Question

How do I find the product of two imaginary numbers?

Answer 1

We are asked to product (multiply) two imaginary numbers. To solve this, we start by understanding about the real number. We will work on what are real numbers and imaginary numbers, we will find the product of some imaginary number, using those examples we will learn how to find the product for the general case.

Complete step by step solution:
We are given that we have to find the product of imaginary numbers, since we have no number so we have to find the product of arbitrary imaginary numbers to learn.
To do this, we will first go into complex numbers.
In general, complex numbers are given as $a+ib$ , where ‘a’ is called real part while ‘b’ is called imaginary part and ‘I’ is called iota and used to denote as $\sqrt{-1}$ .
On the value of ‘a’ and ‘b’, complex numbers show their behavior.
If the value of $a=0$ , then complex numbers are $0+ib=ib$ such form one called an imaginary number.
We are asked to find the product of such numbers, to learn about this, we need to know how iota behaves when they are produced.
So,
$\begin{aligned} & i=\sqrt{-1} \\\ & {{i}^{2}}=-1 \\\ & {{i}^{3}}=-i \\\ & {{i}^{4}}=1 \\\ \end{aligned}$
Let us consider examples of imaginary complex numbers or say imaginary numbers.
Say we have 3i and 4i.
Then the product of $3i\times 4i=3\times 4\times i\times i$ .
As $3\times 4=12$ and $i\times i={{i}^{2}}$ .
So, $3i\times 4i=3\times 4\times i\times i$
$=12{{i}^{2}}$ .
As ${{i}^{2}}=-1$ so –
$=-12$ (As $12{{i}^{2}}=-1\times 12=-12$ )
So, we get a product of $3i\times 4i$ is -12.
So, in general the product of two imaginary numbers is always real and it is given as negative of the product of the number.
So, if we have to say xi and yi then their product is $xi\times yi=-xy$ always.
So, when we produce the two imaginary numbers then we get the result as negative of the product of coefficients if ‘i’.

Note: It is necessary to know about the power of iota as it play vital role, when we multiply by zero imaginary number to non zero imaginary number then product is always zero as
$\begin{aligned} & 0i\times xi=-x\times 0 \\\ & =-x\times 0 \\\ & =0 \\\ \end{aligned}$
And we can write ‘0’ as 0i.
Si, 0 is imaginary as we can write it in the form of $0+0i$ .
Real numbers are those whose imaginary part ‘b’ of the complex number are zero, mean is for any $z=a+ib$ ,$b=0$ , then such 2 are considered as real numbers.