Is 2 an even number or not? Even - odd numbers

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

and odd numbers -

From the divisibility of even numbers by two it follows that every even number can be written in the form , where the symbol denotes an arbitrary integer. When some symbol (like a letter in our case) can represent any element of some specified set of objects (the set of integers in our case), we say that the range of this symbol is the specified set of objects. Accordingly, 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 are of the form , where respectively equals 9, 17, 6 and -31. There is no particular reason to use the letter . Instead of saying that even numbers are integers of the form equals, one could say that even numbers are of the form or or

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

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

The amount is written in the form . From this we can see 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, even divisible by 4), while we need to prove that the sum of any two even numbers is an even number. Therefore, we 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 produces an even number, and we might conclude from this that any odd number is written as

Similarly, we can say that any odd number is written in the form or or etc.

Is it possible to say that every odd number is written in the form Substituting integers into this formula instead

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 is of the form , although every integer of the form is odd. Likewise, it is not true that every even number is written in the form where the range of the symbol k is the set of all integers. For example, 6 is not equal to any integer we 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 involving the phrases “then”, “only then” and “then and only then” (see § 3 of Chapter II).

Exercises

Which of the following statements are true and which are false? (The range of characters is assumed to be the set of all integers.)

1. Every odd number can be represented as

2. Every integer of type a) (see exercise 1) is odd; the same holds for numbers of the form b), c), d), e) and f).

3. Every even number can be represented as

4. Every integer of type a) (see Exercise 3) is even; the same applies to numbers of the form b), c), d) and e).

which doesn't share without remainder: …, −3, −1, 1, 3, 5, 7, 9, …

If m is even, then it can be represented in the form m = 2 k (\displaystyle m=2k), and if odd, then in the form m = 2 k + 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 was often given mystical meaning. In Chinese cosmology and natural philosophy, even numbers correspond to the concept of “yin”, and odd numbers correspond to “yang”.

In different countries there are traditions related to 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 funeral 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 bush flower that have many buds, in which they, in principle, cannot be counted. This is especially true for the larger number of flowers (cuts) given on other occasions.

Practice

According to the Traffic Rules, depending on whether the day of the month is even or odd, parking under signs 3.29, 3.30 may be permitted.
In higher education institutions with complex schedules of the educational process, even and odd weeks are used. Within these weeks, the schedule of training sessions and, in some cases, their start and end times differ. This practice is used to distribute the load evenly across classrooms, academic buildings and to ensure the rhythm of classes in disciplines with a load of 1 time every 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 travel (forward or reverse). For example, the Rossiya train when traveling from Vladivostok to Moscow has the number 001, and from Moscow to Vladivostok - 002;
- Even/odd in railway slang denotes the direction in which a train passes through a station (an example of an announcement: “An odd train will pass along the third track”);
- The even and odd days of the month are linked to the schedules of passenger trains traveling every other day. If two consecutive odd numbers coincide, in order to distribute cars evenly between terminal stations, trains can be scheduled with a deviation from the schedule (in this case, the next train does not go every other day, but two days later or the next day);
- Seats in reserved seat and compartment carriages are always distributed: even - upper, odd - lower.

Even numbers symbolize the material world and systematic work, says numerology.

Odd numbers indicate spiritual quests and attempts at creative transformation of the material world.

Even numbers show that a person will try to solve his problems within himself, in his own family, among his environment, in a familiar and familiar environment; it is always the consolidation of the new, the transformation of the new into the familiar through material and physical effort.

Odd numbers indicate solving problems primarily in the outside world and with its help. They talk about the conflict between the individual and the world. A person resolves it by expanding consciousness, mastering the world of things and feelings and learning the laws of nature. This is learning new things through spiritual efforts.

Even numbers are associated with the resolution of human conflicts:

2 - internal at the level of emotions;

4 - in the family and in small groups;

6 and 8 - between large groups of people, nations, cultures. These are conflicts related to the management of society and information flows.

Odd numbers mean a person’s conflict with the world at the level of: 1 - desires and possibilities;

3 - discovering the world and choosing your place in it;

5 - conquest of the world;

7 - knowledge of the world and the laws of creativity; 9 - understanding the meaning of life.

As the number increases, both conflicts increasingly transform from personal to public, subordinating to social tasks. Numbers determine the evolution of conflicts. All numbers give rise to aggression, but the larger the number, the more intelligent it is. Even numbers contain internal aggression, which is often realized internally.

An odd number tries to open a person to the world, and an even number, on the contrary, tries to hide him from the world. And the meaning of any numerical conflict is to eliminate it through physical or spiritual efforts.

The numbers from 1 to 9 are basic and form all the others, for example: 10 = 1 +0 = 1, which means the first step. Multiple-valued 13 = 7 + 6 - death in an unequal struggle;

13 = 8 + 5 - suicide;

13 = 9+4 - premature death from unsuitable living conditions;

13 = 10+3 - death in childbirth;

13 = 11 + 2 - death from the consciousness of the tragedy of the dual position;

13 = 12+1 - the adept’s transition to another plane as a consequence of the completion of his task on Earth.

In numerology, it emphasizes temptations (from the Prince of Darkness), the karma of fear and laziness.

14 is a number made up of two sevens; ancient Kabbalists considered it lucky and denoted the number of transformations and metamorphoses. A symbol of moderation (if violated, the karma of immoderation is formed).

15 - the number of spiritual ascensions; the fifteenth day of the seventh month was respected and sanctified. It is mysteriously connected with the problems of good and evil, and can imperceptibly make a person a slave of pentagrams (5). For Kabbalists it represented the Genius of Evil.

16 - Pythagoreans was revered as lucky, since it was a perfect quadrangle. Warns of possible pride (if violated, it forms the karma of pride and the inability to resolve love issues).

17 is the number of the Mother of God, patroness of Christians.

18 - due to lack of spirituality - the number of potions and fate, superstition and mistakes, unlucky.

19 - in Kabbalah is considered a favorable number, since it consists of two lucky numbers: 1 and 9, which, when added, give 10 - a perfect number, the number of the law. It is also the number of the sun, gold and the philosopher's stone. Warns against obsessing over one’s minor problems (in case of violations, it forms the karma of fixation).

20 - the number of truth, faith, health. But theologians consider him unhappy, especially in a partnership: this is either a qualitative leap to the highest level of relationships, or a rapid decline. (Don't try to rub others in the face!)

21 - Crown of Magic, connection with the Higher Mind. The number of divinations consisting of three sevens or seven threes. Both combinations have very strong magical properties and provide help from Higher powers to the person asking.

22 - Dominant (Main), the number of the Higher Mind. This number has enough strength to implement major plans. Channeling spiritual and physical forces in the right direction requires wisdom, intelligence and patience, otherwise much can be wasted in boasting, covering up an inferiority complex.

28 is the number of God, the Creator of the Universe. The number of days in the lunar month therefore foretells the favor of the Moon.

30 - The number 30 is remarkable for many mysteries. A mind that knows no limits and barriers. Warns of the possible receipt of a large sum and its possible loss (in case of obvious greed).

31 - the number emphasizes virtue or indicates the root of evil (spiritual corruption).

32 - among the Pythagoreans - the number of justice, since it can be consistently divided into equal parts, without giving any preference. Jewish scholars attributed to him wisdom, loyalty, and mastery of the magic of spells.

33 - The dominant (Main) number in numerology. This combination of numbers gives more effectiveness to the six contained in them and expresses insight, insight, conscious service to people, dedication, trust, which, however, should not turn into self-denial and martyrdom, bordering on irresponsibility.

40 is the number of absolute completeness. According to St. Augustine, it reflects our journey to truth, our path to heaven. We celebrate 40 days after the death of loved ones. It rained for forty days and nights during the flood, Jesus spent 40 days in the desert... The number 40 symbolizes health. Maybe this is where people’s belief comes from that for normal intrauterine development of a child, you need to carry it for 7 x 40 = 280 days - ten (full? and layer) lunar months. The word quarantine literally means a forty-day period. We can also recall the Russian expression forty forty, and many others. In the negative, it indicates unlimited power (despot) in a country or family.

50 means liberation from slavery and complete freedom.

60 - like 3,7,12, has been considered a sacred number since ancient times. The Chaldean magicians, who knew how to perform complex astronomical calculations, used the sexagesimal system along with the decimal system. Fragments of this knowledge have reached us: the circle is divided into 60 degrees, each degree has 60 minutes of 60 seconds each, an hour lasts 60 minutes, etc.

72 - has great similarities with 12.

100 - expresses complete perfection.

1000 (cube of ten) - reflects absolute perfection.

According to many Kabbalists, prime numbers represent divine things, tens represent heavenly things, thousands represent the essence of future ages.

The dominant numbers in numerology are 11,22 and 33.

Let's refresh our memory of the concepts of Universal and Personal years. We will need them in the next topic (see Excursion topic).

The number of the Universal Year (YY) determines the qualities of events and phenomena in the world and is needed to find the number of the Personal Year. Such vibrations affect people, places and other objects. The universal year is determined by adding the digits of any year in question and then converting them to a single digit number (except for the Master Numbers).

The vibrations of the Personal Year (PG) directly affect a person. We all have our own personal vibrations. In the same Universal Year, a person with a certain Personal Number receives vibrations that are different from those received by a person with a different Personal Number. Many people have the same Personal Numbers vibrating for them at the same time, but everyone may use or interpret them differently. The Personal Year is found by the sum of the day, month of birth and the number of the Universal Year.

Parity

If a number is written in decimal form last digit is an even number (0, 2, 4, 6 or 8), then the entire number is also even, otherwise it is odd.
42 , 104 , 11110 , 9115817342 - even numbers.
31 , 703 , 78527 , 2356895125 - odd numbers.

Arithmetic

Addition and subtraction:
- H yotnoe ± H yotnoe = H good
- H yotnoe ± N even = N even
- N even ± H yotnoe = N even
- N even ± N even = H good
Multiplication:
- H× H yotnoe = H good
- H× N even = H good
- N even × N even = N even
Division:
- H yotnoe / H even - it is impossible to clearly judge the parity of the result (if the result is an integer, then it can be either even or odd)
- H yotnoe / N even = if the result is an integer, then it is H good
- N even / H even - the result cannot be an integer, and therefore have parity attributes
- N even / N even = if the result is an integer, then it is N even

History and culture

The concept of parity of numbers has been known since ancient times and was often given mystical meaning. So, in ancient Chinese mythology, odd numbers corresponded to Yin, and even numbers corresponded to Yang.

In 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, it is customary to bring an even number of flowers only to funerals of the dead; in cases where there are many flowers in the bouquet, the evenness or oddness of their number no longer plays such a role.

Notes

Wikimedia Foundation. 2010.

See what “Even numbers” are in other dictionaries:

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So, I'll start my story with even numbers. Which numbers are even? 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 a 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 another type of numbers in the vast kingdom of mathematics - odd numbers. Any number that cannot be divided by two without a remainder, and when divided by two the remainder is one, is usually called odd. Any of them ends with one of the following 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 down any odd numbers, where k is an integer.

Adding and subtracting even and odd numbers

There is a certain pattern in the addition (or subtraction) of even and odd numbers. We have presented it using the table below 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 them rather than add them.

Multiplying 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 presents 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 notation of a number

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

For example: 3.14; 5.1; 6,789 is all

You can do a variety of mathematical operations with decimals, such as comparison, addition, 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, dropping the decimal point, 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 let's write them as integers: 515 and 510, therefore, the first number is greater than the second, which means 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 (for example) the hundredths first, then the tenths, then the whole ones. This rule makes it easy to subtract and multiply decimals.

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

Decimal fractions should also be rounded. To do this, select to what digit 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 was in the range from 1 to 4 inclusive, then the last remaining digit is not changed.