THE CALCULUS OF an n dimensional VOLUME OF AN n dimensional OBJECT / THE CALCULUS of a NUMERIC, Multiple Integral of dimension n - 1, in this case, a triple integral (for 4 dimensions, which means, n = 4)...

Thus, in this last example, that sum expressed also the calculus of the Volume of a hidden Object / function, approached by (m =) 5 different 4 dimensional Parallelipipeds... the values may also be negative, of course (negative n (n=4, here) dimensional Volumes / n-1 (here, triple), multiple, integrals)...

Discrete Maths, Maths Post-Secondary: a Bit of Convolution Product of Cauchy.

Prerequisites

At least complete Maths secondary school (at least 12^th grade), but it's better to have already level 14 completed... To feel more comfortable with the subject, ideally, still 1 or 2 years maths, after the high / secondary school, should be considered.

Some Knowledge of manipulation Double Summations, Series, Power Series could help.

Other Web sites, e.g., the Wikiversité, with subjects such as Series, Power Series, Double Summations, in French (levels 14 and 15), may help to understand better this subject.

Automatic Web translation possible, with Chrome, Edge and Firefox. Please refer to the Web translation instructions of www.christianideas.eu, here, on top of page, which are similar, in order to have a better idea...

Details of the Demo, "Generalized Cauchy Product" of Wikipedia.

As exercise, some details, of the Demo "Generalized Cauchy Product" of Wikipedia, added by me, are now available.

Changes seemed to be approved by Wikipedia (English). But, if, eventually, those changes are removed, a backup copy of those demo details still remains here available.

Practicing Product of Series and Polynomials.

Please enjoy some theory practice / solved exercises over the Convolution Product of Cauchy: Product of Polynomials and Series...

Sorry, for the moment, only draft versions, too much time would be lost, with equation editor details... Comments corrections are welcome, you may send emails, via the Web page www.christianideas.eu/index.php/contact...

Some Practice: Understanding Multi Summations, in the Multinomial Formula.

Please enjoy some theory and practice over the Theorem of the Multinomial, explaining multi Summations in detail, using the Convolution Product of Cauchy, now with small improvements...

Paradox: Cardinality of ℝ Has Order 2^|ℕ| = ℵ₁, But It Has Also Order 2*|ℕ|² = ℵ₀!

This demo exists already, I'm going only to expose it, in my own way. Demonstrations are common in Maths...

How can we represent a real number?

A real number is a positive, negative number (or zero), thus, a number x of the style: x = -289.5872.

Because real numbers may grow, up to the infinite, in absolute value, but also the decimal amount (right to the comma), may have an infinite number of digits, we could represent the real number x, in this way:

x = sign * d_i₁d_i₂d_i₃... d_{i_n}.d_f₁d_f₂d_f₃... d_{f_m}

If the real number x equals -289.5872 this notation means:

the sign of x is '-1', because x is a negative number.

The Integer part of x, left to the decimal point, equals 289. Thus:

d_i₁d_i₂d_i₃=289

with n=3, meaning that the integer value of x is composed of 3 digits.

Right to decimal point '.', we have the decimal part of x, represented here by '5872'.

Thus, here, d_f₁d_f₂d_f₃... d_{f_m} equals 5872

where m = 4 (fractional part has 4 digits), where this 5872 should be seen as 0.5872 × 10^m

and thus d_f₁=5, d_f₂=8, d_f₃=7, d_f₄=2.

The real number x can also be seen as:

x = sign*x_i.x_f=sign * d_i₁d_i₂d_i₃... d_{i_n}.d_f₁d_f₂d_f₃... d_{f_m},

where sign is the applied sign of the real x

and

x_i and x_f are natural positive (or null) numbers, thus,

sign ∈ {-1,1},

x_i ∈ ℕ and x_f ∈ ℕ .

This means that x_i=⌊|x|⌋ can be any natural number.

The same for x_f, the fractional part of real x, right to the comma.

And how many different possibilities we have for ⌊|x|⌋?

Because x_i can be any natural number, the total possibilities is the cardinality of ℕ self, thus,

|ℕ| = ℵ₀ .

In a similar way, the cardinality of x_f is also the cardinality of ℕ self, thus, also |ℕ| = ℵ₀.

Seen in this way, the cardinality of real number x is very easy to estimate. Even the ambiguity 0∈ ℕ is only a less important detail for this purpose.

Thus, the cardinality of a real number x, |x|= |sign ∈ {-1,1} |*| x_i|*|x_f|

Please note that, because x_i = d_i1d_i2d_i3... d_in is a positive integer number (or null) of n digits, and each digit can be any number ∈ {0,1,2,3,4,5,6,7,8,9}, | x_i|= 10ⁿ.

This means that there are 10ⁿ different ways to represent ⌊|x|⌋ (in practice |ℕ|= ℵ₀.

Thus, |x| = 2 * |10ⁿ|*|10^m| = 2*|ℕ|* |ℕ|.

Assuming 0 is counted twice, because 0 ∈ x_i and 0 ∈ x_f ⇒ we have to remove 1 to the value of |x| calculated above.

Thus, |x| = 2 * |10^n+m|-1 = 2 * |ℵ₀|² -1 .

Because n ∈ ℕ and m ∈ ℕ ⇒ n+m ∈ ℕ also. They may have all cardinality of |ℕ| also^*. Specially irrational numbers like ϖ or √2 have decimal part with infinite digits... others are much smaller like e. g. 0.5=1/2 ...

Thus, |x| = 2*|10^|ℕ|| -1.

Because 10 = 2^log₂10, we have also:

Thus, |x| = 2*2^{log₂10*|ℕ|} -1.

Thus, |x| = 2^{1+ log₂10*|ℕ|} -1.

Thus, we can estimate the order of |x| = 2^ℵ₀, which defines the cardinality of ℝ.

* - Watch out: contradiction possible, because |ℕ| = 2^log₂|ℕ| and |x| = 2*|ℕ|*|ℕ|-1 = 2* 2^2log₂|ℕ|-1 = 2^{1+ 2log₂|ℕ|} -1. Or log₂|ℕ| should be different from ℵ₀, because ℵ₁= 2^ℵ₀ ⇒ ℵ₀= log₂ ℵ₁.

This results from concept of infinite, which may introduce such ambiguities and may lead afterwards to paradoxes, e.g., ∞ x ∞ = ∞² = ∞. Why not consider |ℕ|² >|ℕ|? It should be if |ℤ|>|ℕ|... Not to accept this while consider ℵ₁ > ℵ₀ may lead to incoherencies, even to paradoxes...

The easiest is to keep accepting theory of infinite sets as it is... but in this case, one should watch out however the coherency of results, when dealing with the infinite. E.g., I don't accept that |ℤ|=|ℕ|.

It's evident |ℤ|=2*|ℕ| + c, where c is a correction factor (e.g. -1, according to definition of the moment for |ℕ|)...
We can also see it in this way: |ℤ| is of same order of |ℕ|, which is ℵ₀. This reasoning helps to understand official cardinality theory ⇒ ∞/2 = ∞... but it's not the only logic...

There are different logics when we use the concept of infinite... Because of this, the logic to be used should be seen case per case... present theory is already valid in most of the cases; only watch out incoherencies...

It could also be possible that I introduced errors, when claiming also the possibility that |ℝ|= ℵ₀ ... Officially it is ℵ₁, and we can also demonstrate it here... eventually, further analysis needed? or it doesn't worth it... ???

Demonstrations, eventually, better explained over | ℝ|, exist, e. g., at Wikipedia, 'Cardinality of the Continuum'. But it's worth analyzing and practicing it, specially for those, with university scientific profile...

Maths Post-Secondary: Complex Analysis.

Prerequisites

Post secondary knowledge of Maths is required, to understand better Hyper-geometric series and functions. It's recommended to have knowledge of the 1^st 2 years of general Maths, of a scientific high school course or equivalent (except, eventually, Math's specialization's).

Practice, Over the Hypergeometric Function, to Get Cos(z).

As reference, you may begin with the Wikipedia article "Generalized Hypergeometric Function".

Please enjoy the Solved Exercise: proof ₀F₁ ( ; 1/2; -z²/4) = Cos(z), using the rising factorial formula, for hypergeometric series and functions.

I tried to set this in Wikipedia, "Generalized Hypergeometric Function: The series ₀F₁". It was not approved, but a backup of the Wikipedia's resolution remains available here.

Sorry, for the moment, the draft versions remain: too much time would be lost, with more rigorous details, using an Equation editor... even so, several hours were already lost, to try to show the exercises + demonstrations, with a ± adequate presentation. Comments / corrections are welcome. You may send emails, via the page www.christianideas.eu/index.php/contact...

Divine Nomenclature For the Number Sets.

The Current names for the Number Sets is not adequate...

Indeed, presently, we have:

ℕ - the set of Natural Numbers. ℕ = {1, 2, 3, ...}
ℤ - the set of Entire Numbers. ℤ = {..., -1, 0, 1, ...}
ℚ - the set of Rational Numbers. ℚ = {..., -1/4, ...}
ℝ - the set of Real Numbers.

ℝ = { ..., -1, ..., √2, ...}
ℝ \ ℚ - the set of Irrational Numbers.
ℂ - the set of Complex Numbers.
ℂ \ ℝ - the set of Imaginary Numbers.

Some of these names are not adequate. Indeed, the correct name for numbers of the style

i = √-1

is not 'Imaginary Numbers', but 'Supernatural Numbers'...

As consequence, the Set of the Real Numbers ℝ should be renamed, for coherency, not to 'Real Numbers' anymore but to 'Natural Numbers'!

The problem is that, the set of Natural Numbers, ℕ, already exists, but it designs something else: the set of the Positive Entire Numbers, ℤ⁺.

This shall become confusing for the current generations, used to this kind of nomenclature, but it could be easily learned by the future generations...

Thus, the "old set of 'Natural Numbers'" should be simply renamed 'Set of Positive Entire Numbers', identified only by ℤ⁺, and not by ℕ anymore.

The name Irrational Numbers is not adequate neither. Those numbers, like √2, are not Irrational, but Indivisible, says God.

As such, the ℚ Number Set shall be renamed as the 'Set of the Divisible Numbers'...

The correct Number Set Nomenclature shall, thus, be:

ℤ⁺ - the Set of Positive Entire Numbers.

ℤ⁺ = {1, 2, 3, ...}

ℤ⁺, "previously" named ℕ...
ℤ - the Set of Entire Numbers.

ℤ = {..., -1, 0, 1, ...} = ℤ^- U {0} U ℤ⁺
D - the set of Divisible Numbers.

D = {..., -1/4, ...}

D , "previously" named Rational Numbers, ℚ...
ℕ - the set of Natural Numbers,

ℕ = {..., -1,..., √2, ...}

ℕ, "previously" named ℝ...
ℕ \ D - the set of Indivisible Numbers.

ℕ \ D = {..., -3√2,..., √3, ...}

ℕ \ D, "previously" named Irrational Numbers, ℝ\ℚ.
ℂ - the set of Complex Numbers.

ℂ = {..., -1, ..., -1/3 + 2√-1, ...}
ℂ \ ℕ - the set of Supernatural Numbers.

ℂ \ ℕ = {..., -2i, ..., (4√3)i, ...}, with i = √-1

ℂ \ ℕ, "previously" named Imaginary Numbers, ℂ\ℝ.

Being, thus, ℂ = Set of Natural Numbers (ℕ) U Set of Supernatural Numbers (ℂ \ ℕ) ...

This section was inspired by God.

Introduction to the Theory of the Supernatural Numbers of ℂ, the Complex Analysis of 2^nd Order: Another graphical representation for i = √-1.

With the help of a Cartesian Axis, we can represent a square, with area = -1, and, as such, we can, thus, represent the supernatural number i = √-1 (called imaginary, in the official Math's nomenclature), in another way. All this can be very intuitive and easy to understand.

Watch out: the Theory of Supernatural Numbers, the Complex Analysis of 2^nd Order, may further develop and become dangerous and unethical, like Artificial Intelligence. This is allowed, because we approach the end of age. It may introduce temptation, and, thus, as such, it is recommended good behavior, pro God.

Warning, over the Dangers of the Supernatural Sciences: one may develop "superpowers", and help themselves and / or the others, e.g., via subconscious "prayers", but the concerned "champions" should first ask permission to God for doing that, as Enoch and Elijah did, in order to avoid traps in the style 'Aseneth = Isis' / ' husband Joseph = Osiris', even if the demands seem "serious", "pro good", "pro God".

More information over these subjects, in the Chapter 'Danger of Apocalypse' and in the section 'Isis and Osiris: How the Study of Supernatural Sciences Made Them Gods of the Powerful Witchcraft of the Evil Religion of the Ancient Egypt and the Freemasonry / Illuminati' of the Chapter 'Islam, Protestants, etc.'

The basic idea is very simple: i * i = √-1 * √-1 = -1, which is simply a square, with area -1.

Normally, we think that's not possible in the real world: there are no squares, with negative areas...

However, with the theory of the Integrals, we learn that we may have integral calculus, which may produce negative numbers...

As we know, the simple integral calculus computes areas of figures / surfaces... It's relatively easy to compute integrals of simple functions and see the respective area in the Cartesian graphic (from: generic Maths, end secondary)...

The area given by the integral of a function is the area, between the x axis of the Cartesian graphic, and the line, which represents the function, in a given interval of values...

This means that we can also easily represent, in the Cartesian graphic, a square, with area -1...

It's simply a square, with length 1, in the 'x' axis, and -1 in the 'y' axis.

Please note that the alternative square, with length 1 in the 'y' axis and '-1' in the 'x' axis, may indeed exist, but it doesn't follow the normal integral calculus applied to functions and Areas, for the normal 2D Coordinate Systems. Indeed, even having negative values for x, the adequate integral has always the lower integral index smaller (or more negative) than the upper integral index (bigger or plus positive).

For more information over this kind of details, it's recommended then to study the chapter 'integral calculus' of Maths (level: end secondary)...

Please note that '-1' is not only '1*(-1)', but it is also -1 = i * i = √-1*√-1, and, as such, the Cartesian Coordinate System may be represented with alternative equivalent axes, where '1', in the 'x axis' (the horizontal axis), and '-1', in the 'y axis' (the vertical axis), may be replaced by the values i = √-1 in both axes, in order to represent the same square, with area = -1. This is similar to translate the same text in another language...

The following image shows this, in other to give a better idea:

Left: Traditional Representation of an Area of value -1,
according to the Simple Integral Theory.
Right: First Equivalent Representation,
with alternative ℂ Supernatural (Imaginary) Numbers, i = √-1.

As can see, from the picture above, we can represent an equivalent Cartesian graphic, with 2 main axes, the Yi, vertical, and the Xi, horizontal. In this case, we analyzed the 4^th Quadrant only.

In order to observe how the units of i= √-1 evolve with the axis, it's worth repeating this same kind of reasoning for the 4 quadrants. In this way, we can clearly regulate the axes Yi and Xi.

The following figure shows this:

Traditional Representation of an Area of value -1,
according to the Simple Integral Theory Plus Equivalent Representation, with axis Yi and Xi,
with alternative ℂ Supernatural (Imaginary) Numbers, i = √-1, for the 4 Quadrants.

After repeating the same procedure for the 4 Quadrants, according to the rules for the adequate integral calculus, we could regulate the axes of our Cartesian Graphic: in the new horizontal equivalent axis, Xi, if we depart from zero to the right, where each unit means a 'i' value, giving, thus: 0, i, 2i, 3i, ..., where i= √-1...

The square is the same, and, as such, the norm of this 'i', ||i|| = ||√-1|| = 1.

Similarly, when depart from 0, in the Xi axis (equivalent to the xx axis), in the other sense, the normal x negative sense, the values become negative, as we could expect, similarly to the normal 'x' horizontal axis...

The Vertical, equivalent axis, is a bit bizarre: it goes up side down!

It has to be this way, otherwise, the calculus doesn't match with the Calculus of the Simple integrals, applied to the simple functions!

Thus, in the vertical axis, Yi, when we go up, from zero, from the center of the Cartesian Coordinate System, the values become increasingly negative:

0, -i, -2i, -3i, ..., where i= √-1...

And when we go down, from zero, from the center of the Cartesian Coordinate System, the values become increasingly positive:

0, i, 2i, 3i, ..., where i= √-1...

The Yi axis behaves up side down!

The second important conclusion is that there's always an equivalency between real numbers and supernatural (imaginary) numbers!

This allows a perspective over complex analysis completely different of the traditional one, of Cauchy, where the supernatural (imaginary) numbers are always considered as being part of another dimension, the i (imaginary) axis, "impossible" to control from the real axis...

With this new perspective / complex analysis, it's not like that anymore! And this is an important change of perspective, when working with supernatural / imaginary numbers...

Everything remains coupled with real world: only the language / code must be decrypted...

With the preceding figure, we have, thus:

Code for the Horizontal Axis Equivalence:

- From zero, to the right:

0, 1, 2, 3, ... matches with 0, i, 2i, 3i, ...

Thus, 1 = i = √-1...

- From zero, to the left, xx axis / Xi axis:

0, -1, -2, -3, ... matches with 0, -i, -2i, -3i, ...

Thus, -1 = -i = -√-1...

Code for the Vertical Axis Equivalence:

- From zero, to the top, yy axis / Yi axis:

0, 1, 2, 3, ... matches with 0, -i, -2i, -3i, ...

Thus, 1 = -i = -√-1...

- From zero, to the bottom, yy axis / Yi axis:

0, -1, -2, -3, ... matches with 0, i, 2i, 3i, ...

Thus, -1 = i = √-1...

These are the first equivalences noticed, between real numbers and supernatural (imaginary) numbers, for the 2D Cartesian Coordinate System, according to the axes and the senses...

Please note that the same kind of reasoning may be applied to 3D Cartesian Coordinate Systems:

A cube, with Volume = -1, is simply the Volume of a cube, with valid coordinates of values 1 and -1, e.g., x = 1, y = 1 and z = -1... or other similar, alternative correct formulas...

An alternative Cartesian Coordinate System, to represent the same cube, could use the formula:

-1 = ∛-1 *∛-1 * ∛-1 = (-1) * (-1) * (-1)

Or, either:

-1= i * i * 1 = √-1 * √-1 * 1, where i= √-1...

Normally there's much more than this, but, for the moment, we stop in this point, in this Introduction to the Theory of the Supernatural (Imaginary) Numbers of ℂ...

This section was inspired by God.

Analyzing Complex Numbers, as z = 2 + 2i, in Different Cartesian Coordinate Systems and Plans.

The next step is to set Complex Numbers in the different Plans: the New Plans and the Traditional Complex Plan of 1^st Degree.

We begin with a simple example for that: the Complex Number 2 + 2i, with i = √-1. The following image shows that:

Setting the Complex Number z = 2 + 2i in the Plans, with i = √-1.

Complex Analysis II of 2 ^nd Order: Infinite Equivalent Possibilities to Represent Complex Numbers, in a Bidimensional Coordinate System.

There are infinite possibilities, for the representation of Complex Numbers, in equivalent bidimensional Cartesian Coordinate Systems.

Indeed, if -1 = √-1 * √-1 = i * i, this relation defines the area of a square, of sides measuring √-1...

But -1 could also define the area of Rectangles, given by the relation -1 = xy...

This is the function y = -1/x, given by f(x)=-1/x...

This function is similar to the function g(x) = 1/x... a graph of this other function is available at mathway.com. Please scroll down (in the Web site mathway.com)....

Indeed, the graph of the function y = -1/x is of the same kind, but up side down (from: Secondary Maths)...

This means that there are infinite possibilities, for bidimensional Cartesian coordinate systems, respecting the relation -1 = xy...

Pairs of pure real axis or pairs of pure supernatural (imaginary) axis are only 2 possible combinations for Cartesian bidimensional coordinate systems...

Please note that the traditional Complex plan, having the Imaginary (supernatural) Yi axis, and the real axis, doesn't respect this Area criteria, -1 = xy, because -1 is not equal to √-1 * x, where x is a pure real number...

That's why the associated Complex Analysis are not of the same order anymore...

Watch out the minus sign in the Multiplication of supernatural numbers! It's better to replace √-1 by i, to avoid the sign trap...

E.g., -1 = √-1 * √-1 = √(-3) * √(-1/3) = i√3*i√(1/3) = i*i*√1=(-1)*1=-1

because, otherwise, we could do the mistake,

-1 = √(-3) * √(-1/3) = √((-1)*(-1)*3*(1/3)) = 1 ⇒ False!

This error could appear, e.g., if we want to find another equivalent coordinate system to the one, given by the square, given by the Area relation, -1 = √-1 * √-1...

Thus, an alternative rectangle is given by the relation -1 = √(-3) * √(-1/3) = i√3 * i√(1/3), generating an alternative bidimensional Coordinate System, which is, in practice, isomorphic, to the one introduced here, in these sections, 'Complex Analysis I of 2^nd Order', identical or scaled...

Setting the Interval of Complex Numbers 0 ≤ z ≤ 2 + 2i, in the Different Cartesian Coordinate Systems and Plans.

Recommended to review first the section 'Analyzing Complex Numbers, as z = 2 + 2i, in Different Cartesian Coordinate Systems and Plans'.

We could see, thus, that z = 2 + 2i represents, at least, a Cartesian product, with cardinality 3 × 3, where 3 points in the plan cross with 3 different rectangular areas, in order to give 9 different possible regions of the plans supernatural and real, but it gives only a single point in the traditional complex plan.

Indeed, there are infinite alternative 2D plans, to represent -1 as a product of 2 factors, -1 = xy, and, as such, z = 2 + 2i could scaled / deformed in alternative complex plan, based on -1 = √(-3)*√(-1/3)...

We're going to analyse the inequality 0 ≤ z ≤ 2 + 2i only in the previous plans.

The figure follows:

Setting the inequality 0 ≤ z ≤ 2 + 2i in the Plans, with i = √-1.

Theory of the Supernatural Numbers of ℂ, Complex Analysis II of 2 ^nd Order: Another Graphical Representation for i = ⁴√-1, for √i = ⁴√-1, for ⁴√i = ¹⁶√-1...

The idea is precisely the same as for the Complex Analysis I of 2 ^nd Order, but applied to more dimensions: dimension 4, dimension 8, dimension 16, dimension 2ⁿ...

Why Are the powers of 2 In Advantage?

The idea is to get the identity:

ⁿ√-1 * ⁿ√-1 * ... * ⁿ√-1 = -1

If we do this with the Cube Root, we have:

³√-1 * ³√-1 * ³√-1 = -1

But one solution for ³√-1 is -1, ³√-1 = -1,

because (-1)³= (-1) * (-1) * (-1) = -1...

But we want to find other kind of solutions, and not the -1...

We want to find, thus, solutions like for the Complex numbers ℂ, isomorphic to ℝ², in the style i = √-1...

That's why n should be a power of 2, when we analyze numbers of type ⁿ√-1...

Afterwards, the reasoning is precisely of the same kind... E.g, for n=4, for analysis of numbers of type:

⁴√-1 * ⁴√-1 * ⁴√-1 * ⁴√-1 * = -1

A multiplication with 4 terms define a multidimensional paralelipiped with 4 dimensions: e.g., length, width, height, and time... (please refer to sections 'Integrals for Kids', namely the section 'Multiple Integrals For Kids', to better understand this...

Afterwards, we can proceed in the same way, as for the Complex Analysis I of Second Order: we create a Cartesian Coordinate System, with 4 (supernatural numbers) axis, one per supernatural dimension of type √i = ⁴√-1, where i = √-1...

Naturally, there's an equivalent Cartesian Coordinate System, with 4 (Real numbers) axis, because the 4 dimension paralelipiped, with hyper volume -1 may be given also by a hyperparalelipiped, with dimensions (sides) equal to:

Hyper Volume_{Hyper Paralelipiped} = -1 = Length_HyperParal. * Width_HyperParal. * Height_HyperParal. * Time_HyperParal.

Hyper Volume_{Hyper Paralelipiped} = -1 = 1 * 1 * 1 * (-1)

Thus, all sides may have integer (real) measures...

There are, of course, other integer (real) combinations, in order to get this Hyper Volume equal to -1, e.g.:

Hyper Volume_{Hyper Paralelipiped} = -1 = Length* Width * Height * Time

Hyper Volume_{Hyper Paralelipiped} = -1 = (-1) * (-1) * (-1) * 1, etc.

Thus, all the theory applied to the bidimensional Coordinate Systems of 'Complex Analysis I of Second Order' may be generalized to 4 Dimensional Coordinate Systems, to 8 Dimensional Coordinate Systems, to m Dimensional Coordinate Systems, where m is a power of 2, i.e., m = 2ⁿ, where n ∈ ℕ...

Section Computer Science.

Some Solutions for the Exercises of Threads.

In this section, I'm going to expose some resolutions for the exercises proposed by the excellent book, from Badr Benmammar, over Threads and Concurrent Programming (Java).

It is assumed, implicitly, that you have bought the book, in order to correctly benefit from these resolutions.

You should try to solve it first and / or review the book in case of difficulties and only look the resolutions afterwards...

The Resolutions here are based on the French version of the book. I hope he converts quickly, quickly, to true faith... not so easy in Muslim lands... here, www.christianideas.eu, section 'Summary', in top of Web page, tries to help...

Anyhow, the pedagogic quality of the book is good... it's excellent, in order to begin (less superficially) with Threads, if you have already some knowledge of the Java programming language, of 'Introduction to Computer Science' and some knowledge of 'Object Oriented Programming'...

If you find any error remark, etc., you can contact me, via the Web page Contact...

With the ruse of evil, in these modern times, it's better to master some post secondary subjects, and, if possible, to have an university degree, like LEIC IST Lisbon, in order to try to counter the evil sophisms, against the true faith...

Resolution of Exercise 3, page 132: Counters.

Resolution of the Exercise 3a, page 132: Compteurs.java (.pdf) Result Sample

Resolution of the Exercise 3b, page 132: Compteurs2.java (.pdf) Compteur2.java (.pdf) compteurs2Result Sample

Resolution of Exercise 5, page 134: Table of 7.

Resolution of Exercise 5, page 134: TableOf7Task.java (.pdf) TableOf7Main.java (.pdf)

You should try to solve it on your own. Anyhow it's interesting to run it, in an adequate platform, e.g., eclipse for Java, and see the results (already +- available from the book)...

Resolution of Exercise 7b, page 137: Factorials.

Resolution of Exercise 7b page 137: Factorial4.java (.pdf) Results of Factorial4 (.pdf)

Java Threads Example: Car Park (Based Leic94 IST).

This exercise is based on a problem proposed in Distributed Systems, 1993, LEIC (Computer Science Engineering), LEIC94, IST Lisbon.

It goes over a car park with 2 main rooms, park1 and park2, separated by a narrow hall, with only 1 sense.

It is proposed to simulate cars arriving to the car park, remaining for some time in park1, using the hall to go to park2, remaining some time in park2, before exiting the park.

Here we assume park1 has place for 3 cars. 2 cars may be in the intermediate hall. 3 cars may be parked, at the same time, in park2.

The Car Park program proposed here is done in Java, using threads.

A basic sample output is also proposed, simulating an arrival to the car park of 15 cars. No sophisticated graphical interface, sorry. Maybe in the next version... ???

"Another Barber Saloon" Resolution of the Exercise Using Threads.

Still a resolution for the Sleeping Barber Problem. I had it difficult, and I had to check different bibliography, books, examples in the Internet.

Finally, I did my own resolution, available here, but with the help of this example on-line, from the University of Helsinki, SleepingBarber.java.

The Resolution is based on the existing examples, but it is not the same anymore; I tried to use my own logic; it follows here below. Another resolution, thus. Some incoherencies of synchronization were noticed. That's why an improved version follows, a bit below.

Another check, for bugs + analysis of results, is maybe still required, but several of the incoherencies of the 1^st versions were already corrected.

the java file BarberSaloon.java (.pdf).

Results: file resultsBarberSaloon.txt (.pdf)
(3 outputs for the Sleeping Barber: Saloon a. 'many clients' ; b. 'normal' ; c. 'calm').

Improved version, BarberSaloon3:

the java file BarberSaloon3.java (.pdf).

Results: file resultsBarberSaloon3.txt (.pdf)
(~10 outputs for the Sleeping Barber3 java threads application: versions 'many clients', 'saloon normal' , 'calm' ...).

Another Java Solution for the Dining Philosophers Problem.

Some Considerations, over the Solution Presented here:

You may find in the Internet and in the literature available (Threads, Operating Systems...) different solutions for the Dining Philosophers problem. I implemented a solution, in Java, with basic text results, and you can find here the files, in .pdf format.

This solution takes into account that if N Philosophers are sit around a round table, only maximum ⌊ N/2 ⌋ may eat at same time and no 2 neighbors philosophers may eat at the same time (because there are not enough available 'legitimate' chopsticks for both).

In case of 7 Philosophers, thus, maximum 3 philosophers may be eating at a time and if Philosopher0 is eating, neighbors Philosopher6 and Philosopher1 have to wait a bit.

An anti starvation correction factor is added.

The chopsticks are ignored in this solution. Why? They may be used, yes, but they're not absolutely needed to implement the solution. We need one solution without deadlock and without starvation, in order to get the philosophers eating and thinking in a round table. Imposing that both neighbors are not eating, and anti circular wait criteria (maximum ⌊ N/2 ⌋ philosophers eating at same time...) + anti starvation correction, should be enough to get a correct solution.

Watch out: the solution presented here is not necessarily the most efficient... but it happens in a somewhat "spontaneously", "random" way... A mutex is used, in order to access / change the shared variable NrPhilosophersEating, while a resource semaphore is added to impose a maximum of ⌊ N/2 ⌋ Philosophers eating at same time...

the java file Philosopher.java (.pdf).

Results: file resultsPhilosophers.txt (.pdf).
( some outputs for the Dining Philosophers problem with 5, 6 and 9 Philosophers ...).

Other resolutions are available on the Internet, e. g., GeeksforGeeks: Dining Philosopher Problem, Using Semaphores or The Dining Philosophers Problem, at Indiana University...

Previous resolution was a bit improved / optimized in version 2, but now it became also a bit more complex... that's why the previous files remain... no spin locks used but I used sleep + wait synchronous (semaphores) + threads in Java; anti starvation measures were improved: a minimal time criteria was added... extra synchronization practice added... ⇒ main thread is waked up, when last thread (Philosopher) finishes...

This exercise may help for the exercise, of page 144, from the book of Benmammar, 'Threads' (in french)... for solutions, using the chopsticks, please refer to the other solutions, in the available literature: books, Internet... the solutions, using the chopsticks, are not necessarily more efficient, because the mutex here is only used to update the shared global variable...

The improved files follow here below, in .pdf format, in "Catholic Russia Converted Ideas" style...

the java file Philosopher2.java (.pdf)

Results: file resultsPhilosophers2.txt (.pdf)
(outputs for the Dining Philosophers problem, with 5, 6, 7 and 9 Philosophers ...).

Producer Consumer: Resolution of Exercise 14, French Version of Book Benmammar, Threads (Page 147).

14.1. cond1: nbObj==taille

(number of objects == length)

cond2: nbObj==0

(number of objects == 0)

14.2 "last=(last+1) % length " allows circular increment of index buffer (for Producer Thread).

14.3 type of object buffer: Object [], eventually Object[length]... we could go even further here up to Integer[length]...

Object [] is more generic...

File Main.java + small sample outputProducerConsumer.txt are available here, in .txt format...
the rest of the code is available in pages 146 + 147 of the book of Benmammar, except cond1 and cond2, mentioned here above...
Enjoy!
(exercise compatible Computer Science Engineering, 2^nd year, Operating Systems, e.g. LEIC IST Lisbon (now Bologna version). Done by 5 year LEIC, licensed in 1994 (Leic94).

Thread Synchronization: a Resolution of Exercise 15, From the (French) Book of Benmammar, Threads (Page 148).

Here goes a 1^st resolution of the problem of a "one way Street", with a sample (text) output, in .pdf format.

Files available (exercise implemented in java):

onewayStreet.java (.pdf)

FabricCars.java (.txt)

FabricCarsFromSense1.java (.txt)

FabricCarsFromSense2.java (.txt)

Car.java (.txt)

CarFromSense1.java (.txt)

CarFromSense2.java (.txt)

onewayStreetResults.txt (.pdf)

Enjoy!

Some Interesting Links, to Try to Help to Studies.

If you subscribe to Chegg.com, you may have access on-line to many solved exercises, from books as, e.g., Apostol Maths, or Operating Systems Silberschatz.

Christian Ideas Background sample.

Candle: Our Lady of the Roses Honored / Exposed and, possibly, the Death doesn't enter Inside.

Jesus of the Divine Mercy: Those who Honor Jesus, with full Trust, through this Picture, shall be Saved* from Hell.
* - Please refer to the home page from www.christianideas.eu for further Details...

Jesus of the Divine Mercy: Those who Honor Jesus, with full trust, through the preceding Picture, shall be Saved* from Hell.

* - WATCH OUT: IF YOU ACCEPT THE MARK 666 IN the BODY (an electronic chip in the hand or in the forehead) OR IF YOU WORSHIP THE ANTICHRIST OR HIS IMAGE, YOU FINISH IN HELL, as the book of the Apocalypse 14:9-11 says, EVEN IF YOU HONOR A DIVINE MERCY JESUS of Saint Faustina, with full trust...

Please Scroll further Down, to Fully See the Summary...

I try to pray for the protection of all the pro God of the planet, until 00:00h, to try to disinfest, but it's better to complement it with a 24h protective prayer, as the daily Rosary (praying it attentively), or the daily Rosary 2× (or more).

Summary of the Web Site

Catholic Russia Converted Ideas...

Made in Belgium, E.U.

www.christianideas.eu:

DISCERNMENT: e.g., the True Faith is Catholic Conservative Non Naive. The Incorruptible Miraculous Corpses of the Catholic Saints (Padre Pio) + the Holy Blood of Jesus show the Will of God for the Doctrine and the Authority of the Time. Doctrine: Catholic Roman Conservative and Authority: still the Pope.
Idea to Become Rich, in a Pro God Way.

Traps for Rich and for the Holy Host.
Danger of Apocalypse (Prophecies of S. Malachy): risk of Sect 'New Age' in Rome ⇒ the Apocalypse... ⇒ it's better to be in state of grace and / or to go to Confession, asap, to Catholic Priest (after Benedict XVI), in case of sin, + to pray a Rosary, in the required conditions (to get a plenary* indulgence), to be a good practicing Catholic Conservative, to pray the daily Rosary (against the rest of evil).

* - with reject of all sins, for the good intentions of the pope, in church, community or family. If it's not in church, community or family, pray it several times, to have the same effect...

Risk of Great Famine ⇒ News from Heaven to, e. g., Wings of Prophecy, L. de Maria, J. Leary...

Ideas (pro God) to (try) to resist the Great Famine:
- St Onofre, with a Crumb of Bread.
- Blessed Grapes of L. de Maria + S. Damian...
MODERN LIES Unmasked
(Lies of Darwinism, Fake Dates...).

History + Demographics, since ~2000BC, unmask the fake dates: the Earth was almost Empty, after the Great Flood.

Hominids were Hybrids, which existed before the Great Flood, confirms Jesus at the old Web site from V. De Tarlé apparitionsmariales.org…

Proactive Darwinism: ~mouse evolves to ⇒ bat ⇒ vampire, thus, the ancestor of the man / whale etc., evolves to ⇒ Batman ⇒ Dracula..., in the same ~millions of years...
Migrations based in Y Haplogroups (DNA) tend to proof that Atlantis sank ⇒ Ice ages, before the time of Moses? Chapter 'Atlantis'.
DANGER OF Apocalypse: CHIP 666 IN BODY, WW3, dangerous modern vaccines, “good” sect New Age, THE GLOBAL EMPIRE SOCIALIST COMMUNIST OF the ANTICHRIST... ⇒ lots of apocalyptic prophecy, at www.tldm.org (for scenario 'Russia Not Converted').

ICON OF the AUTOMATIC Eternal SALVATION*
(also in the beginning of this section):

THOSE WHO TRUST 100% IN JESUS, through THIS PICTURE, SHALL BE SAVED* (a Jesus of St Faustina, with rays blue and red).

* - However, if the Concerned Person Submits to The Antichrist, this Promise is not Valid anymore...
PROTECTIONS against EVIL, but some suffering needed, sorry!

A Summary, a bit further.
IMMORTALITY SUGGESTION:

TO PRAY GOD, for the ETERNAL SALVATION, of THOSE WHO NEED IT MOST, 'only' CALMS DANGERS for LIFE.

You have nothing to lose, with a correct use, if you feel dying, even if the formula is limited... but: Abuses Risk to Be Severely Punished!

After: begin discreet fast + daily prayer (as the Rosary), prayed attentively, until it calms / improves... ⇒ Ok, not so easy, but it's better than cancer operations. 1-4 days should calm the Flu, + Whiskey, in the bottom of the glass, to disinfect...
LGBT + Modern Sex BEHAVIORS: Unhealthy, they even KILL...

It’s possible to attack the subconscious, e.g., via discreet witchcraft… ⇒ Nero Case Study... Risks of DETACHED INTESTINE...
~3.5 BILLIONS MURDERS, SINCE ~1960:

The ABORTION Horrors...
IDEAS AGAINST Evil, e.g., against AGING & FLU / PLAGUE:

Fast & Prayer (good against Evil ⇒ Medjugorje), until it calms... (Discreet Fast & Prayer, in secret, or in solidarity group, prayer as the (daily) Rosary, prayed attentively, or the daily Rosary 2×, or the 7 sorrows)... some food may help: avoid white sugars (fruits, honey, Stevia, are better, if sweet food needed), but 1/8 glass whiskey or vodka (⇒ let it disinfect, a few minutes, in the throat…), 1/8 glass vinegar, 2 cloves raw, well chopped garlic mixed, e.g., in the salad, to help to counter plague... + Immortality Suggestion, in dramatic cases, to try to calm a bit, and, in place of dying, to win forces to pray a Rosary (attentively)... + Our Lady of the Roses Honored / Exposed and the Death (possibly) doesn’t Come Inside. + details: preceding and following links.

MODERN VACCINES TO BE AVOIDED. Alternatives: Protections against the Apocalypse + Formula 'Fast & Prayer' (please see preceding text).
IDEAS FOR A BETTER LOOK (Ladies & Veil).

The Dream Lady is the Beautiful Blond Lady with Blue Eyes. Now it is Aesthetics. To Marry with Brunette woman it's like the woman who marries with poor man or poorer man. It's better than nothing, but it bothers.

Aesthetic Care Helps, as Money, for Marriages... Comparative Tables, in the section 'Our Lady of Fatima Deserves a Better Look'.
Eden Garden Located. Noah's Ark Found.
MODERN WOMAN IDEOLOGY UNMASKED:

The Demographic Fall in the West (+ Russia)...

Statistics (1950 or) 1970 - 2024: Germany × India...
LINKS TO some HEAVEN NEWS.
Censured on X / Twitter: @_ChristianIdeas and on Facebook (before: www.facebook.com/christian.ideas).
Possible: T Shirts Posters “Songs” Christian Ideas and / or Russia Converted Catholic Ideas (‘made in Belgium’)...
The .pdf Book "Mysteries of the Bible Revealed, Christian Ideas and Discernment" is already available in:

English French Portuguese Spanish.
Ideas Against Cancer.
School
(e.g. Maths, Computer Science: multiple integrals for kids, exercises, complements to Wikipedia...).
www.christianideas.eu Near Invisible in the Internet:

Small Probabilities to find this Web site, e.g., via Google...
Christian Ideas at Linkedin
(https://be.linkedin.com/in/christian-ideas-104ba893).

If You Want to Defend Good,

PROTECTIONS against EVIL + to Take Up the Cross Recommended:

IMMORTALITY SUGGESTION, correct use, at the 1^st Sign of Danger. More information, in the Chapter ‘Against Death’.
To Do a CONTRACT WITH GOD (ICON of the AUTOMATIC SALVATION*, THROUGH DIVINE MERCY, available, e.g., in the Home page, on the top of the Web page).

To Pray, everyday (better: Attentively):
To be clean of sins or to go to Confession, to the Catholic priest and to pray a Rosary, under the conditions required, for a plenary indulgence**, as soon as possible...

* - However, if the Concerned Person Submits to the Antichrist, this Promise is not Valid anymore...

** - it's better to pray the Rosary, in church, family or community, with rejection of all the sins, in state of grace, praying for the (good) intentions of the pope, in order to try to get a plenary indulgence...

More details, in the chapter 'Protections Against Evil', from www.christianideas.eu.

I try to pray for the protection of all the pro God of the planet, until 00:00h, to try to disinfest, but it's better to complement it, with 24h protective prayer, e.g., the Rosary or the daily Rosary 2× (better: praying it attentively, in the required conditions, to get a plenary indulgence).

Satan demanded to God 75 - 100 years, + Money + Power, in order to destroy the Vatican ⇒ www.tldm.org.

Satan had that time, that money and that power.

~140 years later, despite the dangers + the evil, the Vatican still resists.

Let us see evolution... ??? Apocalyptic? Reversal pro God (again)???

Evil doesn't triumph anymore, since ~2023, but Apocalyptic temptation persists. There's still, thus, Danger of Apocalypse, in the next ~50 years (if the Apocalypse doesn't happen).

The best prepared (good prepared) are protected. The best is to prepare your self, asap, to try to be protected. More information, in the section 'How To Be Ready to Enter in the Paradise of God?' from the Chapter 'Discernment'.

Note: www.christianideas.eu began in ~2009 and keeps improving. IT REMAINS INCOMPLETE + NOT PERFECT.

If God helps, I'll try to improve it and / or to correct eventual detected errors, contents + design, little by little.

A backup, in .zip format exists (you may download it also, via the left column, above, using for that, e.g., the browser Firefox). This backup contains the most important part of this Web site.

You may print it, copy it, according to the content's license, above left, in this Web page...

Good intentioned Pub, pro God, pro True Faith, Welcome!!!
Protections from God recommended then! Evil doesn't like it...

Good News: God decided to collaborate with www.christianideas.eu and became already the main Author, the Author 1 of this Web site, even if some weak points remain. Yes!

Below: Our Lady of the Roses Honored / Exposed and the Death possibly doesn't Enter Inside.
(source: Jesus, at the old Web site apparitionsmariales)

Right Click + "Save as", to download the images (in the computer)...

Freemasonry, if with Sabotage Purposes, against God's interests, is Forbidden...

Catholic Russia Converted Translations

العربية (Arabic)

العودة إلى الأعلى

خدمات الترجمة العامة لمتصفحات الويب

على سبيل المثال، في المتصفحات (Chrome، Edge...)،

Windows:

Chrome: انقر بزر الماوس الأيمن، حيث يكون مؤشر الماوس على شكل سهم ← ترجمة (صفحة الويب هذه) إلى اللغة (المحددة)... بعد ذلك، تظهر نافذة لغة صغيرة، أعلى متصفح الويب، حيث يمكنك اختيار اللغة المراد ترجمتها.

Edge: الرجاء النقر على الجانب الأيمن من شريط عنوان الويب ، لغة الرمز بعد ذلك ، تظهر نافذة لغة صغيرة أعلى متصفح الويب ، حيث يمكنك تحديد اللغة المراد ترجمتها.

ملحوظة: إذا كنت تريد ترجمة صفحة ويب أخرى، من فضلك العودة إلى النسخة الأصلية (الإنجليزية) من هذا / موقع الويب المعني، افتح الصفحة التي تريد ترجمتها (لا تزال باللغة الإنجليزية) وترجمتها بعد ذلك، كما ذكرنا أعلاه.

Android (Chrome): الجزء العلوي من الشاشة ← انقر فوق 3 نقاط (عمودي) ← يترجم...

বাংলা (Bangla)

উপরে ফিরে যাও

ওয়েব ব্রাউজারের সাধারণ অনুবাদ পরিষেবা

যেমন, ব্রাউজারে (Chrome, Edge...),

Windows:

ক্রোম: মাউস দিয়ে রাইট ক্লিক করুন, যেখানে মাউস কার্সারের তীর→ (প্রদত্ত) ভাষায় (এই ওয়েব পৃষ্ঠা) অনুবাদ করুন... এর পরে, এটি ওয়েব ব্রাউজারের শীর্ষে একটি ছোট ভাষা উইন্ডো প্রদর্শিত হবে, যেখানে আপনি অনুবাদ করার জন্য ভাষা নির্বাচন করতে পারেন।

Edge: অনুগ্রহ করে ওয়েব ঠিকানা বারের ডান পাশে, আইকন ভাষাতে ক্লিক করুন তারপরে, এটি ওয়েব ব্রাউজারের শীর্ষে একটি ছোট ভাষা উইন্ডো প্রদর্শিত হবে, যেখানে আপনি অনুবাদ করার জন্য ভাষা নির্বাচন করতে পারেন।

দ্রষ্টব্য: আপনি যদি অন্য ওয়েব পৃষ্ঠা অনুবাদ করতে চান, অনুগ্রহ করে এটির (ইংরেজি) মূল সংস্করণে ফিরে যান সংশ্লিষ্ট ওয়েব সাইট, আপনি যে পৃষ্ঠাটি অনুবাদ করতে চান সেটি খুলুন (এখনও ইংরেজিতে) এবং উপরে উল্লিখিত হিসাবে পরে এটি অনুবাদ করুন।

Android (Chrome): পর্দার শীর্ষে → 3 পয়েন্টে ক্লিক করুন (উল্লম্ব) → অনুবাদ করা...

中文 (Chinese)

回到顶部

网络浏览器的通用翻译服务

例如，在浏览器（Chrome、Edge...）中

Windows:

Chrome：用鼠标右键单击，鼠标光标呈箭头形式→ 将（此网页）翻译成（给定的）语言... 然后，在网络浏览器的顶部会出现一个小的语言窗口，您可以在其中选择要翻译的语言。

Edge：请点击网址栏右侧图标语言之后，它会在网络浏览器的顶部出现一个小语言窗口，您可以在其中选择要翻译的语言。

注意：如果您想翻译其他网页，请返回此/的（英文）原始版本相关网站，打开您要翻译的页面（仍然是英文）然后像上面提到的那样翻译它。

Android (Chrome)：屏幕顶部 → 单击 3 点（垂直）→ 翻译...

Deutsch (German)

Zurück nach Oben

Allgemeine Übersetzungsdienste für Webbrowser

Z.B., in den Browsern (Chrome, Edge...),

Windows:

Chrome: Rechtsklick mit der Maus, wobei der Mauszeiger die Form eines Pfeils hat → Übersetzen Sie (diese Webseite) in die (angegebene) Sprache ... Anschließend erscheint oben im Webbrowser ein kleines Sprachfenster. Hier können Sie die zu übersetzende Sprache auswählen.

Edge: Bitte klicken Sie rechts in der Webadressleiste, oben im Webbrowser, auf das Symbol „Sprache“. Anschließend erscheint, oben im Webbrowser, ein kleines Sprachfenster, Hier können Sie die zu übersetzende Sprache auswählen.

Hinweis: Wenn Sie eine andere Webseite übersetzen möchten, wenden Sie sich bitte an uns Zurück zur (englischen) Originalversion dieses / des betroffene Website, Öffnen Sie die Seite, die Sie übersetzen möchten (immer noch auf Englisch). und übersetzen Sie es anschließend, wie oben erwähnt.

Android (Chrome): Oberer Bildschirmrand → Klicken Sie auf 3 Punkte (vertikal) → Übersetzen...

Español (Spanish)

Volver Arriba

Servicios Generales de Traducción de Navegadores Web

Por ejemplo, en los Navegadores (Chrome, Edge...),

Windows:

Chrome: Haga clic derecho, con el mouse, donde el cursor del mouse tiene forma de flecha → Traducir (esta página web) al idioma (dado)... Luego, aparece una pequeña ventana de idioma, en la parte superior del navegador web, donde podrás seleccionar el idioma a traducir.

Edge: haga clic en el lado derecho de la barra de direcciones del navegador Web, icono de idioma Luego, aparece una pequeña ventana de idioma en la parte superior del navegador web, donde puede seleccionar el idioma a traducir.

Nota: Si desea traducir otra página Web, por favor volver a la versión original (inglés) de este / el sitio web en cuestión, abre la página que deseas traducir (aún en inglés) y traducirlo después, como se mencionó anteriormente.

Android (Chrome): Parte superior de la pantalla → haga clic en 3 puntos (vertical) → Traducir...

Nota 2: También puede descargar el libro en Español, “Misterios de la Biblia Revelados, Ideas Cristianas y Discernimiento”, en formato .pdf, que contiene la mayor parte del contenido de este sitio Web.

Français (French)

Retourner au Sommet de Page

Services Généraux de Traduction des Navigateurs Web

Par exemple, dans les Navigateurs Web (Chrome, Edge...),

Windows:

Chrome: clic droit avec la souris, le curseur de la souris ayant la forme d'une flèche → Traduire (cette page Web) dans la langue (donnée)... Ensuite, une petite fenêtre de langue apparaît, en haut du navigateur Web, où vous pouvez sélectionner la langue à traduire.

Edge: veuillez cliquer sur le côté droit de la barre d'adresse du navigateur Web, icône de langue Ensuite, une petite application web de langue apparaît, en haut du navigateur Web, où vous pouvez sélectionner la langue à traduire.

Remarque: Si vous souhaitez traduire une autre page Web, veuillez revenir à la version originale (anglaise) de ce / du site internet concerné, ouvrez la page que vous souhaitez traduire (toujours en anglais) et traduisez-le ensuite, comme mentionné ci-dessus.

Android (Chrome): Haut de l'écran → cliquez sur 3 points (vertical) → Traduire...

Hausa (Hausa)

Koma sama

Gabaɗaya Sabis na Fassarar Masu Binciken Gidan Yanar Gizo

Misali, a cikin Browser (Chrome, Edge...),

Windows:

Chrome: Danna Dama, tare da linzamin kwamfuta, inda siginan linzamin kwamfuta yana da nau'i na kibiya → Fassara (wannan shafin yanar gizon) zuwa harshen (wanda aka bayar)... Bayan haka, yana bayyana ƙaramin tagar harshe, a saman burauzar gidan yanar gizon, inda zaku iya zaɓar yaren don fassarawa.

Edge: da fatan za a danna a gefen dama na sandar adireshin gidan yanar gizo, alamar harshe Bayan haka, yana bayyana ƙaramin tagar harshe a saman burauzar gidan yanar gizon, inda zaku iya zaɓar yaren don fassarawa.

Lura: Idan kuna son fassara wani shafin yanar gizon, don Allah komawa zuwa asalin (Turanci) na wannan / da gidan yanar gizon damuwa, bude shafin da kake son fassarawa (har yanzu a Turanci) kuma a fassara shi daga baya, kamar yadda aka ambata a sama.

Android (Chrome): saman allo → danna maki 3 (a tsaye) → Fassara...

हिन्दी (Hindi)

वापस शीर्ष पर

वेब ब्राउज़र की सामान्य अनुवाद सेवाएँ

उदाहरण के लिए, ब्राउज़र में (क्रोम, एज...),

Windows:

क्रोम: माउस से राइट क्लिक करें, जहां माउस कर्सर में तीर का आकार होता है → (इस वेब पेज का) दी गई भाषा में अनुवाद करें... इसके बाद, वेब ब्राउज़र के शीर्ष पर एक छोटी भाषा विंडो दिखाई देती है, जहां आप अनुवाद करने के लिए भाषा का चयन कर सकते हैं।

Edge: कृपया वेब एड्रेस बार, आइकन भाषा के दाईं ओर क्लिक करें बाद में, यह वेब ब्राउज़र के शीर्ष पर एक छोटी सी भाषा विंडो दिखाई देती है, जहां आप अनुवाद करने के लिए भाषा का चयन कर सकते हैं।

नोट: यदि आप किसी अन्य वेब पेज का अनुवाद करना चाहते हैं, तो कृपया इसके (अंग्रेज़ी) मूल संस्करण पर वापस लौटें संबंधित वेब साइट, वह पृष्ठ खोलें जिसका आप अनुवाद करना चाहते हैं (अभी भी अंग्रेजी में) और बाद में इसका अनुवाद करें, जैसा कि ऊपर बताया गया है।

Android (Chrome): स्क्रीन का शीर्ष → 3 बिंदुओं पर क्लिक करें (ऊर्ध्वाधर) → अनुवाद करना...

日本語 (Japanese)

トップに戻る

Webブラウザの翻訳サービス全般

たとえば、ブラウザ (Chrome、Edge...) では、

Windows:

Chrome: マウスカーソルが矢印の形になっている場所をマウスで右クリックします → (この Web ページを) (指定された) 言語に翻訳します... その後、Web ブラウザの上部に小さな言語ウィンドウが表示されます。翻訳する言語を選択できます。

Edge: Web アドレスバーの右側にあるアイコンの言語をクリックしてくださいその後、Web ブラウザの上部に小さな言語ウィンドウが表示されます。翻訳する言語を選択できます。

注: 別の Web ページを翻訳したい場合は、この/の元の (英語) バージョンに戻る関係するウェブサイト、翻訳したいページを開きます（まだ英語です）上で述べたように、後で翻訳します。

Android (Chrome)：画面上部 → 3 点（垂直）をクリック → 翻訳...

Melayu (Malay)

Kembali ke atas

Perkhidmatan Terjemahan Umum Penyemak Imbas Web

Cth., dalam Penyemak Imbas (Chrome, Edge...),

Windows:

Chrome: Klik Kanan, dengan tetikus, di mana kursor tetikus mempunyai bentuk anak panah → Terjemah (halaman web ini) ke bahasa (diberikan)... Selepas itu, ia muncul tetingkap bahasa kecil, di bahagian atas pelayar web, di mana anda boleh memilih bahasa untuk diterjemahkan.

Edge: sila klik di sebelah kanan bar alamat web, bahasa ikon Selepas itu, ia muncul tetingkap bahasa kecil di bahagian atas pelayar web, di mana anda boleh memilih bahasa untuk diterjemahkan.

Nota: Jika anda ingin menterjemah halaman web lain, sila kembali ke versi asal (Bahasa Inggeris) ini / the laman web berkenaan, buka halaman yang ingin anda terjemahkan (masih dalam bahasa Inggeris) dan terjemahkannya selepas itu, seperti yang dinyatakan di atas.

Android (Chrome): Bahagian Atas Skrin → klik 3 mata (menegak) → Terjemah...

Português (Portuguese)

De Volta ao Topo

Serviços Gerais de Tradução de Navegadores da Web

Por exemplo, nos Navegadores Web (Chrome, Edge...),

Windows:

Chrome: clique, com o botão direito do rato, por ex., aqui → Traduzir (esta página da Web) para a língua proposta... Aparece então, no topo do navegador Web, uma pequena aplicação de idioma, onde poderá selecionar o idioma a traduzir.

Edge: clicar no lado direito da barra de endereço do navegador Web, no ícone de idioma. De seguida, aparece uma pequena aplicação de idioma no topo do navegador web, onde pode selecionar o idioma para traduzir.

Nota: Se quiser traduzir outra página da Web, por favor retornar à versão original em inglês deste / do site Web em questão, abra a página web que deseja traduzir (ainda em inglês) e traduza-a depois, como mencionado acima.

Android (Chrome): Parte superior do écran → clique em 3 pontos (vertical) → Traduzir...

Nota2: Também pode descargar o livro em português, "Mistérios da Biblia Revelados, Ideias Cristãs e Discernimento", em formato .pdf, que contém a maior parte do conteúdo deste site Web.

پارسی (Persian)

بازگشت به بالای صفحه

خدمات ترجمه عمومی مرورگرهای وب

به عنوان مثال، در مرورگرها (Chrome، Edge...)،

Windows:

Chrome: با ماوس کلیک راست کنید، جایی که نشانگر ماوس شکلی از فلش دارد ← ترجمه (این صفحه وب) به زبان (داده شده)... پس از آن، یک پنجره زبان کوچک در بالای مرورگر وب ظاهر می شود، جایی که می توانید زبان مورد نظر را برای ترجمه انتخاب کنید.

لبه: لطفاً روی نماد زبان در سمت راست نوار آدرس وب کلیک کنید. پس از آن، یک پنجره زبان کوچک در بالای مرورگر وب ظاهر می شود که می توانید زبانی را برای ترجمه انتخاب کنید.

توجه: اگر می خواهید صفحه وب دیگری را ترجمه کنید، لطفاً بازگشت به نسخه اصلی (انگلیسی) این / the وب سایت مربوطه، صفحه ای را که می خواهید ترجمه کنید باز کنید (هنوز به زبان انگلیسی) و پس از آن همانطور که در بالا ذکر شد ترجمه کنید.

Android (Chrome): بالای صفحه ← روی 3 نقطه (عمودی) کلیک کنید ← ترجمه کردن...

Русский (Russian)

Вернуться к началу страницы

Общие услуги перевода веб-браузеров

Например, в браузерах (Chrome, Edge...)

Windows:

Chrome: щелкните правой кнопкой мыши там, где курсор мыши имеет форму стрелки → Перевести (эту веб-страницу) на (данный) язык... После этого в верхней части веб-браузера появится небольшое языковое окно. где вы можете выбрать язык для перевода.

Edge: нажмите справа от строки веб-адреса, язык значков После этого в верхней части веб-браузера появляется небольшое языковое окно, где вы можете выбрать язык для перевода.

Примечание. Если вы хотите перевести другую веб-страницу, пожалуйста, вернуться к (английской) оригинальной версии этого / соответствующий веб-сайт, откройте страницу, которую хотите перевести (все еще на английском языке) и переведите его потом, как упоминалось выше.

Android (Chrome): Верхняя часть экрана → нажмите 3 точки (по вертикали) → Переводить...

Tagalog (Tagalog)

Bumalik sa tuktok ng pahina

Pangkalahatang Serbisyo sa Pagsasalin ng mga Web Browser

Hal., sa Mga Browser (Chrome, Edge...),

Windows:

Chrome: Right Click, gamit ang mouse, kung saan ang mouse cursor ay may anyo ng arrow → Isalin (ang web page na ito) sa (ibinigay) na wika... Pagkatapos, lumilitaw ang isang maliit na window ng wika, sa tuktok ng web browser, kung saan maaari mong piliin ang wikang isasalin.

Edge: mangyaring mag-click sa kanang bahagi ng web address bar, sa tuktok ng web browser, sa wika ng icon Pagkatapos, lumilitaw ang isang maliit na window ng wika sa tuktok ng web browser, kung saan maaari mong piliin ang wikang isasalin.

Tandaan: Kung gusto mong magsalin ng isa pang web page, mangyaring bumalik sa (Ingles) orihinal na bersyon nito / ang kinauukulang web site, buksan ang pahinang gusto mong isalin (sa Ingles pa rin) at isalin ito pagkatapos, gaya ng nabanggit sa itaas.

Android (Chrome): Itaas ng Screen → i-click ang 3 puntos (vertical) → Isalin...

Türkçe (Turkish)

Sayfanın başına dön

Web Tarayıcılarının Genel Çeviri Hizmetleri

Örneğin, Tarayıcılarda (Chrome, Edge...),

Windows:

Chrome: Fare imlecinin ok biçiminde olduğu yere fareyle sağ tıklayın → (Bu web sayfasını) (belirtilen) dile çevirin... Daha sonra web tarayıcısının üst kısmında küçük bir dil penceresi görünür. tercüme edilecek dili seçebileceğiniz yer.

Edge: lütfen web adres çubuğunun sağ tarafında, simge dilini tıklayın Daha sonra, web tarayıcısının üst kısmında küçük bir dil penceresi belirir, çevrilecek dili seçebileceğiniz yer.

Not: Başka bir web sayfasını çevirmek istiyorsanız lütfen bunun (İngilizce) orijinal versiyonuna geri dön / ilgili web sitesi, çevirmek istediğiniz sayfayı açın (hala İngilizce) ve daha sonra yukarıda belirtildiği gibi tercüme edin.

Android (Chrome): Ekranın Başı → 3 noktayı tıklayın (dikey) → Çevirmek...

Other Languages: Swedish...

General Translation Services of Web Browsers, e.g., to Other Languages: Greek...

E.g., in the Browsers (Chrome, Edge...),

Windows:

Chrome: Right Click with the mouse, e.g., Here → Translate (this web page) to the (given) language... Afterwards, it appears a small language window, on the top of the web browser, where you can select the language to translate.

Edge: please click at the right side of the web address bar, on the top of the web browser, in the icon language Afterwards, it appears a small language window on the top of the web browser, where you can select the language to translate.

Note: If you want to translate another web page, please return to the (English) original version of (this) web site, open the page you want to translate (still in English) and translate it afterwards, as mentioned above.

Android (Chrome): Top of Screen → click 3 points (vertical) → Translate...

Below: Our Lady of the Roses Honored / Exposed and the Death possibly doesn't Enter Inside.
(source: Jesus, at the old Web site apparitionsmariales)

Right Click + "Save as", to download the images (in the computer)...

Freemasonry, if with Sabotage Purposes, against God's interests, is Forbidden...

Content's License.

School: Maths...

Annex: School. Summary.

Section Physics...

Physics for Kids: Electromagnetism, Production of Electricity, etc., for Everybody...

Understanding the Idea Behind the Dynamos, the Dams and the Cars...

Section Maths...

Maths for Kids: Multiple Integrals, Etc., for Everybody...

Understanding Fractions, i.e., Numbers of the Type '¾'.

Negative Numbers and Negative Areas, for Kids...

What About Negative Areas?

Basic Equations for Kids.

Matrices for Teenagers.

Going Further With Matrix Notation...

Square Roots for Kids: Understanding Square Roots, i.e., Numbers of the Type i = '√-1'.

Integrals for Kids: Simple and Multiple Integrals...

Simple Integrals for Kids...

Computing the Numeric Integral, for a Half Circle. Auxiliar Explanations.

The Simple Numeric Integral, As a Sum of Products of 2 Factors.

Double Integrals for Kids: The Double Numeric Integral, As a Sum of Products of 3 Factors.

Multiple Integrals for Kids: The Multiple Numeric Integral, As a Sum of Products of n Factors.

Discrete Maths, Maths Post-Secondary: a Bit of Convolution Product of Cauchy.

Details of the Demo, "Generalized Cauchy Product" of Wikipedia.

Practicing Product of Series and Polynomials.

Some Practice: Understanding Multi Summations, in the Multinomial Formula.

Paradox: Cardinality of ℝ Has Order 2|ℕ| = ℵ1, But It Has Also Order 2*|ℕ|2 = ℵ0!

Maths Post-Secondary: Complex Analysis.

Practice, Over the Hypergeometric Function, to Get Cos(z).

Divine Nomenclature For the Number Sets.

Introduction to the Theory of the Supernatural Numbers of ℂ, the Complex Analysis of 2nd Order: Another graphical representation for i = √-1.

Analyzing Complex Numbers, as z = 2 + 2i, in Different Cartesian Coordinate Systems and Plans.

Complex Analysis II of 2 nd Order: Infinite Equivalent Possibilities to Represent Complex Numbers, in a Bidimensional Coordinate System.

Setting the Interval of Complex Numbers 0 ≤ z ≤ 2 + 2i, in the Different Cartesian Coordinate Systems and Plans.

Theory of the Supernatural Numbers of ℂ, Complex Analysis II of 2 nd Order: Another Graphical Representation for i = 4√-1, for √i = 4√-1, for 4√i = 16√-1...

Section Computer Science.

Some Solutions for the Exercises of Threads.

Resolution of Exercise 3, page 132: Counters.

Resolution of Exercise 5, page 134: Table of 7.

Resolution of Exercise 7b, page 137: Factorials.

Java Threads Example: Car Park (Based Leic94 IST).

"Another Barber Saloon" Resolution of the Exercise Using Threads.

Another Java Solution for the Dining Philosophers Problem.

Producer Consumer: Resolution of Exercise 14, French Version of Book Benmammar, Threads (Page 147).

Thread Synchronization: a Resolution of Exercise 15, From the (French) Book of Benmammar, Threads (Page 148).

Some Interesting Links, to Try to Help to Studies.

Summary of the Web Site

Catholic Russia Converted Ideas...

www.christianideas.eu:

Catholic Russia Converted Translations

العربية (Arabic)

বাংলা (Bangla)

中文 (Chinese)

Deutsch (German)

Español (Spanish)

Français (French)

Hausa (Hausa)

हिन्दी (Hindi)

日本語 (Japanese)

Melayu (Malay)

Português (Portuguese)

پارسی (Persian)

Русский (Russian)

Tagalog (Tagalog)

Türkçe (Turkish)

Other Languages: Swedish...

Main MENU

Better to Be Vigilant.

Paradox: Cardinality of ℝ Has Order 2^|ℕ| = ℵ₁, But It Has Also Order 2*|ℕ|² = ℵ₀!

Introduction to the Theory of the Supernatural Numbers of ℂ, the Complex Analysis of 2^nd Order: Another graphical representation for i = √-1.

Complex Analysis II of 2 ^nd Order: Infinite Equivalent Possibilities to Represent Complex Numbers, in a Bidimensional Coordinate System.

Theory of the Supernatural Numbers of ℂ, Complex Analysis II of 2 ^nd Order: Another Graphical Representation for i = ⁴√-1, for √i = ⁴√-1, for ⁴√i = ¹⁶√-1...