Overview
An odd number is an integer that is not divisible by two without a remainder. In elementary terms, when an integer is divided by two it either leaves a remainder or does not produce another integer. This distinguishing feature is part of the broader concept of integers and their classification.
Characteristics and simple tests
An odd integer can be written in the form 2k+1 where k is any integer. Equivalently, its least significant binary digit is 1. A common practical test is to divide the number by two: if a nonzero remainder occurs or the quotient is not an integer (i.e. a fraction), the original number is odd. The sequence of positive odd numbers begins with 1, and includes 3, 5, 7, 9 and so on; negative integers with the same form (…, −5, −3, −1) are odd as well, illustrating that negative numbers can be odd.
Arithmetic and parity rules
Parity describes whether an integer is odd or even. An even number is divisible by two, producing another integer; an odd number is not. Simple arithmetic rules follow: the sum or difference of two integers with the same parity is even, while the sum or difference of integers with opposite parity is odd. The product of an odd integer by any odd integer remains odd; multiplying an integer by an even integer yields an even result.
Examples, notation and common uses
- Examples of odd integers: 1, 3, 5, 7, 9, 11, and also −1, −3, −5.
- Algebraically: if n is odd, write n = 2k + 1 for some integer k.
- Oddness appears in combinatorics (parity arguments), number theory (properties of primes except 2), and computer science (bitwise operations on least significant bits).
History and notable facts
The distinction between odd and even numbers dates back to antiquity and arises naturally from counting and dividing objects into pairs. In number theory, odd primes (all prime numbers except 2) have special roles in many theorems. Parity arguments are a basic and widely used technique in proofs and problem solving, valued for their simplicity and power.