## 1.12.2019

### Achilles and the tortoise

 Achilles and the tortoise

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporia, the most famous of which is the Achilles and the Tortoise aporia. Here’s how it sounds:

Suppose Achilles runs ten times faster than a turtle, and is a thousand paces behind it. During the time that Achilles has run this distance, the tortoise will crawl a hundred steps in the same direction. When Achilles runs a hundred paces, the tortoise crawls ten more steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

Another version of the aporia "Achilles and the tortoise":

In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

This reasoning was a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Hilbert ... All of them in one way or another viewed the aporia of Zeno. The shock turned out to be so strong that discussions are continuing at the present time; the scientific community has not yet succeeded in reaching a general opinion on the essence of paradoxes. Mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the question; none of them has become a generally accepted solution. Everyone understands that this is a hoax, but no one understands what it is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from quantity to an inverse quantity. This transition implies the use of variable units of measurement instead of constants. As far as I understand, the mathematical apparatus for using variable units of measurement has either not yet been developed or has not been applied to the aporia of Zeno. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like a slowing down of time to its full stop at the moment when Achilles approaches the turtle. If time stops, Achilles can no longer overtake the turtle.

If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on its overcoming is ten times less than the previous one. If you use the concept of "infinity" in this situation, then it will be correct to say "Achilles will catch up to the tortoise infinitely quickly".

How to avoid this logical trap? Stay in constant time units and don’t go to inverse quantity. In the language of Zeno, it looks like this:

In the time that Achilles has run a thousand steps, the tortoise will crawl a hundred steps in the same direction. Over the next interval of time equal to the first, Achilles will run a thousand more steps, and the turtle will crawl a hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution. On the aporia of Zeno "Achilles and the Turtle" is very similar to Einstein's statement about the irresistible speed of light. We still have to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells of a flying arrow:

A flying arrow is motionless, because at each moment of time it is motionless, and since it is motionless at each moment of time, it is always motionless.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment in time a flying arrow is stationary at different points in space, which, in fact, is movement.

## 2.10.2018

### Trapezoid

In Wikipedia there is a page of a trapezoid. In the drawing it is shown that any trapezoid can be turned into a rectangle.

 Trapezoid
Let's look at algebra. These are formulas for calculation of lengths of diagonals of a trapezoid.

 Diagonals of trapezoid
We substitute in these formulas data for a rectangle.

 Diagonals of rectangle
If to trust these formulas, the rectangle has no diagonals. Even schoolboys or schoolgirls can make what mathematicians couldn't make – to execute verification of the solution of a task. It is the actual level of modern mathematics – any statement of mathematicians can be false.

Height of a trapezoid

Height of a trapezoid is determined by the area of a triangle. The area of a triangle is calculated on Heron's formula. The sizes of the parties of a trapezoid allow to receive a triangle which has the same height as a trapezjid. The cunning trick of mathematicians allows to calculate length of diagonals of a trapezoid.

 Height of a trapezoid
When the legs of a trapezoid are parallel, the triangle disappears and the cunning trick ceases to work. If to determine height of a trapezoid by the area of a trapezoid, then no problems arise upon transition to a rectangle or a parallelogram.

 Height of a rectangle
Conclusion: cunning tricks of mathematicians can result in false results.

## 3.02.2017

### Inverse transformations

Last time we transformed the law of cosines and a Pythagorean theorem to the sum of a line segments. Now we will execute inverse transformations.

 Inverse transformations

In inverse transformations I made everything very simply. The minus sign from a formula disappeared. The problem is that we are not able to measure angles correctly. Than differ a angle 0 degrees from a angle of 180 degrees?

 Measurement of a angle
It is possible to assume that if a line segment one, then a angle is equal 0 degrees. If a line segments two, then a angle is equal 180 degrees.

The transformations executed by us show that the mathematics has no separate areas of mathematics: "arithmetic", "algebra", "geometry" or "trigonometry". The mathematics is a single whole.

The mathematics is DNA of the nature. Further we will continue to study a cosine gene in a cosine law.

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## 2.27.2017

### We use the law of cosines

I was always interested in a question: how the Pythagorean theorem turns into the sum of a line segments? What I speak about? Here you look.

 The Pythagorean theorem and the sum of a line segments
In geometry everything is very prime. The first time we draw a right triangle and we write down a Pythagorean theorem. The second time we draw two a line segments and we write down the sum of a line segments.

 Right triangle and two a line segments
How one formula turns into other formula? To see it, we use the law of cosines. We will draw the picture, we will write down conditions, we will execute transformations.

 Triangle and the law of cosines
 Right triangle and the Pythagorean theorem
 The law of cosines and sum of two a line segments
We use the law of cosines and turned the Pythagorean theorem into the sum of two a line segments. Further we will consider an inverse transformation.

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## 2.25.2017

### Law of sines

The law of sines (sine law, sine formula, or sine rule) is an equation relating the lengths of the sides of a triangle (any shape) to the sines of its angles.

 The law of sines
Where:
a, b, c - are the lengths of the sides of a triangle;
α, β, γ - are the opposite angles;
T - are the area of triangle;
R - are the radius of the triangle's circumcircle.

How to use this monster? Use dress-making courses. Cut the law of sines on a part.

 Cut the law of sines on a part
Make a necessary formula of two parts. Use properties of proportions.

 Example of use of the law of sine

## 2.13.2017

### Transformations of trigonometric functions

 Transformations of trigonometric functions
This table shows how one trigonometric functions can be transformed to other trigonometric functions. Sin, cos, tan, cot, sec, csc - all these functions can be transformed.

## 2.10.2017

### Number to the power minus one

What to do if number to the power minus one? Write down this number in a fraction denominator. Here several examples with numbers to the power minus one.

 Number to the power minus one
Negative exponents of numbers mean that these numbers are in a fraction denominator.

 Negative exponents