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### Identity Element

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

What is the definition of "identity element"?
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```
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"

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/
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Associated Topics:
Elementary Multiplication