INTRODUCTION TO COMPLEX NUMBERS (SIMPLIFICATION)

There are no real number for the solution of:
      $y=\sqrt{-1}$

To obtain the solution to the above equation, we extend the real number to include an imaginary number.

An imaginary number is simply the square root of $–1$. That is:
$i=\sqrt{-1}$

Using the properties of imaginary numbers,
$i^2=(\sqrt{-1})^2$
$i^2=-1$.

Using this fact, we can write the square root of any negative number.

Example 1
Solve $\sqrt{-64}$

Solution:
$\sqrt{-64}=\sqrt{64\times-1}$
                   $=\sqrt{64}\times\sqrt{-1}$
                   $=±8\times i$ ......Recall that $i=\sqrt{-1}$
                   $=±8i$

This is the idea of complex numbers. A complex number is the sum of a real number and an imaginary number.

There are many ways of writing complex numbers, but the most common is the one written in standard form. A complex number is standard form when it is written as:
 $a+bi$

Where $a$ is the real part and b is the imaginary part.

We will discuss complex numbers extensively when we begin to add, subtract, multiply and divide complex numbers.

But, For purpose of this post, we are just going to be simplifying the powers of imaginary numbers

Example 2
Simplify $i^6$

Solution:
Using the idea of indices.
$i^6=i^2\times i^2\times i^2$

Recall that $i^2=-1$, hence,
$i^6=-1\times-1\times-1$
$i^6=-1$

Example 3
Simplify $i^7$

solution:
$i^7=i^2\times i^2\times i^2\times i$
$i^7=-1\times-1\times-1\times i$
$i^7=-i$

Example 4
Simplify $i^{16}$

Solution:
$i^{16}=i^4\times i^4\times i^4\times i^4$

If $i^2=-1$, then $i^4=1$,
Using this fact.
$i^{16}=1\times1\times1\times1$
$i^{16}=1$

Example 5
Simplify $i^9$

Solution:
$i^9=i^2\times i^2\times i^2\times i^2\times i$
$i^9=-1\times-1\times-1\times-1\times i$
$i^9=i$.

There you have it. In our next post, we will be discussing addition and subtraction of complex numbers in our next post.

Meanwhile, if you have got question related to this post, do well to ask me in the comment box and i promise to answer you as quickly as possible
 
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