What is the multiplication identity property?

Feb 28, 2015

For a set of elements, $S$ and an operation (called multiplication and indicated by the symbol $\times$ in this explanation).

If for all $x$ which are members of $S$ if there is one element $\phi$ of $S$ for which

$\phi \times x = x$ and $x \times \phi = x$
(for all $x \epsilon S$)

Then $\phi$ is called the multiplicative identity
and
$\phi \times x = x$ is called the multiplicative identity property .

For Integers, Rational Numbers, Real Numbers and Complex Numbers the multiplicative identity is $1$.

That is
(any number) $\times 1 =$ (the same number).

For matrices the multiplicative identity is the Identity Matrix

For a more complex set and an operation that we might not normally think of as "multiplication",
the multiplicative identity $\phi$ might be something quite different provided it satisfies the multiplicative identity property for that set and operation.