Odd numbers

Odd numbers
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Odd_numbers".

In mathematics, the parity of an object states whether it is even or odd.

This concept begins with integers. An even number is an integer that is "evenly divisible" by 2, i.e., divisible by 2 without remainder; an odd number is an integer that is not evenly divisible by 2. (The old-fashioned term "evenly divisible" is now almost always shortened to "divisible".) A formal definition of an odd number is that it is an integer of the form n = 2k + 1, where k is an integer. An even number has the form n = 2k where k is an integer.

Examples of even numbers are −4, 8, 0, and 42. Examples of odd numbers are −3, 9, 1, and 5. A fractional number like 1/2 or 3.141 is neither even nor odd.

The set of even numbers can be written:

{Evens} = 2Z = {..., −6, −4, −2, 0, 2, 4, 6, ...},

where Z is the set of all integers. The set of odd numbers can be shown like this:

{Odds} = 2Z + 1 = {..., −5, −3, −1, 1, 3, 5, ...}.

A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it's odd; otherwise it's even. The same idea will work using any even base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1 and even if its last digit is 0. In an odd base, the number is even according to the sum of its digits - it is even if and only if the sum of its digits is even.

content

1 Arithmetic on even and odd numbers
2 History
3 Music theory
4 Higher mathematics
5 Parity for other objects
6 See also

Arithmetic on even and odd numbers

The following laws can be verified using the properties of divisibility. They are a special case of rules in modular arithmetic, and are commonly used to check if an equality is likely to be correct by testing the parity of each side.

Addition and subtraction

even ± even = even;
even ± odd = odd;
odd ± odd = even;

Rules analogous to these for divisibility by 9 are used in the method of casting out nines.

Multiplication

even × even = even;
even × odd = even;
odd × odd = odd.

Division

The division of two whole numbers does not necessarily result in a whole number. For example, 1 divided by 4 equals 1/4, which isn't even or odd, since the concepts even and odd apply only to integers. But when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor.

History

The ancient Greeks considered 1 to be neither fully odd nor fully even. Some of this sentiment survived into the 19th century: Friedrich Wilhelm August Froebel's 1826 The Education of Man instructs the teacher to drill students with the claim that 1 is neither even nor odd, to which Froebel attaches the philosophical afterthought,

It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two relatively different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and even numbers one number (one) which is neither of the two. Similarly, in form, the right angle stands between the acute and obtuse angles; and in language, the semi-vowels or aspirants between the mutes and vowels. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws.

In the 18th century, some mathematicians wrote that infinity was neither even nor odd while arguing that Grandi's series 1 − 1 + 1 − 1 + · · · equaled 1/2.

Music theory

In wind instruments which are cylindrical and in effect closed at one end, such as the clarinet at the mouthpiece, the harmonics produced are odd multiples of the fundamental frequency. (With cylindrical pipes open at both ends, used for example in some organ stops such as the open diapason, the harmonics are even multiples of the same frequency, but this is the same as being all multiples of double the frequency and is usually perceived as such.) See harmonic series (music).

Higher mathematics

The even numbers form an ideal in the ring of integers, but the odd numbers do not — this is clear from the fact that the identity element for addition, zero, is an element of the even numbers only. An integer is even if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, and odd if it is congruent to 1 modulo 2.

All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even; it is unknown whether any odd perfect numbers exist.

The squares of all even numbers are even, and the squares of all odd numbers are odd. Since an even number can be expressed as 2x, (2x)² = 4x² which is even. Since an odd number can be expressed as 2x + 1, (2x + 1)² = 4x² + 4x + 1. 4x² and 4x are even, which means that 4x² + 4x + 1 is odd (since even + odd = odd).

Goldbach's conjecture states that every even integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to be true for integers up to at least 4 × 10¹⁴, but still no general proof has been found.

The Feit-Thompson theorem states that a finite group is always solvable if its order is an odd number. This is an example of odd numbers playing a role in an advanced mathematical theorem where the method of application of the simple hypothesis of "odd order" is far from obvious.

Parity for other objects

Parity is also used to refer to a number of other properties.

The parity of a permutation (as defined in abstract algebra) is the parity of the number of transpositions into which the permutation can be decomposed. For example (ABC) to (BCA) is even because it can be done by swapping A and B then C and A (two transpositions). It can be shown that no permutation can be decomposed both in an even and in an odd number of transpositions. Hence the above is a suitable definition. See the article on even and odd permutations for an elaboration.

The parity of a function describes how its values change when its arguments are exchanged with their negatives. An even function, such as an even power of a variable, gives the same result for positive or negative arguments. An odd function, such as an odd power of a variable, gives the negative of its result when given the negative of an argument. It is possible for a function to be neither odd nor even, and for the case f(x) = 0, to be both odd and even.

Contents