Even and odd numbers. The concept of decimal notation of a number

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

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See what "Even and Odd Numbers" is in other dictionaries:

Parity in number theory is a characteristic of an integer that 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

A slightly redundant number, or a quasi-perfect number, is a redundant 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

Integer positive numbers equal to the sum of all their 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 ... ...

Integer (0, 1, 2,...) or half-integer (1/2, 3/2, 5/2,...) numbers that define 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; - Counting in pairs; - Exercises for addition and subtraction. Includes 20…

What do even and odd numbers mean in spiritual numerology. This is a very important topic in the study! What is the difference between even numbers and odd numbers?

Even numbers

It is well known that even numbers are those that are divisible by two. That is, the numbers 2, 4, 6, 8, 10, 12, 14, 16, 18 and so on.

What do even numbers mean relative to ? What is the numerological essence of dividing by two? And the bottom line is that all numbers that are divisible by two carry some of the properties of two.

Have multiple meanings. Firstly, this is the most "human" figure in numerology. That is, the number 2 reflects the whole gamut of human weaknesses, shortcomings and virtues - more precisely, what society considers to be virtues and shortcomings, "correctness" and "incorrectness".

And since these labels of “correctness” and “incorrectness” reflect our limited views of the world, then the deuce can be considered the most limited, most “stupid” number in numerology. From this it is clear that even numbers are much more “hardheaded” and straightforward than their odd counterparts, which are not divisible by two.

This, however, does not mean that even numbers are worse than odd numbers. They are just different and reflect other forms of human existence and consciousness in comparison with odd numbers. Even numbers in spiritual numerology always obey the laws of ordinary, material, "earthly" logic. Why?

Because another meaning of the deuce: standard logical thinking. And all even numbers in spiritual numerology, one way or another, obey certain logical rules for the perception of reality.

An elementary example: if a stone is thrown up, it, having gained a certain height, will then rush to the ground. This is how even numbers "think". And odd numbers will easily assume that the stone will fly into space; or not fly, but get stuck somewhere in the air ... for a long time, for centuries. Or just dissolve! The more illogical the hypothesis, the closer it is to odd numbers.

Odd numbers

Odd numbers are those that are not divisible by two: the numbers 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, and so on. From the standpoint of spiritual numerology, odd numbers are subject not to material, but to spiritual logic.

Which, by the way, gives food for thought: why the number of flowers in a bouquet for a living person is odd, and for a dead person it is even... Is it because material logic (logic within the framework of “yes-no”) is dead relative to the human soul?

Visible coincidences of material logic and spiritual occur very often. But don't let that fool you. The logic of the spirit, that is, the logic of odd numbers, can never be fully traced on the external, physical levels of human existence and consciousness.

Let's take the love number as an example. We talk about love at every turn. We confess it, dream about it, decorate our lives and other people's lives with it.

But what do we really know about love? About that all-penetrating Love that permeates all spheres of the Universe. Can we agree and accept that there is as much cold in it as warmth, as much hatred as kindness?! Are we able to realize that it is these paradoxes that make up the highest, creative essence of Love?!

Paradoxicality is one of the key properties of odd numbers. AT interpretation of odd numbers It must be understood that what seems to a person is not always really existing. But at the same time, if something seems to someone, then it already exists. There are different levels of Existence, and illusion is one of them...

By the way, the maturity of the mind is characterized by the ability to perceive paradoxes. Therefore, it takes a little more "brains" to explain odd numbers than to explain even numbers.

Even and odd numbers in numerology

Let's summarize. What is the main difference between even and odd numbers?

Even numbers are more predictable (except for the number 10), solid and consistent. Events and people associated with even numbers are more stable and explainable. Quite accessible for external changes, but only for external ones! Internal change is the realm of odd numbers...

Odd numbers are eccentric, freedom-loving, unstable, unpredictable. They always bring surprises. It seems that you know the meaning of some odd number, and it, this number, suddenly begins to behave in such a way that it makes you reconsider almost your entire life ...

Note!

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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 assimilation 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 decimal fractions.

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.

Answers to p. 66

212. What number will turn out: even or odd, if an odd number is divided by an odd number, provided that the division is complete? Give three examples to support your hypothesis.

When dividing an odd number by an odd number, the result will always be an odd number.
45 : 5 = 9 55 : 11 = 5 63 : 7 = 9

213. What number will turn out: even or odd, if an even number is divided by an odd number, provided that the division is complete? Give some examples to support your hypothesis. Discuss the result with a classmate.

Dividing an even number by an odd number will always result in an even number.
54 : 9 = 6 50 : 5 = 10 96 : 3 = 32

214. Can you give an example of such a case of division, when an odd number is completely divisible by an even number? Why? Remember how you can get the dividend from the divisor and the value of the quotient.

The dividend can be obtained by multiplying the divisor by the value of the quotient. By convention, the divisor is an even number. We know that if an even number is multiplied by an even or an odd number, the result will always be an even number. In our case, the dividend must be an odd number. This means that no value of the quotient can be chosen in this case, and it is impossible to give an example of such a case of division.

215. Imagine the number 2873 as the sum of round tens and a single digit. Is each of the terms an even or odd number? Is the value of their sum an even or odd number? What digit can an even number end with? What about odd?

2873 = 2870 + 3
The first term is an even number, the second term is an odd number.
2873 is an odd number.
The odd number 2873 ends with an odd number 3, the even number 2870 ends with an even number 0.
An even number can end with even numbers (0, 2, 4, 6, 8), and an odd number can end with odd numbers (1, 3, 5, 7, 9).

216. Write the even numbers in one column and the odd numbers in the other.

2844 57893
67586 9231
10050 9929

217. How many even two-digit natural numbers are there? How many such odd numbers?

The smallest two-digit even number is 10, and the largest is an odd number 99. There are 99 in total - 10 + 1 = 90. Even and odd numbers in the natural series alternate, therefore there are as many even two-digit numbers as odd ones, that is, 45, since 90 : 2 = 45.

218. Write down the largest even six-digit number.