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

What they are and how to simplify

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 around 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.

Table 1
• $$\red{i^ \textbf{2}} = \sqrt{-1} \cdot \sqrt{-1} =$$
$$\red{ \textbf{ -1 }}$$
• $$\red{i^ \textbf{3}} = i^2 \cdot i = -1 \cdot i =$$
$$\red{ \textbf{-i} }$$
• $$\red{ i^ \textbf{4} } = i^2 \cdot i^2 = -1 \cdot -1 =$$
$$\red{ \textbf{ 1}}$$

Note: Make sure that you understand Table 1 above , because your work with imaginary numbers will continually refer back to it.

#### What is the larger pattern/formula?

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

• $$\red{i^ \textbf{5}} = i^4 \cdot i =$$
$$1 \cdot i = \red{ \textbf{ i }}$$
• $$\red{i^ \textbf{6}} = i^4 \cdot i^2 =$$
$$1 \cdot -1= \red{ \textbf{-1}}$$
• $$\red{ i^ \textbf{7} } = i^4 \cdot i^3 =$$
$$1 \cdot -i= \red{ \textbf{ -i}}$$
• $$\red{ i^ \textbf{8} } = i^4 \cdot i^4 =$$
$$1 \cdot 1= \red{ \textbf{ 1}}$$

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

• $$\red{i^ \textbf{9}} = i^4 \cdot i^4 \cdot i =$$
$$1 \cdot 1 \cdot i = \red{ \textbf{ i }}$$
• $$\red{i^ \textbf{10}} = i^4 \cdot i^4 \cdot i^2 =$$
$$1 \cdot 1 \cdot i^2= \red{ \textbf{-1}}$$
• $$\red{ i^ \textbf{11} } = i^4 \cdot i^4 \cdot i^3 =$$
$$1 \cdot 1 \cdot -i= \red{ \textbf{ -i}}$$
• $$\red{ i^ \textbf{12} } = i^4 \cdot i^4 \cdot i^4 =$$
$$1 \cdot 1 \cdot 1= \red{ \textbf{ 1}}$$

### The General Formula

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

### Practice Problems

Step 1

calculate remainder

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

Step 1

rewrite

$$i^3$$

Step 1

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

Step 1

calculate remainder

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

Step 1

rewrite

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

Step 1

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

Step 1

calculate remainder

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

Step 1

rewrite

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

Step 1

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

Step 1

calculate remainder

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

Step 1

rewrite

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

Step 1

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

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 1

rewrite

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

Step 1

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

Step 1

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

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 1

rewrite

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

Step 1

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

Step 1

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

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 1

rewrite

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

Step 1

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

Step 1

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