tag:blogger.com,1999:blog-37090524521612558162024-11-08T01:11:11.637+02:00Mathematics For BlondesMathematics should be done by blondes - they don't know how to lie.Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.comBlogger195125tag:blogger.com,1999:blog-3709052452161255816.post-26385604173583703872024-05-15T09:25:00.007+03:002024-09-26T22:26:26.308+03:00Area of a trapezoid and division by zeroTrapezoid
Once I already considered the formula for the diagonals of a trapezoid. When moving from a trapezoid to a rectangle, I ended up with an indefinite expression - zero divided by zero under the square root sign.
Now I propose to consider another formula. In the 7th century, the Indian mathematician Bhaskara I derived a formula for determining the area of a trapezoid with successive Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-41760638847507323942024-03-13T13:31:00.003+02:002024-03-13T14:43:12.342+02:00Number spirals and prime numbersNumber spirals and prime numbers
Beginning: Number spirals introduction
All primes of the form P are located at the beginning of the number rays on all number spirals, with the exception of the prime number a, located on the main axis of the a-spiral. The relative position of the prime numbers will change depending on the number a, which underlies the construction of the spiral. The Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-33955613578007675022024-03-12T15:19:00.004+02:002024-03-13T14:42:48.135+02:00Number spirals and numeral systemsNumber spirals and numeral systems
Beginning: Number spirals introduction
Numbers on number spirals can be represented in different number systems. For example, consider the numbers on the main axis of some spirals in different number systems. Table 1 shows the numbers on the main axis of the 2-spiral in binary, decimal and hexadecimal numeral systems.
Main axis of 2-spiral
Table 2 presents Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-88503280030646098552024-03-12T10:42:00.003+02:002024-03-13T14:42:20.086+02:00Analysis of number spiralsAnalysis of number spirals
Beginning: Number spirals introduction
Movement along the turns of any a-helix is an addition table with the sequential addition of one unit to the previous number, starting from one. Movement along the rays of any a-spiral is a multiplication table with sequential multiplication of the number located at the beginning of the ray by the number a. These two movements Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-58365262647620091072024-03-12T10:16:00.005+02:002024-03-13T14:56:46.955+02:00Description of number spiralsDescription of number spirals
Beginning: Number spirals introduction.
2-spiral
2-spiral
On a 2-spiral the main axis is formed by successive powers of the number 2. All odd numbers are located at the beginning of the numerical N-rays, all even numbers are located on the N-rays of the spiral.
On the 2.0-turn there is one single arc of size 360°, at the beginning of this turn there is the Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-80877357523565246492024-03-12T10:04:00.003+02:002024-03-13T14:41:55.404+02:00Number spiralsNumber spirals
Beginning: Number spirals introduction
If you use angle units of measurement and an arbitrary spiral to visually display natural numbers, then all numbers can be ordered according to the following rules:
1. On the main axis of the a-spiral there are numbers of the form an in ascending order, where а>1, n≥0.
2. The main axis coincides with the zero ray of angle units of Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-91685668012497706522024-03-12T09:44:00.004+02:002024-03-14T19:40:29.743+02:00Number spirals introductionNumber spirals
preprint
Annotation
Number spirals are a representation of natural numbers on a spiral in units of measurement of angles. Following certain rules for placing numbers allows you to get an infinite spiral multiplication table for any natural number а>1. The appearance and structure of a particular a-spiral will be the same for any units of measurement of angles in any Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-50013213818157868052024-01-27T12:51:00.007+02:002024-02-06T15:36:34.625+02:00One to the power of zeroBefore, I had never paid attention to the strange equality - one to the power of zero is equal to one to the power of one. Any number to the power of zero is equal to one. Any number to the first power is equal to the same number. If the number is equal to one, then a strange equality results.
Оne to the power of zero
In the process of studying number theory, I got to a place where the generallyНиколай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-50861794425013609642023-12-28T13:35:00.003+02:002023-12-28T13:48:19.555+02:00Prime number XPrime 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. Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com2tag:blogger.com,1999:blog-3709052452161255816.post-4202434845977284762023-12-02T12:58:00.002+02:002023-12-02T12:59:22.563+02:00Why 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 Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-61112933938091026592023-10-28T08:27:00.001+03:002023-10-28T08:28:47.088+03:002 to the power of 0 to 1002 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, Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-68907335442825423262023-10-18T10:22:00.004+03:002023-11-02T09:27:17.805+02:00Negative numbers: numbers gameYou 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 Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-67536698957231643862023-10-17T15:16:00.004+03:002023-10-18T10:34:35.323+03:00Negative numbers: it's debtAuthor 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 Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-13175495767771017792023-10-17T11:43:00.002+03:002024-03-13T14:54:07.011+02:00Table of primesA 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 Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-41877502120976785842023-05-22T10:54:00.009+03:002024-04-02T10:56:42.043+03:00Negative numbers: subtractionMathematicians 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. Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-76294221263596170852023-04-14T11:28:00.003+03:002023-05-01T15:08:10.321+03:00Аverage percentageAverage 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 Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-51645274617945370332023-03-12T11:11:00.008+02:002023-03-12T11:12:58.618+02:00 Sine and cosine tableThis 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 Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-44538778548498508402023-03-11T11:29:00.007+02:002023-05-01T15:00:49.779+03:00Sine and cosine of 45 degreesWe 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 Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com1tag:blogger.com,1999:blog-3709052452161255816.post-10215164956214773212023-03-03T12:37:00.003+02:002023-05-01T15:00:13.663+03:00What 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.
Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-31734311073475504262023-02-21T23:44:00.003+02:002023-05-01T14:59:45.409+03:00Where 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". MyНиколай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-9783428471147249552023-02-12T19:43:00.007+02:002024-04-13T19:26:53.246+03:00Why 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
EquilateralНиколай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-86709243346844415772022-11-26T19:40:00.004+02:002023-08-25T22:46:29.636+03:00Sums classificationAnnotation
If you take any sum, then all other sums can be divided into three groups: identical sums, decomposition sums, and other sums. The classification criteria are the terms and the result of the addition.
Identical sums
Identical sums are combined into a separate group by the commutativity property. Sums from this group have pairwise identical terms and an equal number of terms. The Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-2163769157463417442022-11-22T22:55:00.005+02:002023-08-25T22:44:54.854+03:00Shift in arithmeticAnnotation
The use of a shift in arithmetic leads to the addition of terms with different units of measurement.
Consider the results of a shift in arithmetic addition. A shift is possible when performing addition in a column. For example, let's take a three-digit number and add to it the same number without a shift, with a shift by one position and with a shift by two positions. At the same timeНиколай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0tag:blogger.com,1999:blog-3709052452161255816.post-55315289881704093942022-11-12T13:20:00.004+02:002023-08-25T23:00:28.494+03:00Trigonometry value table
Trigonometry value table for children
In trigonometry value table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
Trigonometry value table
Why is this table of trigonometry values for kids (and mathematicians)? Firstly, the values of trigonometric functions are not given in numerical form, but in the form of mathematical expressions using square roots. Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com2tag:blogger.com,1999:blog-3709052452161255816.post-68905384492996489452022-08-13T18:44:00.006+03:002023-08-25T22:43:37.952+03:00Mathematical operations with infinite seriesAnnotation
When performing mathematical operations with infinite series, it is necessary to follow the rules of the positional number system.
Number of terms
The number of terms of an infinite series, presented in the visible part of the series, must be considered as an analogue of writing a number using numbers in the positional number system. The number of terms in the visible part of the Николай Хижняк, Nikolay Hizhniakhttp://www.blogger.com/profile/12714206257364077985noreply@blogger.com0