Parity in number theory is a characteristic of an integer that determines its ability to divide completely into two .
Content
Definitions
- An even number is an integer that is divisible by 2 without a remainder: ..., −4, −2, 0 , 2, 4, 6, 8, ...
- An odd number is an integer that cannot be divided by 2 without a remainder : ..., −3, −1, 1, 3, 5, 7, 9, ...
If m is even, then it can be represented as , and if it’s odd, then in the form where .
From the point of view of comparison theory , even and odd numbers are elements of the residue classes [0] and [1] modulo 2, respectively.
Arithmetic
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- Division:
- Even / Even - it is impossible to clearly determine the evenness of the result (if the result is an integer , then it can be either even or odd)
- Even / H even = if the result is an integer , then it is Even
- Odd / Even - the result cannot be an integer, and therefore cannot have parity attributes
- H even / H even = if the result is an integer , then it is H even
Sign of parity
In decimal notation
If in decimal notation of a number the last digit is even (0, 2, 4, 6 or 8), then the whole number is even, otherwise it is odd.
- 4 2 , 10 4 , 1111 0 , 911581734 2 - even numbers.
- 3 1 , 7 5 , 70 3 , 7852 7 , 235689512 5 - odd numbers.
In other number systems
For all numeral systems with an even base (for example, for hexadecimal ), the same parity criterion applies: a number is divisible by 2 if its last digit is divisible by 2. For numeral systems with an odd base there is another parity criterion: a number is even if and only if when the even sum of its digits [1] [2] . For example, the number denoted by the entry “136” is even in any number system, starting with the seven-digit [1] .
History and Culture
The concept of parity of numbers has been known since ancient times, and it was often given mystical significance. In Chinese cosmology and natural philosophy, even numbers correspond to the concept of “ yin ”, and odd numbers correspond to “ yang ” [3] .
In different countries, there are traditions associated with the number of flowers presented. For example, in the United States , Europe and some eastern countries, it is believed that an even amount of flowers presented 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 when there are a lot of flowers in a bouquet (usually more than 11 ), the evenness or oddness of their number no longer plays any role. For example, it is perfectly acceptable to give a lady a bouquet of 12, 14, 16, etc., of flowers or sections of a bush flower that have many buds , for which they, in principle, are not counted. Moreover, this applies to a greater number of flowers (slices), given in other cases.
Practice
- According to the Rules of the Road , depending on the evenness or oddness of the month, parking under the signs 3.29 , 3.30 may be allowed.
- In higher education institutions with complex educational process schedules, even and odd weeks are used. Within these weeks, the timetable for training sessions and, in some cases, their start and end times are different. This practice is used to evenly distribute the load across classrooms, academic buildings and for the rhythm of classes in disciplines with a load of once every 2 weeks.
- Parity / oddness of numbers is widely used in railway transport:
- When the train is moving, it is assigned a route number, which can be even or odd, depending on the direction of travel (direct or reverse). For example, the train " Russia " when traveling from Vladivostok to Moscow has the number 001, and from Moscow to Vladivostok - 002;
- Evenness / oddness on the slang of railway workers indicates the direction in which the train passes through the station (example of the announcement “An odd train will pass along the third route”);
- Schedules of movement of passenger trains following one day are linked to even and odd dates of the month. If two consecutive odd numbers coincide, for even distribution of cars between terminal stations, trains can be assigned with a deviation from the schedule (in this case, the next train runs not every other day, but two days or the next day);
- Seats in reserved seats and compartment cars are always allocated: even - top, odd - bottom.
See also
- Parity zero
Notes
- ↑ 1 2 Jacob Perelman . Odd or even? // Interesting arithmetic: riddles and wonders in the world of numbers. - The eighth edition, abridged. - M .: Detgiz , 1954. - S. 66-68.
- ↑ Ruth L. Owen. Divisibility in bases // The Pentagon: A Mathematics Magazine for Students: Journal. - 1992. - Vol. 51 , iss. 2 . - P. 17–20 . Archived on September 9, 2015.
- ↑ Riftin B.L. Yin and Yang. Myths of the peoples of the world. Volume 1, Moscow: Sov.Encyclopedia, 1991, p. 547.
Links
- Sequence
A005408 in OEIS : odd numbers
- Sequence
A005843 at OEIS : even numbers
- Sequence
A179082 in OEIS : even numbers with an even sum of decimal digits