We will learn about identity and inverse elements here. Let us start with the simplest operation of all, that is the addition. For our purposes an element would mean any number. Although, later on we will see that these concepts are also applicable for other mathematical entities like matrices, functions, etc.
II. Identity Element
What do we get when we take any number and add zero to it? We get the
same number
as the result.
So, adding zero does not change the identity of any other number. We say that zero is the $Identity\ Element$ for addition.
Now we take the operation multiplication. What do think will be the identity element for multiplication, that will not change the identity of any other number when multiplied with this number. You most probably guessed it by now, it is the number
$one$
.
III. Inverse Of An Element
Let us take a look at inverses now. Like before, we will start with addition.
The additive inverse of any number $x$ is the number when added to $x$ will give the identity element for addition, which is
$zero$
. For example, what is the additive inverse for $25$?
That is, what do we add to $25$ to get $0$? The answer is
$negative\ 25$
.
We know that:
$25 + (-25) = 0$
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So, the additive inverse of any number is the negative of that number.
Let us take a look at multiplication. The identity element for multiplication is $1$.
Now, what do we multiply $11$ with to get $1$? The answer, of course, is $\dfrac{1}{11}$.
Similarly, what would be the multiplicative inverse of $\dfrac{17}{19}$. The answer is $\dfrac{1}{\frac{17}{19}}$ , because,
$\dfrac{17}{19} \times \dfrac{1}{\dfrac{17}{19}}$
$= \dfrac{17}{19} \times \dfrac{19}{17} = 1$
So, we see that the multiplicative inverse of a number is the reciprocal of that number (or $1$ divided by that number). Multiplicative inverse of all finite numbers, except
$zero$
are defined and well known.
However the inverse of
$zero$
is not defined because
$\dfrac{1}{0}$
does not give us any defined number.
Now let us try the problem in the following page:
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--------- Reference to question: 65082049-7f2b-4e20-8689-e8fcdb480419 ---------