imaginary numbers real world

Imaginary Numbers Explained with Formula: $$ \sqrt{-1} \text{= ?} $$

What they are and how to simplify

Part I. (this Page)

Simplifying Powers of $$i$$

$$ i^5 = ? \\ i ^ {21} = ? $$

Part II. Simplifying negative radicands

$$ \sqrt{-25} = ? \\ \sqrt{-18} = ? $$

What is an imaginary number anyway?

Imaginary numbers are based on the mathematical number $$ i $$

$$ i \text { is defined to be } \sqrt{-1} $$


From this 1 fact, we can derive a general formula for powers of $$ i $$ by looking at some examples.

$ \text{ Table 1} \\ \begin{array}{ccc|c} \hline Expression & & Work & Result \\\hline \red{i^ \textbf{2}} & = & i \cdot i = \sqrt{-1} \cdot \sqrt{-1} & \red{ \textbf{ -1 }} \\\hline \red{i^ \textbf{3}} & = & i^2 \cdot i = -1 \cdot i & \red{ \textbf{-i} } \\\hline \red{ i^ \textbf{4} } & = & i^2 \cdot i^2 -1 \cdot -1 = & \red{1} \\\hline \end{array} $
You should understand Table 1 above .

Table 1 above boils down to the 4 conversions that you can see in Table 2 below . You should memorize Table 2 below because once you start actually solving problems, you'll see you use table 2 over and over again!

$ \text{Table 2} \\ $

What is the larger pattern?

In order to understand how to simplify the powers of $$ i $$, let's look at some more examples, and we'll soon see a formula emerge!

$ \begin{array}{c|c|c} Expression & Work & Result \\\hline \red{i^ \textbf{5}} & \blue{i^4} \cdot i^1 = \blue{1} \cdot i & \red{ \textbf{ i }} \\\hline \red{i^ \textbf{6}} & \blue{i^4} \cdot i^2= \blue{1} \cdot -1 & \red{ \textbf{-1}} \\\hline \red{ i^ \textbf{7} } & \blue{ i^4} \cdot i^3 =\blue{1} \cdot -i & \red{ \boldsymbol{ -i}} \\\hline \red{ i^ \textbf{8} } & = \blue{ i^4} \cdot \blue{ i^4}= \blue{1} \cdot \blue{1} = & \red{ \textbf{ 1}} \\\hline \end{array} $

Do you see the pattern yet? Let's look at 4 more and then summarize

$ \begin{array}{c|c|c} Expression & Work & Result \\\hline \red{i^ \textbf{9}} & = \blue{i^4} \cdot \blue{i^4} \cdot i^1 = \blue{1} \cdot \blue{1} \cdot i = & \red{ \textbf{ i }} \\\hline \red{i^ \textbf{10}} & = \blue{i^4} \cdot \blue{i^4} \cdot i^2 = \blue{1} \cdot \blue{1} \cdot i^2 = & \red{ \textbf{ -1 }} \\\hline \red{i^ \textbf{11}} & = \blue{i^4} \cdot \blue{i^4} \cdot i^3 = \blue{1} \cdot \blue{1} \cdot i^3 = & \red{ \textbf{ -i }} \\\hline \red{i^ \textbf{12}} & = \blue{i^4} \cdot \blue{i^4} \cdot \blue{i^4} = \blue{1} \cdot \blue{1} \cdot \blue{1}= & \red{ \textbf{ 1 }} \\\hline \end{array} $

The General Formula

$$ i^k$$ is the same as $$ i^\red{r} $$ where $$ \red{r} $$ is the remainder when k is divided by 4.

Whether the remainder is 1, 2, 3 , or 4 , the key to simplifying powers of i is the remainder when the exponent is divided by 4 . simplifying the powers of the imaginary number i

Practice Problems

Problem 1
$$ i^ {23} $$
Step 1

Calculate the remainder

$$ 23 \div 4 $$ has a remainder of $$ \red{3} $$

Step 2

rewrite

$$i^3$$

Step 3

Simplify using Table2

$$ i^{ \red{3}} = -i $$

Problem 2
$$ i^ {18} $$
Step 1

Calculate the remainder

$$ 18 \div 4 $$ has a remainder of $$ \red{2} $$

Step 2

rewrite

$$ i^{ \red{2}} $$

Step 3

Simplify using Table2

$$ i^{ \red{2}} = -1 $$

Problem 3
$$ i^ {41} $$
Step 1

Calculate the remainder

$$41 \div 4 $$ has a remainder of $$ \red{1} $$

Step 2

rewrite

$$ i^{ \red{1}} $$

Step 3

Simplify using Table2

$$ i^{ \red{1}} = i $$

Problem 4
$$ i^ {100} $$
Step 1

Calculate the remainder

$$ 100 \div 4 $$ has a remainder of $$ \red{0} $$

Step 2

rewrite

$$i^{ \red{0}} $$

Step 3

Simplify using Table2

$$i^{ \red{0}} = 1 $$

Problem 5
$$ 5i^ {22} $$

Remember your order of operations. Exponents must be evaluated before multiplication so you can think of this problem as $$ 5 \cdot (\color{Blue}{i^ {22}}) $$

Step 1

calculate remainder

$$22 \div 4 $$ has a remainder of $$ \red{2} $$

Step 2

rewrite

$$i^{ \red{2}} $$

Step 3

$$i^{ \red{2}} = -1 $$

Step 4

$$ 5 \cdot ( \color{Blue}{-1} ) = -5 $$

Problem 6
$$ 12i^ {36} $$

Remember your order of operations. Exponents must be evaluated before multiplication so you can think of this problem as $$ 12 \cdot ( \color{Blue}{i^ {36}}) $$

Step 1

calculate remainder

$$22 \div 4 $$ has a remainder of $$ \red{2} $$

Step 2

rewrite

$$ 36 \div 4 $$ has a remainder of $$ \red{0} $$

Step 3

$$ i^{ \red{0}} = 1 $$

Step 4

$$ 12 \cdot ( \color{Blue}{1} ) = 12 $$

Problem 7
$$ 7 i^ {103} $$

Remember your order of operations. Exponents must be evaluated before multiplication so you can think of this problem as $$ 7 \cdot ( \color{Blue}{i^ {103}}) $$

Step 1

calculate remainder

$$ 103 \div 4 $$ has a remainder of $$ \red{3} $$

Step 2

rewrite

$$i^{ \red{3}} $$

Step 3

$$i^{ \red{3}} = -i $$

Step 4

$$ 7 \cdot ( \color{Blue}{-i} ) = -7i $$


Ultimate Math Solver (Free)

Free Algebra Solver ... type anything in there!