Question

Use the properties of real to answer each question

If $mn=1$, what is the value of $n$? What is $n$called with respect to $m$?

Short answer

Value of n is $\frac{1}{m}$. n is multiplicative inverse of m

Step-by-step solution

- Write properties of real numbers 

Properties of real numbers are written in following table:

For any real numbers $a,b,$ and $c:$
Property	Addition	Multiplication
Commutative	$a+b=b+a$	$a·b=b·a$
Associative	$\left(a+b\right)+c=a+\left(b+c\right)$	$\left(a·b\right)·c=a·\left(b·c\right)$
Identity	$a+0=a=0+a$	$a·1=a=1·a$
Inverse	$a+\left(-a\right)=0=\left(-a\right)+a$	If $a\ne 0$, then $a·\frac{1}{a}=1=\frac{1}{a}·a$
Distributive	$a\left(b+c\right)=ab+ac\text{and}\left(b+c\right)a=ba+ca$

- Compare with given equation

Compare inverse property of multiplication with given equation to find required value as shown below

$\begin{array}{c}a\cdot \frac{1}{a}=1\\ mn=1\end{array}$

Thus, on substituting $\frac{1}{m}$ for $n$, given equation is satisfied.

- relation between n and m 

Substitute $\frac{1}{m}$for n in given equation

$\begin{array}{c}mn=1\\ m\frac{1}{m}=1\\ 1=1\end{array}$

Thus, n is multiplicative inverse of m