# What is the difference between a reciprocal of a number and a opposite of the number?

May 6, 2018

When you multiply a reciprocal with its original number, the result is 1.
When you add the opposite to its original number, the result is 0.

#### Explanation:

4*¼=1

¼ is the reciprocal of 4

$7 + \left(- 7\right) = 0$

-7 is the opposite of 7

May 6, 2018

The reciprocal of a number is its inverse under multiplication.
The opposite of a number is its inverse under addition.

#### Explanation:

The number $1$ is the identity for multiplication, since for any number $x$, we have:

$1 \cdot x = x \cdot 1 = x$

If $x \ne 0$ then it has a reciprocal $\frac{1}{x}$ which is its (right and left) inverse under multiplication:

$x \cdot \frac{1}{x} = \frac{1}{x} \cdot x = 1$

The number $0$ is the identity for addition, since for any number $x$, we have:

$0 + x = x + 0 = x$

For any $x$, there is an additive inverse $- x$ called the opposite of $x$ such that:

$x + \left(- x\right) = \left(- x\right) + x = 0$

So for example the reciprocal of $2$ would be $\frac{1}{2}$ and its opposite would be $- 2$.

May 6, 2018

A reciprocal is one over the number

Reciprocal of 2$\implies \frac{1}{2}$

$\frac{3}{4} \implies . \frac{4}{3}$

0.45$\implies \frac{1}{0.45} = \frac{100}{45} = \frac{20}{9}$

The opposite of a number is more about the operation:

The opposite of +4 is -4

$\times 4$ is $\div 4$

$\sqrt{4}$ is ${4}^{2}$