Even and odd numbers. The concept of decimal notation of a number. Odd number 3 even number or not

So, I'll start my story with even numbers. What are even numbers? Any integer that can be divided by two without a remainder is considered even. In addition, even numbers end with one of the given number: 0, 2, 4, 6 or 8.

For example: -24, 0, 6, 38 are all even numbers.

m = 2k - general formula writing even numbers, where k is an integer. This formula may be needed to solve many problems or equations in elementary grades.

There is one more kind of numbers in the vast realm of mathematics - these are odd numbers. Any number that cannot be divided by two without a remainder, and when divided by two, the remainder is equal to one, is called odd. Any of them ends with one of these numbers: 1, 3, 5, 7 or 9.

Example of odd numbers: 3, 1, 7 and 35.

n = 2k + 1 is a formula that can be used to write any odd numbers, where k is an integer.

Addition and subtraction of even and odd numbers

There is a pattern in adding (or subtracting) even and odd numbers. We have presented it with the help of the table below, in order to make it easier for you to understand and remember the material.

Operation	Result	Example
Even + Even
Even + Odd	odd
Odd + Odd

Even and odd numbers will behave the same way if you subtract rather than add them.

Multiplication of even and odd numbers

When multiplying, even and odd numbers behave naturally. You will know in advance whether the result will be even or odd. The table below shows all possible options for better understanding of information.

Operation	Result	Example
Even * Even
Even Odd
Odd * Odd	odd

Now let's look at fractional numbers.

Decimal number notation

Decimals are numbers with a denominator of 10, 100, 1000, and so on that are written without a denominator. The integer part is separated from the fractional part with a comma.

For example: 3.14; 5.1; 6.789 is everything

You can perform various mathematical operations with decimals, such as comparison, summation, subtraction, multiplication, and division.

If you want to compare two fractions, first equalize the number of decimal places by adding zeros to one of them, and then, discarding the comma, compare them as whole numbers. Let's look at this with an example. Let's compare 5.15 and 5.1. First, let's equalize the fractions: 5.15 and 5.10. Now we write them as integers: 515 and 510, therefore, the first number is greater than the second, so 5.15 is greater than 5.1.

If you want to sum two fractions, follow this simple rule: start at the end of the fraction and sum first (for example) hundredths, then tenths, then integers. With this rule, you can easily subtract and multiply decimals.

But you need to divide fractions as whole numbers, counting at the end where you need to put a comma. That is, first divide the whole part, and then the fractional part.

Also, decimal fractions should be rounded. To do this, select to what decimal place you want to round the fraction, and replace the corresponding number of digits with zeros. Keep in mind that if the digit following this digit was in the range from 5 to 9 inclusive, then the last digit that remains is increased by one. If the digit following this digit lay in the range from 1 to 4 inclusive, then the last remaining one does not change.

Odd number is an integer that not shared without a remainder: ..., -3, -1, 1, 3, 5, 7, 9, ...

If m is even, then it can be represented in the form $m = 2k$ , and if odd, then in the form $m = 2 k + 1$ , where $k \in \mathbb Z$ .

History and culture

The concept of parity of numbers has been known since ancient times and has often been given a mystical meaning. In Chinese cosmology and natural philosophy, even numbers correspond to the concept of "yin", and odd - "yang".

V different countries there are traditions associated with the number of flowers given. For example, in the USA, Europe and some eastern countries, it is believed that an even number of flowers given brings happiness. In Russia and the CIS countries, it is customary to bring an even number of flowers only to the funerals of the dead. However, in cases where there are many flowers in the bouquet (usually more), the evenness or oddness of their number no longer plays any role. For example, it is quite acceptable to give a lady a bouquet of 12, 14, 16, etc. flowers or sections of a spray flower that have many buds, in which they, in principle, are not counted. This applies even more to the larger number of flowers (cuts) given on other occasions.

Practice

In higher educational institutions with complex graphics educational process odd and even weeks apply. Within these weeks, the schedule of training sessions and, in some cases, their start and end times differ. This practice is used to evenly distribute the load across classrooms, academic buildings and for the rhythm of classes in disciplines with a small classroom load (1 time in 2 weeks)

In train schedules, even and odd train numbers are used, depending on the direction of movement (forward or reverse). Accordingly, even/odd indicates the direction in which the train passes through each station.

Even and odd days of the month are sometimes linked to train schedules, which are organized every other day.

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Notes

An excerpt characterizing even and odd numbers

“So, so,” said Prince Andrei, turning to Alpatych, “tell everything as I told you.” And, without answering a word to Berg, who fell silent beside him, he touched the horse and rode into the alley.

The troops continued to retreat from Smolensk. The enemy was following them. On August 10, the regiment commanded by Prince Andrei passed through high road, past the avenue leading to the Bald Mountains. The heat and drought lasted for more than three weeks. Curly clouds moved across the sky every day, occasionally obscuring the sun; but towards evening it cleared again, and the sun set in a brownish-red mist. Only heavy dew at night refreshed the earth. The bread remaining on the root burned and spilled out. The swamps have dried up. The cattle roared from hunger, not finding food in the meadows burned by the sun. Only at night and in the forests the dew still held, it was cool. But along the road, along the high road along which the troops marched, even at night, even through the forests, there was no such coolness. The dew was not noticeable on the sandy dust of the road, which was pushed up more than a quarter of an arshin. As soon as it dawned, the movement began. Convoys, artillery silently walked along the hub, and the infantry up to their ankles in soft, stuffy, hot dust that had not cooled down during the night. One part of this sandy dust was kneaded by feet and wheels, the other rose and stood like a cloud over the army, sticking to the eyes, hair, ears, nostrils and, most importantly, the lungs of people and animals moving along this road. The higher the sun rose, the higher the cloud of dust rose, and through this thin, hot dust it was possible to look at the sun, not covered by clouds, with a simple eye. The sun was a big crimson ball. There was no wind, and people were suffocating in this still atmosphere. People walked with handkerchiefs around their noses and mouths. Coming to the village, everything rushed to the wells. They fought for water and drank it to the dirt.
Prince Andrei commanded the regiment, and the structure of the regiment, the well-being of its people, the need to receive and give orders occupied him. The fire of Smolensk and its abandonment were an epoch for Prince Andrei. A new feeling of bitterness against the enemy made him forget his grief. He was completely devoted to the affairs of his regiment, he was caring for his people and officers and affectionate with them. In the regiment they called him our prince, they were proud of him and loved him. But he was kind and meek only with his regimental officers, with Timokhin, etc., with completely new people and in a foreign environment, with people who could not know and understand his past; but as soon as he ran into one of his former staff members, he immediately bristled again; became malicious, mocking and contemptuous. Everything that connected his memory with the past repulsed him, and therefore he tried in the relations of this former world only not to be unjust and to fulfill his duty.
True, everything was presented in a dark, gloomy light to Prince Andrei - especially after they left Smolensk (which, according to his concepts, could and should have been defended) on August 6, and after his father, who was sick, had to flee to Moscow and throw away the Bald Mountains, so beloved, built up and inhabited by him, for plunder; but, despite the fact, thanks to the regiment, Prince Andrei could think of something else, completely independent of general issues subject - about his regiment. On August 10, the column, in which his regiment was, caught up with the Bald Mountains. Prince Andrey two days ago received the news that his father, son and sister had left for Moscow. Although Prince Andrei had nothing to do in the Bald Mountains, he, with his characteristic desire to inflame his grief, decided that he should call in the Bald Mountains.
He ordered his horse to be saddled and from the crossing rode on horseback to his father's village, in which he was born and spent his childhood. Passing by a pond, on which dozens of women, talking to each other, beat with rollers and rinsed their clothes, Prince Andrei noticed that there was no one on the pond, and a torn-off raft, half flooded with water, floated sideways in the middle of the pond. Prince Andrei drove up to the gatehouse. There was no one at the stone entrance gate, and the door was unlocked. The garden paths were already overgrown, and the calves and horses were walking through the English park. Prince Andrei drove up to the greenhouse; the windows were broken, and the trees in tubs, some felled, some withered. He called Taras the gardener. Nobody responded. Going around the greenhouse to the exhibition, he saw that the carved board fence was all broken and the plum fruits were plucked with branches. An old peasant (Prince Andrei had seen him at the gate in his childhood) was sitting and weaving bast shoes on a green bench.
He was deaf and did not hear the entrance of Prince Andrei. He was sitting on a bench, on which the old prince liked to sit, and beside him was hung a bast on the knots of a broken and withered magnolia.
Prince Andrei drove up to the house. Several lindens in the old garden were cut down, one piebald horse with a foal walked in front of the house between the roses. The house was boarded up with shutters. One window downstairs was open. The yard boy, seeing Prince Andrei, ran into the house.
Alpatych, having sent his family, remained alone in the Bald Mountains; he sat at home and read the Lives. Upon learning of the arrival of Prince Andrei, he, with glasses on his nose, buttoning up, left the house, hurriedly approached the prince and, without saying anything, wept, kissing Prince Andrei on the knee.

There are pairs of opposites in the universe, which are an important factor in its structure. The main properties that numerologists attribute to even (1, 3, 5, 7, 9) and odd (2, 4, 6, 8) numbers, as pairs of opposites, are as follows:

1 - active, purposeful, imperious, callous, leading, initiative;
2 - passive, receptive, weak, sympathetic, subordinate;
3 - bright, cheerful, artistic, lucky, easily achieving success;
4 - hardworking, boring, lack of initiative, unhappy, hard work and frequent defeat;
5 - agile, adventurous, nervous, insecure, sexy;
6 - simple, calm, homely, arranged; mother's love;
7 - departure from the world, mysticism, secrets;
8 - worldly life; material luck or defeat;
9 - intellectual and spiritual perfection.

Odd numbers have much brighter properties. Next to the energy of "1", the brilliance and luck of "3", the adventurous mobility and versatility of "5", the wisdom of "7" and the perfection of "9", even numbers do not look so bright. There are 10 main pairs of opposites that exist in the universe. Among these pairs: even - odd, one - many, right - left, male - female, good - evil. One, right, masculine and good was associated with odd numbers; many, left, feminine and evil - with even.

Odd numbers have a certain generating middle, while in any even number there is a receptive hole, as it were, a gap within itself. The masculine properties of phallic odd numbers stem from the fact that they are stronger than even numbers. If an even number is split in half, then, apart from emptiness, nothing will remain in the middle. An odd number is not easy to split because there is a dot in the middle. If you add together an even and an odd number, then the odd one wins, since the result will always be odd. That is why odd numbers have masculine properties, imperious and sharp, and even numbers - feminine, passive and perceiving.

Odd numbers odd number: there are five of them. Even numbers an even number - four.

Odd numbers are solar, electric, acidic and dynamic. They are terms; stack them with something. Even numbers are lunar, magnetic, alkaline and static. They are deductible, they are reduced. They remain motionless because they have even groups of pairs (2 and 4; 6 and 8).

If we group odd numbers, one number will always be left without its pair (1 and 3; 5 and 7; 9). This makes them dynamic. Two similar numbers (two odd numbers or two even numbers) are not auspicious.

even + even = even (static) 2+2=4
even + odd = odd (dynamic) 3+2=5
odd + odd = even (static) 3+3=6

Some numbers are friendly, others are opposed to each other. The relationship of numbers is determined by the relationship between the planets that rule them (details in the "Number Compatibility" section). When two friendly numbers touch, their cooperation is not very productive. Like friends, they relax - and nothing happens. But when hostile numbers are in the same combination, they make each other on their guard and encourage active action; thus, these two people work a lot more. In this case, hostile numbers turn out to be really friends, and friends are real enemies, hindering progress. Neutral numbers remain inactive. They do not give support, do not cause or suppress activity.

As we saw above, any permutation decomposes into a product of transpositions. Generally speaking, one and the same permutation can be represented as a product of transpositions by many different ways. For example, it is obvious that

(formulas (1) and (2) express, as is easy to see, the same fact, but in different notation).

Lemma. If the product of several transpositions is equal to the identity permutation, then the number of these transpositions is even.

We will prove this lemma by induction on the number s of different numbers in the records of these transpositions.

The smallest possible value of s is obviously two. If , then the product under consideration is a degree of some transposition and, therefore, is equal to the identity permutation only if the exponent is even (because any transposition has order 2). Thus, in the case the lemma is proved.

Assuming now that the lemma has already been proved for any product of transpositions whose records contain less than s different numbers, consider some product of transpositions equal to the identity permutation

whose records contain exactly s different numbers. Let I be one of these numbers. Using relation (1) and the fact that independent transpositions are permutable, we can “move forward” all transpositions that include the number i, i.e., go from product (3) to an equal product of the form

in which all numbers are different from the number l. If , then, using relation (2) or the relation

we can pass from the product (4) to a product of the same form, but with a smaller . As a result of a series of such transformations, we either completely destroy all transpositions that contain the number l, or we get a product containing only one such transposition:

But this product obviously translates a number into a number l and therefore cannot be an identical permutation. Therefore, the latter case is impossible. Thus, as a result of our transformations, we obtain a product of transpositions, equal to the identity substitution, whose entries do not contain the number l. Obviously, these substitutions do not contain any new numbers. Therefore, by the inductive hypothesis, this product contains an even number of transpositions.

It remains to note that under the described transformations, the number of transpositions either does not change (when we use relations (1), (2)), or decreases by two units (when we use the relation . Therefore, the original product (3) also consists of an even number of transpositions. Thus the lemma is completely proved.

Let now some permutation a be decomposed into a product of transpositions in two ways:

(the first decomposition contains transpositions, and the second q). Then

and, therefore, by the lemma just proved, the number is even.

Thus, the numbers and q are either both even or both odd. In other words, for all expansions of a permutation into a product of transpositions, the parity of the number of these transpositions will be the same.

A permutation is called even if it decomposes into a product of an even number of transpositions, and odd otherwise. According to the proved theorem, the parity of a permutation does not depend on the choice of its decomposition into a product of transpositions.

Any transposition, or in general any cycle of even length, is an odd permutation, and any cycle of odd length, in particular any cycle of length 3, is an even permutation. The identity permutation is obviously even.

Decomposition of the permutation a into a product of transpositions, then

whence it follows that the inverse of an even permutation is even, and the inverse of an odd one is odd.

An integer is said to be even if it is divisible by 2; otherwise it is called odd. So the even numbers are

and odd numbers

From the divisibility of even numbers by two, it follows that every even number can be written as , where the symbol denotes an arbitrary integer. When a symbol (like a letter in our case) can represent any element of some particular set of objects (the set of integers in our case), we say that the range of this symbol is the specified set of objects. In accordance with this, in the case under consideration, we say that every even number can be written in the form , where the range of the symbol coincides with the set of integers. For example, the even numbers 18, 34, 12, and -62 have the form , where they are 9, 17, 6, and -31, respectively. There is no particular reason to use the letter here. Instead of saying that even numbers are integers of the form one could equally say that even numbers are of the form or or

When two even numbers are added together, the result is also an even number. This circumstance is illustrated by the following examples:

However, a set of examples is not enough to prove the general assertion that the set of even numbers is closed under addition. To give such a proof, let's denote one even number by , and the other by . Adding these numbers, we can write

The sum is written as . This shows that it is divisible by 2. It would not be enough to write

since the last expression is the sum of an even number and the same number. In other words, we would prove that twice an even number is again an even number (in fact divisible even by 4), while we need to prove that the sum of any two even numbers is an even number. Therefore, we have used the notation for one even number and for another even number in order to indicate that these numbers can be different.

What notation can be used to write any odd number? Note that subtracting 1 from an odd number results in an even number. Therefore, it can be argued that any odd number is written in the form A record of this kind is not unique. Similarly, we might notice that adding 1 to an odd number results in an even number, and we might conclude from this that any odd number can be written as

Similarly, we can say that any odd number is written as or or, etc.

Can it be argued that every odd number is written as Substituting into this formula instead of integers

we get the following set of numbers:

Each of these numbers is odd, but they do not exhaust all odd numbers. For example, the odd number 5 cannot be written this way. Thus, it is not true that every odd number has the form , although every integer of the form is odd. Similarly, it is not true that every even number is written as where the range of the symbol k is the set of all integers. For example, 6 is not equal to whichever integer you take as A. However, every integer of the form is even.

The relationship between these statements is the same as between the statements "all cats are animals" and "all animals are cats". It is clear that the first of them is true, but the second is not. This relationship will be discussed further in the analysis of statements that include the phrases "then", "only then" and "then and only then" (see § 3 ch. II).