Identity Element

Date: 10/12/2001 at 14:55:48
From: Cami Stein
Subject: Identity elements

What is the definition of "identity element"?

Date: 10/12/2001 at 15:12:54
From: Doctor Ian
Subject: Re: Identity elements

Hi Cami,

An identity element, for some particular operation, returns the input 
unchanged. For example, for multiplication, the identity element is 1, 
because 

  a * 1 = a

For addition, the identity element is 0, because
  
  a + 0 = a

For matrix multiplication, the identity element is

   _         _
  |  1  0  0  |
  |  0  1  0  |
  |_ 0  0  1 _|

because when you multiply any matrix by that, you get the original 
matrix back.  

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 10/12/2001 at 15:20:33
From: Doctor Paul
Subject: Re: Identity elements

An identity element is always associated with a "binary operation" 
such as addition or multiplication.

Formally speaking, a binary operation is a way to take two elements of 
a set, perform some sort of operation on them, and have the result 
also be a member of the set.

For example if the set is the integers and the binary operation is 
addition, then the sum of any two integers is again an integer. So the 
integers together with addition produces a binary operation that is 
well-defined.

Similarly, the integers with multiplication work. I'm sure you can 
think of other examples - the real numbers with addition - and so 
forth.

If the binary operation is denoted by the # key, then an identity 
element satisfies the following condition:

   e # b = b # e = b

Here e is the identity element in the given set and b is any element 
of the given set. The equation must be true for all elements b in the 
given set for e to be declared the identity element of the set.

Pick your favorite set - the integers or the real numbers work fine.  
If the binary operation is addition, then the identity is zero since

   0 + a = a + 0 = a

However if the binary operation is multiplication, then the identity 
is 1 since:

   1 * a = a * 1 = a 

I hope the idea is clear. In general, the identity element depends on 
what the binary operation is. Mathematicians have very interesting 
ways of defining binary operations. But addition and multiplication 
are the two with which are probably most familiar.

- Doctor Paul, The Math Forum
  http://mathforum.org/dr.math/