0 is even or not. Odd numbers

Parity

If a number is written in decimal form 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 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 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 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.
Odd parity

Odd and even functions

See what “Odd numbers” are in other dictionaries: Even and odd numbers

- 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 Numbers - In many cultures, especially Babylonian, Hindu and Pythagorean, number is a fundamental principle underlying the world of things. It is the beginning of all things and the harmony of the universe behind their external connection. Number is the basic principle... ...

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Big family dream book ROOT OF A NUMBER Economic dictionary

Pythagoras and the Pythagoreans- Pythagoras was born on Samos. The heyday of his life was in the 530s BC, and his death at the beginning of the 5th century. BC. Diogenes Laertius, one of the famous biographers of ancient philosophers, tells us: Young and greedy for knowledge, he left his fatherland,... ... Western philosophy from its origins to the present day

litters- (from the Greek soros heap) a chain of abbreviated syllogisms in which either the major or minor premise is omitted. There are two types of S.: 1) S., in which, starting from the second syllogism in the chain of syllogisms, a smaller premise is omitted; 2) S., in which... ... Dictionary of Logic Terms

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Books

I'm doing math. For children 6-7 years old, Sorokina Tatyana Vladimirovna. The main objectives of the manual are to familiarize the child with the mathematical concepts of “addend”, “sum”, “minuend”, “subtrahend”, “difference”, “single/double digit numbers”, “even/odd…

Parity of zero- the question is whether to consider zero an even or odd number. Zero is an even number. However, the parity of zero raises doubts among people who are not sufficiently familiar with mathematics. Most people think longer before identifying 0 as an even number, compared to identifying ordinary numbers like 2, 4, 6 or 8. Some mathematics students, and even some teachers, mistakenly consider zero to be an odd number, or even and odd at the same time , or do not classify it into any category.

By definition, an even number is an integer that is divisible by without a remainder. Zero has all the properties that even numbers have, for example 0 is bordered on both sides by odd numbers, every decimal integer has the same parity as the last digit of that number, so since 10 is even, 0 will also be even. If y (\displaystyle y) is an even number, then y + x (\displaystyle y+x) has such parity that it has x (\displaystyle x), A x (\displaystyle x) And 0 + x (\displaystyle 0+x) always have the same parity.

Zero also follows the patterns that form other even numbers. Parity rules in arithmetic such as even−even=even, assume that 0 must also be an even number. Zero is the additive neutral element of the group of even numbers, and it is the origin from which the other even natural numbers are recursively defined. The application of such graph theory recursion to computational geometry relies on the fact that zero is even. Zero is not only divisible by 2, it is divisible by all powers of two. In this sense, 0 is the "most even" number of all numbers.

Why is zero even?

To prove that zero is even, we can directly use the standard definition of "even number". A number is said to be even if it is a multiple of 2. For example, the reason 10 is even is because it is equal to 5 × 2. At the same time, zero is also an integer multiple of 2, that is, 0 × 2, hence zero is even.

In addition, it is possible to explain why zero is even without using formal definitions.

Simple explanations

Numbers can be represented using points on a number line. If you plot even and odd numbers on it, their general pattern becomes obvious, especially if you add negative numbers:

Even and odd numbers alternate with each other. There is no reason to skip the number zero.

Mathematical context

The numerical results of the theory address the fundamental theorem of arithmetic and the algebraic properties of even numbers, so the above convention has far-reaching consequences. For example, the fact that positive numbers have a unique factorization means that it is possible to determine for a given number whether it has an even or odd number of distinct prime factors. Since 1 is not a prime number and also has no prime factors, it is the empty product of primes; Since 0 is an even number, 1 has an even number of prime factors. It follows from this that the Möbius function takes the value μ (1) = 1, which is necessary for it to be a multiplicative function and for the Möbius rotation formula to work.

In education

The question of whether zero is an even number has been raised in the UK school system. Numerous surveys of schoolchildren's opinions on this issue were conducted. It turned out that students assess the parity of zero differently: some consider it even, some consider it odd, others believe that it is a special number - both at the same time or neither. Moreover, fifth grade students give the correct answer more often than sixth grade students.

As studies have shown, even teachers in schools and universities are not sufficiently aware of the parity of zero. For example, about 2/3 of the teachers at the University of South Florida answered “no” to the question “Is zero an even number?” .

Notes

Literature

Anderson, Ian (2001) A First Course in Discrete Mathematics, London: Springer, ISBN 1-85233-236-0
Anderson, Marlow & Feil, Todd (2005), A First Course in Abstract Algebra: Rings, Groups, And Fields, London: CRC Press, ISBN 1-58488-515-7
Andrews, Edna (1990), Markedness Theory: the union of asymmetry and semiosis in language, Durham: Duke University Press, ISBN 0-8223-0959-9
Arnold, C. L. (January 1919), "The Number Zero", The Ohio Educational Monthly T. 68 (1): 21–22 , . Retrieved April 11, 2010.
Arsham, Hossein (January 2002), Zero in Four Dimensions: Historical, Psychological, Cultural, and Logical Perspectives, . Retrieved September 24, 2007. Archived September 25, 2007 on the Wayback Machine
Ball, Deborah Loewenberg; Hill, Heather C. & Bass, Hyman (2005), "Knowing Mathematics for Teaching: Who Knows Mathematics Well Enough to Teach Third Grade, and How Can We Decide?" American Educator, . Retrieved September 16, 2007.
Ball, Deborah Loewenberg; Lewis, Jennifer & Thames, Mark Hoover (2008), "Making mathematics work in school", Journal for Research in Mathematics Education T. M14: 13–44 and 195–200 , . Retrieved March 4, 2010.
Barbeau, Edward Joseph (2003), Polynomials, Springer, ISBN 0-387-40627-1
Baroody, Arthur & Coslick, Ronald (1998), Fostering Children's Mathematical Power: An Investigative Approach to K-8, Lawrence Erlbaum Associates, ISBN 0-8058-3105-3
Berlinghoff, William P.; Grant, Kerry E. & Skrien, Dale (2001) A Mathematics Sampler: Topics for the Liberal Arts(5th rev. ed.), Rowman & Littlefield, ISBN 0-7425-0202-3
Border, Kim C. (1985), Fixed Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, ISBN 0-521-38808-2
Brisman, Andrew (2004), Mensa Guide to Casino Gambling: Winning Ways, Sterling, ISBN 1-4027-1300-2
Bunch, Bryan H. (1982), Mathematical Fallacies and Paradoxes, Van Nostrand Reinhold, ISBN 0-442-24905-5
Caldwell, Chris K. & Xiong, Yeng (27 December 2012), "What is the Smallest Prime?", Journal of Integer Sequences T. 15 (9) ,
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Crumpacker, Bunny (2007), Perfect Figures: The Lore of Numbers and How We Learned to Count, Macmillan, ISBN 0-312-36005-3
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Dehaene, Stanislas; Bossini, Serge & Giraux, Pascal (1993), "The mental representation of parity and numerical magnitude", Journal of Experimental Psychology: General T. 122 (3): 371–396, doi:10.1037/0096-3445.122.3.371 , . Retrieved September 13, 2007.
Devlin, Keith (April 1985), "The golden age of mathematics", New Scientist T. 106 (1452)
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Odd number- an integer that not shared without remainder: …, −3, −1, 1, 3, 5, 7, 9, …

If m is even, then it can be represented in the form $m = 2k$ , and if odd, then in the form $m = 2 k + 1$ , Where $k \in \mathbb Z$ .

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 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 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 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

In higher education institutions with complex schedules of the educational process, even and odd weeks are used. The schedule is different within these weeks. training sessions and in some cases their start and end times. This practice is used to distribute the load evenly across classrooms, academic buildings and to ensure the rhythm of classes in disciplines with a low classroom load (once every 2 weeks)

Train schedules use even and odd train numbers, depending on the direction of travel (direct or reverse). Accordingly, even/odd denotes the direction in which the train passes through each station.

Even and odd days of the month are sometimes associated with train schedules that are organized every other day.

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Notes

Excerpt describing Even and Odd Numbers

“Well, well,” said Prince Andrei, turning to Alpatych, “tell me everything, as I told you.” - And, without answering a word to Berg, who fell silent next to him, he touched his horse and rode into the alley.

The troops continued to retreat from Smolensk. The enemy followed them. On August 10, the regiment, commanded by Prince Andrei, passed along the high road, past the avenue leading to Bald Mountains. The heat and drought lasted for more than three weeks. Every day, curly clouds walked across the sky, occasionally blocking the sun; but in the evening it cleared again, and the sun set in a brownish-red haze. Only heavy dew at night refreshed the earth. The bread that remained on the root burned and spilled out. The swamps are dry. The cattle roared from hunger, not finding food in the sun-burnt meadows. Only at night and in the forests there was still dew and there was coolness. But along the road, along the high road along which the troops marched, even at night, even through the forests, there was no such coolness. The dew was not noticeable on the sandy dust of the road, which had been pushed up more than a quarter of an arshin. As soon as dawn broke, the movement began. The convoys and artillery walked silently along the hub, and the infantry were ankle-deep in soft, stuffy, hot dust that had not cooled down overnight. One part of this sand dust was kneaded by feet and wheels, the other rose and stood as a cloud above the army, sticking into the eyes, hair, ears, nostrils and, most importantly, into the lungs of people and animals moving along this road. The higher the sun rose, the higher the cloud of dust rose, and through this thin, hot dust one could look at the sun, not covered by clouds, with a simple eye. The sun appeared as a large crimson ball. There was no wind, and people were suffocating in this still atmosphere. People walked with scarves tied around their noses and mouths. Arriving at the village, everyone rushed to the wells. They fought for water and drank it until they were dirty.
Prince Andrei commanded the regiment, and the structure of the regiment, the welfare of its people, the need to receive and give orders occupied him. The fire of Smolensk and its abandonment were an era for Prince Andrei. A new feeling of bitterness against the enemy made him forget his grief. He was entirely devoted to the affairs of his regiment, he was caring for his people and officers and affectionate with them. In the regiment they called him our prince, they were proud of him and loved him. But he was kind and meek only with his regimental soldiers, with Timokhin, etc., with completely new people and in a foreign environment, with people who could not know and understand his past; but as soon as he came across one of his former ones, from the staff, he immediately bristled again; he became angry, mocking and contemptuous. Everything that connected his memory with the past repulsed him, and therefore he tried in the relations of this former world only not to be unfair and to fulfill his duty.
True, everything seemed to Prince Andrei in a dark, gloomy light - especially after they left Smolensk (which, according to his concepts, could and should have been defended) on August 6, and after his father, sick, had to flee to Moscow and throw the Bald Mountains, so beloved, built and inhabited by him, for plunder; but, despite this, thanks to the regiment, Prince Andrei could think about another subject completely independent of general issues - about his regiment. On August 10, the column in which his regiment was located reached Bald Mountains. Prince Andrey received news two days ago that his father, son and sister had left for Moscow. Although Prince Andrei had nothing to do in Bald Mountains, he, with his characteristic desire to relieve his grief, decided that he should stop by Bald Mountains.
He ordered a horse to be saddled and from the transition rode on horseback to his father’s village, in which he was born and spent his childhood. Driving past a pond, where dozens of women were always talking, beating rollers and rinsing their laundry, Prince Andrei noticed that there was no one on the pond, and a torn raft, half filled with water, was floating sideways in the middle of the pond. Prince Andrei drove up to the gatehouse. There was no one at the stone entrance gate, and the door was unlocked. The garden paths were already overgrown, and calves and horses were walking around the English park. Prince Andrei drove up to the greenhouse; the glass was broken, and some trees in tubs were knocked down, some withered. He called out to Taras the gardener. Nobody responded. Walking around the greenhouse to the exhibition, he saw that the wooden carved fence was all broken and the plum fruits were torn from their branches. An old man (Prince Andrei saw him at the gate as a child) sat and weaved bast shoes on a green bench.
He was deaf and did not hear Prince Andrei's entrance. He was sitting on the bench on which the old prince liked to sit, and near him was hung a stick on the branches of a broken and dried magnolia.
Prince Andrei drove up to the house. Several linden trees in the old garden had been cut down, one piebald horse with a foal walked in front of the house between the rose trees. The house was boarded up with shutters. One window downstairs was open. The yard boy, seeing Prince Andrei, ran into the house.
Alpatych, having sent his family away, remained alone in Bald Mountains; he sat at home and read the Lives. Having learned about the arrival of Prince Andrey, he, with glasses on his nose, buttoned up, left the house, hastily approached the prince and, without saying anything, began to cry, kissing Prince Andrey on the knee.

Definitions

Even number- an integer that shares without remainder by 2: …, −4, −2, 0, 2, 4, 6, 8, …
Odd number- an integer that not shared without remainder by 2: …, −3, −1, 1, 3, 5, 7, 9, …

According to this definition, zero is an even number.

If m is even, then it can be represented in the form , and if odd, then in the form , where .

In different countries there are traditions related to the number of flowers given.

In Russia and the CIS countries, it is customary to bring an even number of flowers only to 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 bush flower, if they 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.

Notes

Wikimedia Foundation.

Maardu
Superconductivity

See what “Even and odd numbers” are in other dictionaries:

Odd numbers

Even numbers Even and odd numbers

Odd Even and odd numbers

Odd number Even and odd numbers

Odd numbers Even and odd numbers

See what “Odd numbers” are in other dictionaries: Even and odd numbers

Even numbers Even and odd numbers

Slightly redundant numbers- A slightly redundant number, or a quasi-perfect number, is a redundant number whose sum of its proper divisors is one greater than the number itself. To date, no slightly redundant numbers have been found. But since the time of Pythagoras,... ... Wikipedia

Perfect numbers- positive integers equal to the sum of all their regular (i.e., smaller 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 number numbers can be... ...

Quantum numbers- 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, Tatyana Aleksandrovna Barchan, Anna Samodelko. The set includes: 10 puzzles and 10 mathematical labyrinths on the topics: - Number series; - Even and odd numbers; - Composition of numbers; - Counting in pairs; - Addition and subtraction exercises. Includes 20...