Investigation: Complex Arithmetic



Investigation: Complex Numbers

Not all numbers are numbers that exist in our everyday world. These numbers are called imaginary numbers.

[pic]

All numbers are considered complex numbers, whether they are imaginary, real, or both. Complex numbers can be written in the for a + bi

Part 1: Simplify the following complex numbers

a. [pic] b. [pic] c. [pic]

d. [pic] e. [pic] f. [pic]

g. [pic] h. [pic] i. [pic]

Part 2: Types of numbers

Step 1: Family tree of numbers:

Step 2: Classify each number

a. 0 b. 4.453423… c. 3+4i

d. [pic] e. [pic] f. [pic]

g. -5.4 h. [pic] i. [pic]

Part 2: Graphing complex numbers

The x-axis is now the real number, the y-axis represents the imaginary.

Graph each number on the complex plane on the left. Then name each complex number on the complex plane on the right.

a. 3+4i b. -2-i c. 3 d. -1+5i

Part 3: Find the value of each expression. Remember [pic]

i =[pic] i2 = i3 = i4 =

a. i24 = b. i35 = c. i50 = d. i13 = e. i102 = f. i37 =

Investigation: Complex Arithmetic

Part 1: Add these complex numbers.

a. (2 – 4i) + (3 + 5i) b. (7 + 2i) + (-2 + i)

c. (2 – 4i) – (3 + 5i) d. (4 – 4i) – (1 – 3i)

Part 2: Now multiply these binomials. Express your products in the form a + bi. Remember what is i2?

a. (2 – 4i)(3 + 5i) b. (7 + 2i)(-2 + i)

c. (2 – 4i)2 d. (4 – 4i)(1 – 3i)

Part 3: The conjugate of a + bi is a – bi. Let’s see what happens when we add or subtract them together.

a. (2 – 4i) + (2 + 4i) b. (7 + 2i) + (7 – 2i)

c. (2 – 4i)(2 + 4i) d. (-4 + 4i)(-4 – 4i)

Part 4: Recall rationalizing the denominator with radicals

[pic]

We will use a similar technique to change the complex denominator to a real number by using conjugates. Once you have a real number in the denominator, divide to get an answer in the form a + bi.

a. [pic] b. [pic]

c. [pic] d. [pic]

e. [pic] f. [pic]

Part 5 Solve each equation.

a. x2 = -36 b. x2 = -28 c. -(x – 3)2 = 25

d. (2x + 7)2 – 15 = -28 e. 4(x – 11)2 + 27 = 3 e. -5(5x – 1)2 = 18

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