Parity (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 28/01/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Parity_(mathematics)


00:00:53 1 Arithmetic on even and odd numbers
00:03:09 1.1 Addition and subtraction
00:04:02 1.2 Multiplication
00:04:22 1.3 Division
00:04:26 2 History
00:04:35 3 Higher mathematics
00:04:51 3.1 Higher dimensions and more general classes of numbers
00:05:02 3.2 Number theory
00:05:40 3.3 Group theory
00:07:01 3.4 Analysis
00:07:11 3.5 Combinatorial game theory
00:09:08 4 Additional applications
00:09:23 5 References
00:10:35 Group theory
00:11:54 Analysis
00:12:36 0, to be both odd and even. The Taylor series of an even function contains only terms whose exponent is an even number, and the Taylor series of an odd function contains only terms whose exponent is an odd number.
00:12:55 Combinatorial game theory
00:13:05 In combinatorial game theory, an evil number is a number that has an even number of 1's in its binary representation, and an odious number is a number that has an odd number of 1's in its binary representation; these numbers play an important role in the strategy for the game Kayles. The parity function maps a number to the number of 1's in its binary representation, modulo 2, so its value is zero for evil numbers and one for odious numbers.  The Thue–Morse sequence, an infinite sequence of 0's and 1's, has a 0 in position i when i is evil, and a 1 in that position when i is odious.
00:13:53 Additional applications



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SUMMARY
=======
In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd. An integer is even if it is divisible by two and odd if it is not even. For example, 6 is even because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of even numbers include −4, 0, 82 and 178. In particular, zero is an even number. Some examples of odd numbers are −5, 3, 29, and 73. 
A formal definition of an even number is that it is an integer of the form n = 2k, where k is an integer; it can then be shown that an odd number is an integer of the form n = 2k + 1.  It is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2, 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings.
The sets of even and odd numbers can be defined as following:
Even 
  
    
      
        =
        {
        2
        k
        :
        k
        ∈
        
          Z
        
        }
      
    
    {\displaystyle =\{2k:k\in \mathbb {Z} \}}
  
Odd 
  
    
      
        =
        {
        2
        k
        +
        1
        :
        k
        ∈
        
          Z
        
        }
      
    
    {\displaystyle =\{2k+1:k\in \mathbb {Z} \}}
  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 is odd; otherwise it is 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.