**Middle School Number Sense Lesson 76: Imaginary Numbers**

A few months ago we introduced the idea of imaginary numbers in

**ComplexNumber**. That concept is found on high school tests, but today's concept is found in both middle school and high school. It showed up

**twice**last year in middle school, with a median placement at

**question # 75**.

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As we discussed in the previous lesson, an imaginary number is the opposite of a real number, and is represented by

**, which is**

*i***√(-1)**. So

**. The questions found in middle school are pretty straightforward, but those found on high school tests require a little more calculation.**

*i*x*i*= -1**The Basics**

Let's look at the first several powers of

*:*

**i**

*i*^{1}= √(-1) =*i**i*^{2}= -1*i*^{3}= -√(-1) = -*i**i*^{4}= 1*i*^{5}= √(-1) =*i**i*^{6}= -1*i*^{7}= -√(-1) = -*i**i*^{8}= 1*and the pattern continues indefinitely...*

**How to Solve:**

- Look at the exponent. If it is > 3, subtract the greatest possible multiple of 4
*less than*the exponent. - You'll be left with an exponent of 1, 2, 3, or 4. Memorize the pattern above to determine the result.

**Example 1:**

*i*^{12}= ___1. Look at the exponent: 12. Subtract the largest multiple of 4 (less than 12) from it, which is 8. 12 - 8 = 4.

2.

*i*

^{4}=

**1**.

**Example 2:**

*i*^{26}= ___1. Look at the exponent: 26. Subtract the largest multiple of 4 (less than 26) from it, which is 24. 26 - 24 = 2.

2.

*i*

^{2}=

**-1**.

Example 3:

Example 3:

*i*^{64}= ___1. Look at the exponent: 64. Subtract the largest multiple of 4 (less than 64) from it, which is 60. 64 - 60 = 4.

2.

*i*

^{4}=

**1**.

**Example 4:**

*i*^{6}x*i*^{8}= ___1.

*i*

^{6}x

*i*

^{8}=

*i*

^{14}

2. Look at the exponent: 14. Subtract the largest multiple of 4 (less than 14) from it, which is 12. 14 - 12 = 2.

3.

*i*

^{2}=

**-1**.

**Example 5:**

**(3**

*i*)^{4}= ___1. Break this up into 3

^{4}x

*i*

^{4}

2. 3

^{4}=

**81**.

*i*

^{4}=

**1**.

3. 81 x 1 =

**81**.

**Check back soon for a free worksheet to help you practice ImaginaryNum.**

**Up Next for Middle School: SubstRatio**