Question

Is $\left( 8\div 2 \right)\div 4=8\div \left( 2\div 4 \right)$ ? Is there any associative property for division? Check.

Answer 1

Hint: Association law doesn’t hold for division. We can calculate LHS and RHS of the given equation and show that $LHS\ne RHS$ and thus there is no any associative law for division.

Complete step by step answer:
Given an equation to check: $\left( 8\div 2 \right)\div 4=8\div \left( 2\div 4 \right)$ ?
$\begin{align}
  & LHS=\left( 8\div 2 \right)\div 4 \\
 & =4\div 4 \\
 & =1 \\
 & And, \\
 & RHS=8\div \left( 2\div 4 \right) \\
 & =8\div 0.5 \\
 & =16 \\
\end{align}$
LHS = 1 and RHS = 16
$\Rightarrow LHS\ne RHS$
Associative law for an operator $'*'$ is defined as, $\left( a*b \right)*c=a*\left( b*c \right)$ if a, b, c be any integer.
But in case of division, associative law doesn’t hold true.
In the above example, we proved that,
$\left( 8\div 2 \right)\div 4\ne 8\div \left( 2\div 4 \right)$
So, we can say that there is no any associative law defined for division.

Note: There are some cases of division in which associative law holds true.
Such as if a = 2, b = 2 and c = 1.
$\begin{align}
  & LHS=\left( a\div b \right)\div c=\left( 2\div 2 \right)\div 1 \\
 & =1\div 1 \\
 & =1 \\
 & RHS=a\div \left( b\div c \right)=2\div \left( 2\div 1 \right) \\
 & =2\div 2 \\
 & =1 \\
\end{align}$
Here LHS = RHS.
But for a law to be defined, all the possible values of a, b, and c should satisfy the law. Even if only one possible case won’t satisfy the law, we will say that the law doesn’t hold.
As here we can see that some cases are satisfying associative law of division and some are not. So, Associative law for division doesn’t hold and we say that there is no any associative law defined for division.