**High School Number Sense Lesson 53: Complex Numbers**

Today's concept requires a bit of imagination, because it introduces the idea of

**imaginary numbers**. An imaginary number is the opposite of a real number, and is represented by

**, which is**

*i***√(-1)**. So

**. A**

*i*x*i*= -1**complex number**is some real number plus an imaginary number. It is expressed in the form

**a + b**, where

*i***a**and

**b**are real numbers and

**is the imaginary unit.**

*i*This concept appeared

**15**

**times**last year, with a median placement at

**question # 52**.

**Number Dojo Level: TBD**

Most of these questions require a bit of

**FOILing**. I will jump right into the examples so you can see how they work.

**Example 1: (5 + 6i)(5 - 6i) = (a + bi). Find (a + b). ___**

- Simplify (5 + 6i)(5 - 6i) by FOILing. You get (5 x 5) + (5 x -6i) + (5 x 6i) + (6i x -6i).
- The middle two terms cancel each other out, so you have
**(5 x 5) + (6i x -6i)**. Remember that**i x i = -1**, so you are left with**25 - 36(-1) = 25 + 36 = 61**. - You were asked for (a + b). a =
**61**and b =**0**(there are no**i**'s left), so (a + b) =**61**.

**Example 2: (2 + 3i)(4 - 5i) = (a + bi). Find ab. ___**

- Simplify by FOILing. You get (2 x 4) + (2 x -5i) + (4 x 3i) + (3i x -5i).
- This reduces to
**8 - 10i + 12i - 15ii**, which is**8 + 2i - 15(-1)**. - You now have
**8 + 2i + 15**, or**23 + 2i**. a =**23**and b =**2**. ab = (23)(2) =**46**.

**Example 3: (4 + 2i)(1 - 3i) = a + bi. Find b. ___**

- Since you are asked only for b, worry about FOILing only the terms that have
**i**in them (which would be OI). - (4 x -3i) + (1 x 2i) = -12i + 2i =
**-10i**. b =**-10**.

**Example 4: If (5 - 2i)(3 - 4i) = a + bi, then a = ___**

- Since you are asked only for a, and the OI terms will not cancel, worry only about FL.
- (5 x 3) + (-2i x -4i) = 15 + 8ii = 15 + (8)(-1) = 15 - 8 =
**7**.

Example 5: (2 + 3i) ÷ 5i = (a + bi). Find (a + b). ___

Example 5: (2 + 3i) ÷ 5i = (a + bi). Find (a + b). ___

- This one is pretty tricky. Since we are dividing by 5i, let's also multiply everything by
**5i/5i**(which is equal to 1) to keep i only in the numerator. - (2 + 3i) ÷ 5i x 5i/5i = (2 + 3i)(5i) divided by (5i)(5i). This simplifies to
**(10i + 15ii)/(25ii)**. - Simplified further, we get
**(10i - 15)/(-25)**, which is the same as**(-15 + 10i)/(-25)**. - Factor out a
**-5**from the top and bottom. You are left with**(3 - 2i)/5**. This is the same as**3/5 - 2/5 i**. - You were asked for (a + b). a =
**3/5**and b =**-2/5**. (a + b) = 3/5 - 2/5 =**1/5**.

**Here's a free worksheet to help you practice ComplexNumber:**

complexnumber.pdf |

**Up Next for High School: UnitsDigit**