Let's start with a little spoiler
Yes, I know that if you write a surname with a capital letter, the incident will not work. Further translation.
Mathematics is one of the few fields of knowledge that can be objectively called true because its theorems are based on pure logic. But at the same time, these theorems often turn out to be very strange and counterintuitive.
Some people find math boring. The following examples show that she is anything but not like that.
5. Random datasets
Oddly enough, random data isn't actually all that random. In the data shown, which is everything from stock prices to city populations, building heights and river lengths, about 30 percent of all numbers start with one. Fewer start with 2, even fewer start with 3, and so on, only every twentieth number starts with 9. And the larger the data set, the wider the order of magnitudes covered, the stronger this pattern manifests itself.
4. Spirals of prime numbers
Because prime numbers are indivisible (other than by 1 and themselves) and because all other numbers can be represented as their product, prime numbers are often viewed as "atoms" in the world of mathematics. Despite its importance, the distribution of prime numbers is still a mystery. There is no such rule that would unambiguously say which numbers will be prime and after how long the next prime number will occur.
The apparent randomness of prime numbers makes the facts found in Ulam's Tablecloth very strange.
In 1963, mathematician Stanisław Ulam discovered a surprising pattern when he was painting his notebook during a presentation: if you write integers in a spiral, prime numbers line up along diagonal lines. In itself, this is not very surprising if we remember that all prime numbers except two are odd, and the diagonal lines in the spirals of integers are alternately odd. More unusual was the tendency for prime numbers to lie predominantly on some diagonals and virtually absent on others. Moreover, the regularity was observed regardless of the number from which the spiral began (from one or any other).
Even if you scale the spiral to accommodate a much larger number of numbers, you can see that the accumulation of primes on some diagonals is much denser than on others. There are mathematical assumptions that explain this pattern, but so far they have not been proven.
3. Eversion of the sphere
In one important area of mathematics called topology, two objects are said to be equivalent or homeomorphic if one of them can be transformed into the other by twisting or stretching the surface. Objects are considered different if the transformation requires cuts or breaks in the surface.
As an example, consider a torus, a doughnut-shaped object. If you put it vertically, expand one side and push in the top of the same side, you get a cylindrical object with a handle. There is a classic joke among mathematicians that topologists can't tell a donut from a cup of coffee.
On the other hand, Mobius strips - loops with a single kink are not homeomorphic loops without kinks (cylinders), because you cannot straighten a Mobius strip without cutting it, flipping one side and gluing it again.
Topologists have long been interested in the question, will a sphere be homeomorphic to itself when turned inside out? In other words, is it possible to invert a sphere? At first glance, this seems impossible, because you can't poke a hole in a sphere. But it turns out that the inversion of the sphere is possible. How it's done is shown in the video. .
It is striking that the topologist Bernard Morin, who is the main developer of the above method of turning the sphere, is blind.
2. Mathematics of the walls
While walls can be decorated with an infinite number of swirls, mathematically speaking, there are a finite number of distinct geometric patterns. All of Escher's periodic drawings, wallpapers, tile designs, and in general all two-dimensional repeating groups of figures, can be assigned to one or another of the so-called "flat crystallographic group". And do you know how many such groups exist? Exactly 17.
1. Sonnet
"Like a Shakespearean sonnet captures the very essence of love, or a painting reveals the inner beauty of a person, Euler's equation penetrates the very depths of existence."
Stanford mathematician Keith Devlin wrote these words about the equation in a 2002 essay called "The Most Beautiful Equation." But why does Euler's formula take your breath away? And what does she even mean?
First, the letter “e” is an irrational number (with an infinite number of digits) that begins with 2.71828… Opened in the context of continuously compounding interest, it describes the rate of exponential growth from colony insect populations to radioactive decay. In mathematics, a number has a number of unexpected properties, for example, it equals the sum of inverse factorials from zero to infinity. Ultimately, the constant e took over mathematics, appearing out of nowhere, but ending up in a large number of important equations.
Further. i represents the so-called imaginary unit - the square root of minus 1. "So-called" because in reality there is no number that, when multiplied by itself, results in a negative number (because negative numbers do not have real square roots) . But in mathematics, there are a large number of situations where you have to take the square root of a negative number. The number i is used as a kind of marking of the place where such an operation was performed.
Pi is the ratio of the circumference of a circle to its diameter, one of the favorite and most interesting constants in mathematics. Like e, it has appeared in a large number of mathematical and physical formulas as if from nowhere.
The constant e, raised to the power of the imaginary unit, multiplied by pi equals minus one. It follows from the Euler equation that adding one to this gives zero. It's hard to believe that all these strange numbers, one of which doesn't even belong to the real world, can be combined so easily. But this
Numbers, functions and geometric shapes are pure pleasure. Yes, and mathematics itself is just a very successful joke. When you understand this, then be sure to fall in love with the “queen of sciences” with all your heart. So says Alex Bellos, author of Beauty Squared. Here are some interesting facts from it that will help you dive into the insanely interesting world numbers and graphs.
How to incinerate a boar with a paraboloid
Parallel rays of light entering the paraboloid are reflected by its surface into focus. Therefore, paraboloids are widely used in solar energy technology.
For example, the Scheffler reflector, a parabolic metal bowl, is commonly used in developing countries for cooking. It is directed at the sun and slowly turns following its movement in order to catch as many sun rays as possible, reflecting them to the same point (focus) where the plate is located.
The most powerful solar oven is a parabolic mirror 45 meters high, located in the French Pyrenees, near Odeillo.
Due to its huge size, the mirror itself does not move, but receives reflected sunlight from 63 small flat rotating mirrors. At the focus of the mirror is a round shield that, on sunny days, heats up to 3,500°C, hot enough to boil lead, melt tungsten, or reduce a wild boar to ashes.
Queen's Secret
One of the most interesting mathematical puzzles comes down to rolling one coin around another. Place two identical queen coins next to each other on the table, placing them crown-side up. Scroll the left coin around the right one. In which direction will the crown point when the coin is on the right side?
Are you suggesting that the coin will end up upside down since it has only traveled halfway around the stationary coin? This is mistake. The queen makes a full turn, which at first glance contradicts common sense. The fact is that a coin revolves around itself and around another coin. Movement occurs in two independent directions. For every degree of movement of the left coin around the right, there are two degrees of its rotation around itself.
Why an even number can't be mystical
The Sumerians came up with names for numbers, using the words available in their language. For example, the word ges (“gesh”) was used to designate a unit, the second meaning of which is a man or a phallus. The deuce was denoted by the word min ("min"), also symbolizing the feminine. Perhaps this emphasized the fact that a man occupies a dominant position, and a woman is only an addition to him, or characterized the male penis and female breasts.
The Greek thinker Pythagoras, who lived in the 6th century BC, proclaimed odd numbers to be masculine, and even numbers to be feminine, thereby confirming the associative relationship noted by the Sumerians between a unit and a man, as well as a deuce and a woman. He argued that the unwillingness to divide by two is a sign of strength, while the tendency to such a division is a sign of weakness. In Christianity, this is reflected in the myth of the creation of the world: God created Adam first, and Eve second.
These prejudices have persisted to this day. Only odd numbers are still considered mystical.
Focus with numbers
If you count the frequency of the first digits in all the numbers that you find on the front page of any newspaper, you can notice an interesting pattern. You'll see that numbers starting with 1 are the most common; then numbers follow, the first digit of which is 2, then 3 - and so on until the number 9, which is used least often at the beginning of numbers. It's really incredible. Try it yourself!
In 1938, General Electric physicist Frank Benford discovered the phenomenon of the first digit by noticing the frayed pages in books with logarithm tables. He studied the distribution of leading digits based on data such as the population of US cities, the addresses of the first few hundred people from the biographical directory of American scientists American Men of Science, atomic weight chemical elements, river basin area and baseball game statistics. In most cases, the results were close to the expected distribution.
The method of analyzing numbers for their compliance with Benford's law is increasingly being used to detect data manipulation, not only in the context of financial fraud, but in all those cases to which this law is applicable.
In 2006, Scott de Marchi and James Hamilton of Duke University wrote that the provided industrial enterprises information on the level of lead emissions and nitric acid do not satisfy Benford's law, and this indicates the likelihood of information distortion.
On the basis of the Benford law, University of Michigan political scientist Walter Meebane claimed possible falsification of the results of the presidential elections in Iran. Scientists use Benford's law as a diagnostic tool as well. So, during earthquakes, the upper and lower values of the seismograph readings obey this law.
How to sell a house for more
Cornell University psychologist Manoy Thomas argues that because of the discomfort generated by large non-circular numbers, their value seems to us smaller than it really is: “We tend to believe that small numbers are more accurate, therefore, seeing an exact large number, instinctively We assume that it is less than in reality. As a result, according to Manoy Thomas, we pay more for an expensive product if its price is represented by a non-round number.
In one experiment, Thomas gave subjects pictures of several houses that also listed their prices, randomly represented as either a round number (say, $390,000) or a slightly higher exact number (say, $391,534).
When asked which price they considered higher or lower, respondents on average rated the exact price as lower, when in fact it was the other way around. Tip for those who are going to sell a house: if you want to bail out for it more money, its price must not end with zero.
In the world of prime numbers
Jerry Newport is a former taxi driver from Tucson who suffers from Asperger's syndrome, a mental disorder in which a person has difficulty in interpersonal communication, but has unique talents. When Jerry sees a big number, he immediately divides it into prime numbers - 2, 3, 5, 7, 11... that is, numbers that are only divisible by themselves and one.
“I only pay attention to numbers that have more than four digits. if there are fewer of them, it is like an animal crushed on the road. Yes exactly! he declares indignantly. “Come on, show me something new!”
Sometimes Jerry fails to factor a large number into prime factors, which means that the number itself is prime.
“When you meet a new prime number, it's like looking at stones and finding something unusual among them. Something like a diamond that you can take home and put on the shelf,” Jerry explains. “A new prime number is like a new friend.”
Paradox of infinity
The philosopher Zeno warned against using such a concept as infinity in a series of paradoxes. The most famous of them, "Achilles and the Tortoise", demonstrated that adding an infinite number of quantities leads to an absurd result.
Imagine, said Zeno, that Achilles is trying to catch up with the tortoise. When the athlete reaches the place where she was when he started his run, the turtle will crawl a little further. When he gets to the second position, the tortoise will move further again. Achilles can continue his run as long as he likes, but each time he reaches the place where the tortoise was, he will already be a little ahead.
This is a very interesting, important science - mathematics.
It is possible not to be a mathematician, not to know it at a very high level, but it is difficult to argue with the fact that we meet mathematics almost everywhere.
Mathematics is found both at work and in Everyday life, the numbers are chasing us everywhere.
So I suggest you familiarize yourself with interesting, unusual facts from the world of this serious science. There is a place for the frivolous or simply fascinating in any exact science. The main thing is the desire to find it.
1. Among all figures with the same perimeter, the circle will have the largest area. Conversely, among all figures with the same area, the circle will have the smallest perimeter.
2. In fact, a moment is a unit of time that lasts about a hundredth of a second.
3. The number 18 is the only (except zero) number, the sum of the digits of which is two times less than itself.
4. In a group of 23 or more people, the probability that two of them will have the same birthday exceeds 50%, and in a group of 60 people, this probability is about 99%.
5. In mathematics, there are: braid theory, game theory and knot theory.
6. The cake can be cut with three touches of the knife into eight equal parts. And, in two ways.
7. 2 and 5 are the only prime numbers that end in 2 and 5.
8. Zero is the only number that cannot be written in Roman numerals.
9. The equal sign "=" was first used by the British Robert Record in 1557.
10. The sum of numbers from 1 to 100 is 5050.
11. Since 1995, in Taipei, Taiwan, residents have been allowed to remove the number four, since in Chinese this number sounds identical to the word "death." Many buildings do not have a fourth floor.
12. It is believed that the number 13 became unlucky because of the Last Supper, which was attended by 13 people, including Jesus. The 13th was Judas Iscariot.
13. Charles Lutwidge Dodgson is a little-known British mathematician who devoted much of his life to logic. However, he is a world-famous writer who wrote under the pseudonym Lewis Carroll.
14. The first female mathematician in history is the Greek Hypatia, who lived in Egyptian Alexandria in the 4th-5th centuries AD.
15. American George Dantzig, being a student, was late for class and mistakenly accepted the equations written on the blackboard as homework. With difficulty, but the future scientist coped with them. As it turned out later, these were two "unsolvable" problems in statistics, over the solution of which scientists struggled for many years.
16. Modern genius and professor of mathematics Stephen Hawking claims that he studied mathematics only at school. When teaching mathematics at Oxford, Stephen simply read the textbook ahead of his own students by a couple of weeks.
17. In 1992, like-minded Australians teamed up to win the lottery. There was $27 million at stake. The number of combinations, 6 out of 44, was a little over seven million, with a lottery ticket costing $1. These like-minded people created a fund in which each of the 2,500 people invested three thousand dollars. The result is a win and a return of 9 thousand to everyone.
18. Sofya Kovalevskaya, for the sake of science, had to arrange a fictitious marriage. In Russia, women were forbidden to engage in science. The father was against the departure of his daughter abroad. Marriage was the only way. True, later, a fictitious marriage became actual and Sophia even gave birth to a daughter.
Interesting facts about mathematics are not familiar to everyone. In modern times, mathematics is used everywhere, even in spite of technological progress. The science of mathematics is valuable to people. Interesting facts about her will interest even children.
1. Not always people used the decimal number system. Previously, a system of 20 numbers was used.
2. Rome never had the number 0, despite the fact that the people there are smart and know how to count.
3.Sofya Kovalevskaya proved that you can learn mathematics at home.
4. The records that were found in Swaziland on the bones are the oldest mathematical work.
5. The decimal number system began to be used due to the presence of only 10 fingers on the hands.
6. Thanks to mathematics, it is known that a tie can be tied in 177147 ways.
7. In 1900, all mathematical results could be contained in 80 books.
8. The word "algebra" has the same pronunciation in all languages of the world.
9. Real and imaginary numbers in mathematics were introduced by Rene Descartes.
10. The sum of all numbers from 1 to 100 will be 5050.
11. The Egyptians did not know fractions.
12. Counting the sum of all the numbers on the roulette, you get the number of the devil 666.
13. With three touches of the knife, the cake is divided into 8 identical parts. And there are only 2 ways to do it.
14. You can't write zero in Roman numbers.
15. The first female mathematician is Hypatia, who lived in Egyptian Alexandria.
16. Zero is the only number that has several names.
17. There is a world day of mathematics.
18. The bill was created in the state of Indiana.
19. Writer Lewis Carroll, who wrote Alice in Wonderland, was a mathematician.
20. Thanks to mathematics, logic arose.
21. Moivre at the expense arithmetic progression was able to predict the date of his own death.
22. Solitaire is considered the simplest mathematical solitaire.
23. Euclid was one of the most enigmatic mathematicians. No information about him reached the descendants, but there are mathematical works.
24. Most mathematicians in school years behaved disgustingly.
25. Alfred Nobel decided not to include mathematics in his list of prizes.
26. In mathematics there is braid theory, knot theory and game theory.
27. In Taiwan, you almost never see the number 4.
28. For the sake of mathematics, Sofya Kovalevskaya had to enter into a fictitious marriage.
30. All our life consists of mathematics.
20 fun facts about math for kids
1. It was Robert Record who, in 1557, began to use the equal sign.
2. Researchers from America believe that students who chew gum during a math exam achieve more.
3. The number 13 is considered unlucky because of the biblical story.
4. Even Napoleon Bonaparte wrote mathematical works.
5. Fingers and pebbles were considered the first computing devices.
6. The ancient Egyptians did not have multiplication tables and rules.
7. The number 666 is shrouded in legends and is the most mystical of all.
8. Until the 19th century, negative numbers were not used.
9. If translated from Chinese, the number 4 means "death."
10. Italians don't like the number 17.
11. Large number of people lucky number count exactly 7.
12. The largest number in the world is the centillion.
13. The only prime numbers that end in 2 and 5 are the numbers 2 and 5.
14. The number pi was first introduced into use in the 6th century BC by the Indian mathematician Budhayana.
15. In the 6th century, quadratic equations were created in India.
16. If a triangle is drawn on a sphere, then all its angles will only be straight lines.
17. The first signs of addition and subtraction familiar to us were described almost 520 years ago in the book “Rules of Algebra”, written by Jan Widman.
18. Augustin Cauchy, who is a French mathematician, wrote more than 700 works in which he proved the finiteness of the number of stars, the finiteness of the natural series of numbers and the finiteness of the world.
19. The work of the ancient Greek mathematician Euclid consists of 13 volumes.
20. For the first time, it was the ancient Greeks who brought this science into a separate branch of mathematics.
sin2 + cos2 = 1
or:
orange 2 + apricot 2 = 1
How do you mentally multiply by 11?
How to quickly mentally multiply two-digit numbers by 11? Everything is simple!
Add the first and second digits of the number you are going to multiply by 11 and put the sum of the digits in the middle. The resulting number of three digits is the result. If the sum of the digits turns out to be more than 10, for example 14, then add 1 to the first digit, and put 4 in the middle.
Here are some examples to make things clear:
25 x 11 = 2 (2+5) 5 = 275,
34 x 11 = 3 (3+4) 4 = 374,
48 x 11 = 4 (4+8) 8 = 4 (12) 8 = (4+1) (2) 8 = 528.
Calculator not working :)
Do you know that there is a bug in the Windows calculator?
1. Open the Windows Calculator.
2. Enter 6084.
3. Press the division button [/].
4. Enter 78.
5. Press the equals button [=].
The calculator is not responding. If you click on "equal" again and one more time, then it starts to give out some nonsense.
How to make triangular milk bags
Remember milk in triangular bags? What do you think, if the package is pasted, then what shape will the scan be? It can be assumed that you will get 4 triangles with stripes on the sides for gluing. But actually it is not. The scan will represent nothing more than ... a rectangle. Yes, it's a rectangle. The rectangle is first glued into a cylinder (lateral surface of the cylinder), then along the mutually perpendicular diameters of the bases - into a triangular (or rather, tetrahedral) package. Technologically, this is much easier to implement than gluing a package of triangles.
How many can you count?
Ask a small child: “How many can you count to?” He will answer: "Up to ten!" The older one will answer “up to a thousand” or “up to a million”. What if you ask an adult? Try to answer yourself a simple question: "How many can I count?" Just out of curiosity.
As a rule, adults can count up to several billion or trillion. They don't remember or don't know how. And in general, this is normal. All subsequent orders - clogging the head with "garbage". But the question itself, banal at first glance, makes an adult think for a while. Tested in practice :)
For reference:
ten
hundred
one thousand
million
billion or billion
trillion
quadrillion
quintillion
sextillion
septillion
octillion
etc.
How to compose poetry?
Read the numbers as they are: twenty forty thirty three...
20 40 33
10 18
50 11 03
60 12
Math in jokes
Why do the wheels rattle when the train is moving? Because they are round...
Do you remember the formula for the area of a circle?
- I remember. S = πR 2
- Well... Square, do you understand?! That's exactly what he knocks.
* * *
- What is the date today?
- Pi.
- Why???
- Well, why? 3 month and 14 days... 3.14
About beer...
Surprise your friends and acquaintances with your versatile knowledge of mathematics: beer foam in a glass settles according to the exponential law.
amazing squares
Below is an amazing square: in any row, the sum of the numbers is 66, even adjacent four cells add up to 66. Try to count how many different ways you can get 66 in this square.
Case declension
There is a famous example of using fractions to get a dative question. Sometimes teachers show it to the class to defuse the situation. At one time he was popular on forums on the Internet. However, not everyone has heard of it, so we decided to include it in our article as another unusual way to use mathematics in different areas.
Nominative: who? what?
Genitive: who? what?
Dative: to whom? ...
To get a question for the dative case:
1) accept the question as X.
2) make up the relationship: Whom? / What? = To whom?/x?
3) We express X: X = (To whom? * What?) / Whom?
4) We reduce the numerator and denominator of the fraction by "Ko" and "go"
5) The syllables "mu" and "Che" remaining after the reduction are rearranged
6) We get that X = "What?"
Abbreviations
Shortening words by writing them as letters and numbers is another example of the use of mathematics in everyday life. You have seen them more than once, perhaps you have used them yourself. We list a few:
7th - family
40a - magpie
100 faces - capital
pro100 - simple
etc.
gr8-great
b4 - before
l8 - late
w8 - wait
2day - today
etc.
Guess the number
Think of a number. Add to it the following in order. Add 9 to the result. Divide by 2 (count only whole numbers). Now subtract the number you have in mind. How much did it turn out? Five!
Example.
We take 70.
Add the following: 70 + 71 = 141
Add 9: 141 + 9 = 150
Divide by 2: 150: 2 = 75
Subtract what we intended: 75 - 70 = 5
How to quickly make a multiplication table for 9?
Let's write in a column:
9x1=
9x2=
9x3=
9x4=
9x5=
9x6=
9x7=
9x8=
9x9=
Then, without hesitation, we put the numbers from 0 to 9 after the equal sign from top to bottom:
9x1=0
9x2=1
9x3=2
9x4=3
9x5=4
9x6=5
9x7=6
9x8=7
9x9=8
9x10 = 9
Then we put down the second digit from 0 to 9 from bottom to top:
9x1=09
9x2=18
9x3=27
9x4=36
9x5=45
9x6=54
9x7=63
9x8=72
9x9=81
9x10 = 90
