Identity ElementDate: 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/ |
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