Chapter 1.6, Problem 49CCR

To determine

Whether the given conjecture is true. If not, then give a counterexample.

Answer to Problem 49CCR

  $\text{Not true}$

Explanation of Solution

Given information:

Division of whole numbers is associative.

Concept Used:

Commutative property:

  $a+b=b+a$

  $a\cdot b=b\cdot a$

Associative property:

  $\left(a+b\right)+c=a+\left(b+c\right)$

  $\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$

Distributive property:

  $a\left(b+c\right)=ab+ac$

  $\left(b+c\right)a=ba+ca$

Additive identity:

  $a+0=0+a=a$

Multiplicative identity:

  $a\cdot 1=1\cdot a=a$

Zero property of Multiplication:

  $a\cdot 0=0\cdot a=0$

Calculation:

The division of two different whole numbers is not commutative.

For example, on dividing 9 by 3, it gives the result 3. While on dividing 3 by 9 will result in a fractional number $\frac{1}{3}$ . So, it implies that the quotient of these whole numbers is not associative, that is

  $\frac{9}{3}\ne \frac{3}{9}$

Thus,the conjecture that the division of whole numbers is associativeis not true.