3 is an even number or not. Even and odd numbers
There are pairs of opposites in the universe, which are an important factor in its structure. The main properties that numerologists attribute to odd (1, 3, 5, 7, 9) and even (2, 4, 6, 8) numbers, as pairs of opposites, are as follows:
Odd numbers have much brighter properties. Next to energy "1", brilliance and luck "3", adventurous mobility and versatility "5", wisdom "7" and perfection "9" even numbers they don't look as 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 perceiving 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- solar, electric, acidic and dynamic. They are terms; stack them with something. Even numbers- 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 oppose each other. The relationship of numbers is determined by the relationship between the planets that rule them. 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.
Definitions
- Even number is an integer that is divided without a remainder: …, −4, −2, 0 , 2, 4, 6, 8, …
- Odd number is an integer that not shared without a remainder: ..., -3, -1, 1, 3, 5, 7, 9, ...
If a m is even, then it can be represented in the form m = 2k (\displaystyle m=2k), and if odd, then in the form m = 2k + 1 (\displaystyle m=2k+1), where k ∈ Z (\displaystyle 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".
AT 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
- According to the Rules of the Road, depending on the even or odd number of the month, parking under the signs 3.29, 3.30 may be allowed.
- 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, educational buildings and for the rhythm of classes in disciplines with a load of 1 time in 2 weeks.
- Even / odd numbers are widely used in railway transport:
- When a train moves, it is assigned a route number, which can be even or odd, depending on the direction of movement (forward or reverse). For example the train
As we saw above, any permutation decomposes into a product of transpositions. Generally speaking, the same permutation can be represented as a product of transpositions in 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.
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 is the general formula for 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 assigning 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 add two fractions, follow this simple rule: start at the end of the fraction and add 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.
Definitions
- Even number is an integer that is divided no remainder by 2: …, −4, −2, 0, 2, 4, 6, 8, …
- Odd number is an integer that not shared no remainder by 2: …, −3, −1, 1, 3, 5, 7, 9, …
According to this definition, zero is an even number.
If a m is even, then it can be represented as , and if odd, then as , where .
In different countries, there are traditions associated with the number of flowers given.
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 young lady a bouquet of 12 or 14 flowers or sections of a spray flower if they have many buds, in which they, in principle, are not counted.
This is especially true for the larger number of flowers (cuts) given on other occasions.
Notes
Wikimedia Foundation. 2010 .
- Maardu
- Superconductivity
See what "Even and Odd Numbers" is in other dictionaries:
Odd numbers
Even numbers- Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19). ... ... Wikipedia
odd- Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19). ... ... Wikipedia
Odd number- Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19). ... ... Wikipedia
Odd numbers- Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19). ... ... Wikipedia
Even and odd numbers- Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19). ... ... Wikipedia
Even numbers- Parity in number theory is a characteristic of an integer, which determines its ability to be divided by two. If an integer is divisible by two without a remainder, it is called even (examples: 2, 28, −8, 40), if not odd (examples: 1, 3, 75, −19). ... ... Wikipedia
Slightly redundant numbers- A slightly redundant number, or a quasi-perfect number is an excess number whose sum of its own divisors is one more than the number itself. So far, no slightly redundant numbers have been found. But since the time of Pythagoras, ... ... Wikipedia
Perfect Numbers are positive integers, equal to the sum all its correct (i.e., less than this number) divisors. For example, the numbers 6 = 1+2+3 and 28 = 1+2+4+7+14 are perfect. Even Euclid (3rd century BC) indicated that even S. hours can be ... ...
quantum numbers- integer (0, 1, 2,...) or half-integer (1/2, 3/2, 5/2,...) numbers that determine the possible discrete values of physical quantities that characterize quantum systems (atomic nucleus, atom , molecule) and individual elementary particles. ... ... Great Soviet Encyclopedia
Books
- Mathematical labyrinths and puzzles, 20 cards, Barchan Tatyana Aleksandrovna, Samodelko Anna. In the set: 10 puzzles and 10 mathematical labyrinths on the topics: - Numerical series; - Even and odd numbers; - Composition of the number; - Account in pairs; - Exercises for addition and subtraction. Includes 20…