Question

What operations are not associative for whole numbers?

Answer 1

Hint: In this problem, we have to find the operations, which are not associative for whole numbers. We know that we have four operations in mathematics, they are addition, subtraction, multiplication and division. We also know that associative property is \[a+\left( b+c \right)=\left( a+b \right)+c\], if this condition is satisfied in all cases in addition, then addition is associative else it is not associative. Similarly, we can check for the other operations and find which are not associative.

Complete step by step answer:
Here we have to find the operations, which are not associative for whole numbers.
We know that whole numbers start from 0 to infinity.
We know that we have four operations in mathematics, they are addition, subtraction, multiplication and division.
We can now check the associative property for addition.
We can now take any numbers to be substituted.
We know that associative property is \[a+\left( b+c \right)=\left( a+b \right)+c\] for addition.
\[\Rightarrow 5+\left( 10+15 \right)=\left( 5+10 \right)+15=30\]
Therefore, addition is associative for whole numbers.
We can now take for multiplication, we get
\[\begin{align}
  & \Rightarrow 2\times \left( 3\times 4 \right)=\left( 2\times 3 \right)\times 4 \\
 & \Rightarrow 24=24 \\
\end{align}\]
Therefore, multiplication is also associative.
Similarly, we can subtract, we get
\[\begin{align}
  & \Rightarrow 5-\left( 10-15 \right)=\left( 5-10 \right)-15 \\
 & \Rightarrow 10\ne -20 \\
\end{align}\]
Here we can see that subtraction is not associative for whole numbers.
Similarly, we can take division.
\[\begin{align}
  & \Rightarrow 100\div \left( 25\div 5 \right)=\left( 100\div 25 \right)\div 5 \\
 & \Rightarrow 20\ne 0.8 \\
\end{align}\]
Therefore, subtraction and division are not associative for whole numbers.

Note: We should know that subtraction and division are not associative for whole numbers, because by changing the order while performing these operations, the result would also change in most cases. We should remember that associative property is \[a+\left( b+c \right)=\left( a+b \right)+c\], if this condition is satisfied in all cases in addition, then addition is associative else it is not associative. Similarly, we can check for the other operations.