Prime number X

Prime number X

Elon Musk published the prime number X on his page. As you can see, this prime number actually has 1800 digits. Reddit users have started a fascinating discussion about this prime number. I liked these comments:

Watch out: there’s a sneaky 7 near the end of the tenth row.

It's pretty obvious someone tried a few tweaks until they found a configuration that actually gives a prime. But yeah, the digit 7 is the most visible one and surely the last change made - essentially that's the point where the author gave up on having a nicer solution :)

Now I'll remove all 1 from the drawing and we'll see what's left.

Prime number X without 1

As you can see, the number of eights in the lines does not correspond to the ideal pattern. And also this number 7. Yes, even Elon Musk is powerless against mathematics.

I’m interested in this question: how much money did Elon Musk pay for this work? Personally, I would not regret the largest amount possible. What could be greater than the SUM OF ALL NATURAL NUMBERS? Nothing. Who do I owe -$1/12? Why can you use this in physics, but you can’t pay for work?

Why 13?

No, today I will not tell you why the number 13 is considered unlucky. All numbers are the same. It is we who consider some numbers to be happy or unlucky, positive or negative. We will talk about the sum of two numbers on a watch dial. Why does adding two numbers on a watch dial equal 13? We see six pairs of such numbers on the clock. The explanation is very simple.

Why 13?

If we add 1 to one term and subtract 1 from another term, the sum will remain the same. This can be repeated many times and the result of the addition will not change. This property of addition can be observed not only on the watch dial, but also on the number line.

Sum on a number line

As the initial sum on the number line, you can take two adjacent numbers (top picture) or two numbers located next to the number we chose (bottom picture). I called such sums “Decomposition sums”.

2 to the power of 0 to 100

2 to the power of 0 to 29

The successive powers of two from 0 to 29 are presented in the table above. The table of powers of the number 2 begins with the exponent zero. Any number to the zero power is equal to one. Therefore, 2 to the power of 0 is equal to 1. Any number to the first power is equal to itself. Therefore, 2 to the power of 1 is equal to 2.

If this table is not enough for someone, then you can look at another one, where the powers of the number 2 are presented up to the 100th power.

2 to the power of 0 to 49

2 to the power of 50 to 69

2 to the power of 70 to 89

2 to the power of 90 to 100

I hope programmers will like these tables of powers of the number 2 from 0 to 100. Mathematicians love to stick all sorts of nasty things anywhere. As a worthy student, I could not resist inserting 2 to the power “pi” and 2 to the power “e” into the table. Perhaps one of the child prodigies will find this useful. And now a little piece of theory.

Two to the second power means that the number two must be multiplied by itself. Therefore, 2 to the power of 2 or 2 squared equals four.

2² = 2 x 2 = 4

In general, the exponent shows how many identical numbers are multiplied together. So, two to the third power or 2 cubed means that three 2s are multiplied at the same time and this equals eight:

2³ = 2 x 2 x 2 = 8

Two to the fourth power is equal to four twos multiplied:

2⁴ = 2 x 2 x 2 x 2 = 16

This table of successive powers of the number 2 is very often used in programming, since it uses the binary number system.

Negative numbers: numbers game

You have all seen the game of chess. Adults go on stage, sit at a table opposite each other and start the game.

Let's organize a similar numbers game between two mathematicians. The rule is very simple - the mathematician who correctly depicts the number axis wins.

There is a piece of paper between the players. There is already ONE straight line drawn on the sheet. The players' task is to turn this line into a number axis. Players sit down at the table and start the game.

Players draw a zero, place an arrow in the direction of positive numbers, and choose the size of one unit. After this, positive numbers and negative numbers appear on the playing field. Game over.

Numbers axis. Numbers game.

The game ended in a draw. Both mathematicians drew the number axis correctly, as they were taught. Further, the winner can be determined by the number of scientific publications or through a boxing match.

What did we get? All numbers are both positive and negative at the same time. The sign of the number depends on which side we look at the number аxis.

The secret is that in nature numbers have no sign. The plus and minus signs are a reflection of our own opinion about a particular number. This opinion may be different for two different people.

Negative numbers: it's debt

Author Nikolay Khyzhnjak

You can't take life from the dead.

In one popular science film, a mathematician gave an example about negative numbers. "You have minus one fish. You went fishing, you caught one fish and you have nothing left. Amazing result!".

The mathematical record of this story looks like this:

-1 + 1 = 0

Now suppose I am a fisherman. This is how I see this story. Someone says that I have minus one fish. I don't have anything. I write zero.

0

I go fishing and I catch one fish. Now I have one fish. I add one.

0 + 1 = 1

After that, the one who claimed that I have minus one fish appears and takes my fish for himself.

0 + 1 - 1 = 0

Ups! Where are the negative numbers here? This is an ordinary racket.

Now let's look at what it means to borrow. I urgently need money, 1 dollar. What do I have now? Nothing, zero.

0

A good man (or an evil bank) gave me $1. What do I have now? I have a real, positive $1 that I can spend however I want. No, I won't waste it. I'll keep it as a museum piece.

0 + 1 = 1

After that I earned 1 dollar. How much money do I have?

1 + 1 = 2

I am an honest person and I pay my debts. How much money will I have left?

2 - 1 = 1

I borrowed money, I paid it back and I was left with $1. Or I will still have what I bought. What was it? Were these negative numbers? No. It was a time machine that allows me to buy today what I can earn only tomorrow.

The main rule is this: it is impossible to borrow something that does not exist. Try borrowing a time travel machine, a perpetual motion machine, or a flying saucer. This is impossible to do because these things do not exist. And no matter how many minus signs you promise, you still won’t get these things. You can't take life from the dead.

Next, I suggest playing a numbers game.

Table of primes

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. Prime numbers have exactly two divisors: 1 and the number itself. Below is a table of prime numbers up to 20000.

Table of primes 2 to 1,000

Table of primes 1,000 to 10,000

Table of primes 10,000 to 15,000

Table of primes 15,000 to 20,000

Prime numbers are a favorite toy of mathematicians. As of January 2019, the largest prime number known to mathematicians had 24,862,048 decimal digits. This prime numbers game is so addicting that even I am addicted to this game. What can I say about this? Playing prime numbers is very time-consuming and tedious, but the results can be very interesting. I came up with my own rules of the game, which I will tell you about soon. I hope this will be of interest to mathematicians.

I needed prime numbers up to 20,000. Here you will find prime numbers up to 21,000,000.

You might be interested:

Prime number X.

Number spirals and prime numbers.

Negative numbers: subtraction

Mathematicians tell us that there are positive and negative numbers. Only then do they talk about the absolute value of the number (the modulus of the number). I don't trust mathematicians. Are there negative numbers in nature? What do the plus and minus signs mean in math? We'll try to figure this out.

Negative numbers appear in arithmetic when a larger number is subtracted from a smaller one. That's what mathematicians say and that's how they teach us.

If you always subtract a smaller number from a larger number, then negative numbers will never occur. This rule is no better or worse than the first rule. But we are not taught this rule.

Subtraction

The minus signs fell off like dried mud. What remains is a crystal clear addition. As you can see, negative numbers cannot be obtained by subtraction.

I did everything right. That's how they taught me at school. If minus x equals minus one, then x equals one.

-x = -1
x = 1

What is the secret of this trick? Mathematicians solve the problem of subtraction incorrectly. If we know the result of the addition and one of the terms, then the second term can be found by subtracting the known term from the sum. The minus signs indicate that we are confused.

Let's see what my manipulations on the number axis look like.

Subtraction on the number line

On the left side of the figure, the numerical axis is fixed. I change the position of the segments relative to zero. On the right side of the figure, the segments remain motionless. I am changing the position of zero. In any case, the sign of the number changes when passing through zero. In algebra, this is equivalent to moving numbers through the equals sign. In the future, we will return to the numerical axis and consider it carefully. An equals sign and zero is a mathematical boundary where spies become scouts and scouts become spies.

We are done with subtraction. Next, we'll see if negative numbers come up when we borrow something.

Аverage percentage

Average percentage

On Reddit, a person asked an interesting question about percentages:

Help with explaining negative compounding

Hey guys, not a maths teacher and I need to teach a concept related to work where something is negatively compounded.

For example:

Of a 1000 items 20% are rejected, leaving 800. At the next stage of those 800 30% are rejected, leaving 560. At next stage of the 560 all are rejected at 100% rejection.

The "average" rejection rate is NOT (20+30+100)/3 equalling 50%, it's 42% because "maths". I do actually have the formula and I personally understand why it's 42% instead of 50 but can anyone help me try to explain in a manner that makes it simple to understand WHY it's 42 and not 50?

I wrote my comment:

This is an incorrect application of mathematics. The average failure rate can only be calculated within one process. The average percentage between processes is absurd. Mathematically it looks like this:

1000*(1-0,42)=580
580*(1-0,42)=336,4
336,4*(1-0,42)=195,112

I showed that the proposed concept of "average percentage" does not allow you to get zero at the end of the calculation, which is not true. The man explained how he sees his decision:

I'm not sure I understand.

I understand the formula, as you have written, as it is what I use, and what I demonstrated above in a reply to an answer. But I don't understand what you mean about the average percentage between processes being absurd.

If I use my previous example:

First use = 1 (Used once, no rejection)
Second use = 1*(1-0.2) = 0.8 (20% rejected)
Third use = 0.8*(1-0.3) = 0.56 (30% rejected)
Fourth use 0.56*(1-1) = 0 (End of life product, 100% rejected)

Total average = 1/(1+0.8+0.56) = 0.42

In the end, using the average of 0.42, gives us an approximate product rejection, or if we want to call it something else, it's average "life", before it needs a brand new replacement.

Now I will show you what is wrong in this decision.

Let's go back to 1000 items. Absurdity is the summation of numbers with different units of measure. Percentages are taken from different numbers:

I use     100%=1000   1%=10
II use    100%=1000   1%=10
III use   100%=800    1%=8
IV use   100%=560    1%=5.6

To determine the average percentage, we are actually adding numbers with different units of measure. We received 42%, but it is not known from what number these percentages are to be calculated. It's like adding one dress to one gallon. We can get two. But what does the number two mean? Two gallons, two dresses, or two wet dresses?

1[gallon]+1[dress]=2[?]

To get wet dresses, you need to use multiplication:

1[gallon]*1[dress]=1[gallon*dress]

This math matches reality much better. By the way. In mathematics, the absence of a number is denoted by the number "zero". The absence of a letter in the grammar is indicated by a "space". Mathematicians say that zero is a number. Philologists have never said that a space is a letter. In units of measurement, it is more logical to use a space.

1[gallon*dress]=1[wet_dress]

Now let's look at percentages. To be correct, we must count the number of items rejected at each stage as a percentage of the total number of items.

I use      0 items      0%
II use    200 items   20%
III use   240 items   24%
IV use   560 items   56%

It's even better to count as a percentage the number of items that fail after each stage with a cumulative total. This will be a more realistic picture of what is happening.

I use      0 items      0%
II use    200 items   20%
III use   440 items   44%
IV use   1000 items  100%

Personally, I argue like this. In this case, it is not appropriate to calculate the average percentage.

Sine and cosine table

This is the most unusual table of sines and cosines. The idea to make such a table came to me in the process of writing these articles:

Why is the sine of 30 degrees equal to half one?

What is the sine of 60 degrees?

Sine and cosine of 45 degrees

I strongly recommend that you read these articles. Then you will know how easy it is to calculate the values of sine and cosine yourself. I made one sine and cosine table from these three articles.

Sine and cosine table

The table indicates which sides of the triangle to take as a unit. For 0 and 90 degrees, triangles do not exist, these are ordinary segments.

The third column of the table shows how to use the Pythagorean theorem to calculate the desired value of the sine or cosine. For a cosine of 30 degrees and a sine of 60 degrees, the calculations are the same.

At the bottom of the table there are hints on how to calculate the tangent and cotangent values if you know the sine and cosine values. If you forgot how to divide one fraction by another, use one more hint. You need to multiply the first fraction by the reciprocal of the second fraction.

Sine and cosine of 45 degrees

We have already considered what the sine of 30 degrees is and what the sine of 60 degrees is equal to. Now we will see how to find the sine and cosine of 45 degrees and why they are equal.

Last time we considered an equilateral triangle. Now we will consider an isosceles right triangle. A very rare beast in a herd of triangles. Mathematicians have known it for many thousands of years, and it is simply boring for mathematicians to tinker with it.

Isosceles right triangle

We will consider a tired isosceles right triangle. Why is the triangle tired? He lay down on his side to rest.

Isosceles right triangle

The legs of this triangle are equal to one. Once again I repeat that both legs have the same length. We do not know the length of the hypotenuse of this triangle, but we can easily calculate it using the Pythagorean theorem.

Sine and cosine of 45 degrees

Smart people came to us and said that the sine and cosine of an angle are equal to the ratio of the legs to the hypotenuse. Since the legs of a right triangle with an angle of 45 degrees are equal, then the value of the sine of 45 degrees is equal to the value of the cosine of 45 degrees. We take the math in hand and calculate this value.

Sine and cosine of 45 degrees

Why did I multiply the numerator and denominator of a fraction by the square root of two? Small children do not like lumps in porridge. Mathematicians don't like square roots in denominators. Very capricious uncles and aunts.

What is the sine of 60 degrees?

Before we look for the answer to the question: "What is the sine of 60 degrees?", I strongly recommend reading the post "Why is the sine of 30 degrees equal to half one?".

Height of an equilateral triangle

I again take an equilateral triangle with a side equal to one. I draw height.

Height of an equilateral triangle

After that, I will not turn anything. I'll just remove half of the drawing.

Cosine 60 degrees

I have a right-angled triangle, the diagonal of which is equal to one.

Cosine 60 degrees

The base of this triangle is equal to 1/2 and is also equal to the cosine of an angle of 60 degrees.

Sine 60 degrees

The height of this triangle is equal to the sine of the angle of 60 degrees. We calculate the height using the Pythagorean theorem.

Sine 60 degrees

The sine of 60 degrees is equal to the square root of three divided by two. It is calculated in the same way as the cosine of 30 degrees.

Where does math end?

I was offered to sell my Russian site for 45.000 rubles. This prompted me to a very interesting question: "Where does mathematics end?". And here's my answer: "Math ends where money begins". Try any mathematical theory to apply to money and you will learn a lot of interesting things about this theory.

I love math trolling. The sentence contained the phrase: "The price may be revised upwards". My response was: "$45.000 Price may be revised down". From the point of view of mathematics, the equality 45.000=45.000 is beyond doubt. As soon as I add different units of money to this equality, it disappears.

Where does math end?

We use numbers to measure money. Numbers are written on coins and banknotes. Have you heard of banknote or a coin with the number "zero"? Zero is not a number. The more I study this question, the more I am convinced of the correctness of my statement.

Zero dollars

I'm not even talking about the minus sign on money. Through the prism of money, I considered set theory. The result is very interesting.

Why is the sine of 30 degrees equal to 1/2?

The answer to the question "why is the sine of 30 degrees equal to 1/2?" can be searched in the history of mathematics. These are ancient Mesopotamia, ancient Greece and other ancient civilizations. I am not an expert in this area. Obviously, knowledge has evolved from simpler to more complex. I will show you the most obvious answer to this question as I see it.

Equilateral triangle

Equilateral triangle

I have drawn an equilateral triangle with sides equal to one. I so want. The sum of the angles of a triangle is 180 degrees. An equilateral triangle has three 60 degree angles. I don't know trigonometry yet.

Height of an equilateral triangle

Height of an equilateral triangle

I drew the height in an equilateral triangle. The height is always perpendicular to the base of the triangle. If the height is drawn through the vertex of such a triangle, it will divide it into two equal right-angled triangles. This is always the case in isosceles triangles. An equilateral triangle is a special case of an isosceles triangle, in which the base is equal to the sides.

What happened as a result? The height divided the angle at the apex into two equal angles 60=30+30, it divided the base into two equal segments 1=(1/2)+(1/2). I still haven't heard anything about trigonometric functions.

Sine 30 degrees

After that, someone came up with trigonometric functions. I was told that the sine of an angle in a right triangle is the ratio of the opposite leg to the hypotenuse. How do I find the sine value for a 30 degree angle? I just flip the picture 90 degrees and remove all unnecessary.

Sine 30 degrees

The hypotenuse is equal to one. Any number divided by one does not change. So the length of the opposite leg in my triangle is equal to the sine of the angle of 30 degrees, that is, 1/2.

Cosine 30 degrees

The cosine of 30 degrees I can easily find from the Pythagorean theorem. We take the Pythagorean theorem in our hands and count.

Cosine 30 degrees

The cosine of 30 degrees turned out to be equal to the square root of three, divided by two.

Here's how easy it is to calculate. No tables needed.