Question

State the following statement as true or false: Multiplication and Division of two negative numbers is always a negative number.
(a) True
(b) False

Answer 1

Hint: We start solving the problem by assigning two variables for representing the negative numbers. We then multiply the two assigned variables and use the fact $\left( -\times -=+ \right)$ to check whether the multiplication result is negative or not. We are taking the examples sustaining the proof. Similarly, we also divide the assigned variables and check whether the division result is negative or not. We are given an example of sustaining the proof.

Complete step-by-step solution:
According to the problem, we need to verify whether the Multiplication and Division of two negative numbers always give a negative number.
Let us assume the variables ‘-a’ and ‘-b’ to represent the two negative numbers. Here ‘a’ and ‘b’ are positive numbers.
Let us multiply ‘-a’ and ‘-b’ and assume the result of multiplication by ‘c’.
So, $c=\left( -a\times -b \right)$.
$\Rightarrow c=\left( -\times -\times ab \right)$.
We know that $\left( -\times -=+ \right)$. We use to in calculation of ‘c’.
So, we have $c=ab$. We know that multiplication of two positive numbers is positive which makes c a positive number ---(1).
Example: multiplication of –3 and –2 is 6.
Now, we divide ‘-a’ and ‘-b’ and assume the result of division be ‘d’.
So, $d=\dfrac{-a}{-b}$.
We multiply numerator and denominator with –1
$\Rightarrow d=\dfrac{-a\times -1}{-b\times -1}$.
$\Rightarrow d=\dfrac{a}{b}$.
So, we have $d=\dfrac{a}{b}$. We know that division of two positive numbers is positive which makes d a positive number ---(2).
Example: Division of –4 with –2 gives results as 2.
From (1) and (2), we can see that the claim in our problem Multiplication and Division of two negative numbers is always a negative number is false.
$\therefore$ The correct option for the given problem is (b).

Note: We must make sure that the example given is clearly reflecting the proof that we have just done. We should know that the numbers assumed here are real and there is no meaning in saying negative and positive complex numbers. We can prove the contradiction by giving one counterexample for each statement. Similarly, we can also expect problems to check for addition and subtraction also.