2 is 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 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 relation 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).

Exercises

Which of the following statements are true and which are false? (It is assumed that the range of characters is the collection of all integers.)

1. Every odd number can be represented as

2. Every integer of the form 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 the form a) (see exercise 3) is even; the same holds for numbers of the form b), c), d) and e).

Which 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 (\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".

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 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 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 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 "Russia" when traveling from Vladivostok to Moscow has the number 001, and from Moscow to Vladivostok - 002;
- Odd/even is railroad slang for the direction in which a train passes through a station (example of an announcement “An odd train will pass on the third track”);
- The schedules of passenger trains moving every other day are linked to even and odd days of the month. If two odd numbers in a row coincide, for an even distribution of wagons between the end stations, trains can be assigned with a deviation from the schedule (in this case, the next train goes not in a day, but in two days or the next day);
- Seats in reserved seat and compartment cars are always allocated: even - top, odd - bottom.

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

Odd ones indicate spiritual quests and attempts to creatively transform 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 the solution of problems primarily in the outside world and with its help. They talk about the conflict of the individual with the world. Man resolves it by expanding his consciousness, mastering the world of things and feelings, and cognizing the laws of nature. This is the knowledge of the new 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, peoples, cultures. These are conflicts related to the management of society and the flow of information.

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

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 - comprehension of the meaning of life.

These and other conflicts, with the increase in the value of the number, are increasingly turning from personal into public, submitting to social tasks. Numbers determine the evolution of conflicts. All numbers generate aggression, but the larger the number, the more reasonable it is. Even numbers contain internal aggression, which is often realized inside.

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 lies in its elimination through physical or spiritual efforts.

Numbers from 1 to 9 are basic and form all others, for example: 10 = 1 + 0 = 1, which means the first step. Multiple 13 \u003d 7 + 6 - death in an unequal struggle;

13 = 8 + 5 - suicide;

13 \u003d 9 + 4 - premature death from unsuitable living conditions;

13 \u003d 10 + 3 - death in childbirth;

13 \u003d 11 + 2 - death from the consciousness of the tragedy of a dual position;

13 = 12+1 - the transition of the adept to another plane as a result 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 - this number, made up of two sevens, was considered lucky by the ancient Kabbalists and denoted the number of transformations, metamorphoses. A symbol of moderation (in case of violation, the karma of immoderation is formed).

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

16 - was revered by the Ythagoreans as lucky, as 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 - the number of the Mother of God, the patroness of Christians.

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

19 - in Kabbalah it is considered an auspicious number, since it consists of two lucky numbers: 1 and 9, which, when added together, 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 obsession with one's petty problems (in case of violations, it forms the karma of obsession).

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

21 - Crown of magic, connection with the Higher mind. The number of divinations, consisting of three sevens or seven triples. Both combinations have very strong magical properties, provide the help of the Higher powers to the asking person.

22 - Dominant (Main), the number of the Higher mind. This number has enough strength to realize big ideas. Wisdom, intelligence and patience are required to direct spiritual and physical forces in the right direction, 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 of the lunar month, therefore, portends the favor of the moon.

30 - The number 30 is wonderful in many mysteries. The mind that knows no limits and barriers. Warns of the possible receipt of a large amount and its possible loss (with obvious self-interest).

31 - the number emphasizes virtue or points to 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, fidelity, mastery of the magic of spells.

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

40 is the number of absolute completeness. According to Saint Augustine, it reflects our journey to the truth, our journey to heaven. We celebrate 40 days after the death of loved ones. For forty days and nights it rained during the flood, Jesus spent 40 days in the wilderness... The number 40 symbolizes health. Maybe this is where people's belief comes from, that for the normal intrauterine development of a child, you need to wear it for 7 x 40 = 280 days - ten (full? and layers) 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 the country or family.

50 - means liberation from slavery and complete freedom.

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

72 - has a great resemblance to 12.

100 - expresses complete perfection.

1000 (cube ten) - reflects absolute perfection.

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

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

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

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

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 have the same Personal Numbers vibrating for them at the same time, but each 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 sign

If in decimal notation of a number last digit is an even number (0, 2, 4, 6 or 8), then the whole 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 exact ± H ethnoe = H ethnoe
- H exact ± H even = H even
- H even ± H ethnoe = H even
- H even ± H even = H ethnoe
Multiplication:
- H black × H ethnoe = H ethnoe
- H black × H even = H ethnoe
- H even × H even = H even
Division:
- H ethnoe / H even - it is impossible to unambiguously judge the parity of the result (if the result is an integer, then it can be either even or odd)
- H ethnoe / H even = if the result is an integer, then it H ethnoe
- H even / H parity - the result cannot be an integer, and therefore have parity attributes
- H even / H even = if the result is an integer, then it H even

History and culture

The concept of parity of numbers has been known since ancient times and has often been given a 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 for the funeral 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

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See what "Even numbers" are in other dictionaries:

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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 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 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 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 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.