# Week 3: Complex Numbers – Day 5

Geometry

There are connections between complex numbers and geometry. You can graph complex numbers in the complex plane, where the horizontal axis is the real part, and the vertical coordinate is the complex part. The complex number 2 + 3i, for example, would occupy a point similar to the coordinates (2, 3) in the real plane.

1. Graph the number 1 in the complex plane. (Hint, 1 as a complex number is 1 + 0i, and it occupies a point similar to (1,0) in the real plane). Multiply the number 1 by i and graph the result. Do it again and again. What is the geometrical effect of multiplying by i?

2. Graph the complex number 1 + i in the complex plane. What is the effect of multiplying this number by i? Graph the result.

3. Graph the complex number 2 + i in the complex plane. Multiply this number by the real number 2 and graph the result. What is the effect of multiplying by 2?

4. Graph the complex number -i in the complex plane. Multiply -i by 2 and graph the result. What is the effect of multiplying by 2?

5. Graph the complex number 1 + 2i in the complex plane. Multiply this number by -1 and graph the result. What is the effect of multiplying by -1?

# Week 3: Complex Numbers – Day 4

Roots of Unity

1. The non-real number i raised to what power is one? (Of course, when raised to the zeroth power it is one, but there are other powers as well.)

2. Compute the square of:2-1/2(1 + i)

3. Compute the cube of: 2-1/2(1 + i)

4. What power of 2-1/2(1 + i) is one? In other words, 2-1/2(1 + i) raised to what power is one?

5. Compute the square of: 2-1/2(1 – i)

# Week 3: Complex Numbers – Day 3

Dividing Complex Numbers by Complex Numbers

Addition and multiplication of complex numbers is straightforward. If you divide by a real number it is easy: divide real and non real parts by the number, for example: (2 + 6i)/2 = 1 + 3i.

Division by a non-real number takes a little more work. The trick is to turn division into multiplication, using the complex conjugate.

Example: (2 + 3i) / (1 – i)

Multiply numerator and denominator by the complex conjugate of 1 – i, which is 1 + i:

(2 + 3i) x (1 + i ) = 2 – 3 + 3i + i = -1 + 4i

(1 – i) x (1 + i) = 1 – i + i + 1 = 2

Solution: (-1 + 4i)/2 = -1/2 + 2i

In the following problems, do the division.

1. 1 / i

(1 divided by i).

2. (2 + 3i)/i

3. (1 + i) / (1 – i)

4. i / (1 – i)

5. (3 – 2i) / (3 + 2i)

# Week 3: Complex Numbers – Day 2

Complex Solutions

1. What are the solutions of x2 + 9 = 0?

2. What are the solutions of (x + 1)2 + 4 = 0?

3. What are the solutions of x2 – 2x + 10 = 0?

4. What are the solutions of x3 + 25x = 0?

5. What are the solutions of (x – 1)3 + 36(x – 1) = 0?

# Week 3: Complex Numbers – Day 1

Review of arithmetic of complex numbers

Simplify these expressions. The final result should be a number of the form A + Bi.

1. (-1 + i) + (3 + 2i)

2. (7 + 2i) + 10

3. (2 + 2i) x 3i

4. (5 + 3i) x (5 – 3i)

5. (4 + i) x (1 + 6i)

# Week 2: Equations – Day 5

Deriving Equations

1. Set up the equation for solving for the intersection between the line joining (0, 0) and (1, 4) and the line x + y = 1.

2. Set up the equation for solving for the zeros of the function f(x) = 2x3 + 4x2 – 17x – 3.

3. Set up the equation for solving for the time it takes to drive 432 kilometers at a speed of 65 kilometers per hour.

4. An isosceles triangle has two angles that are the same: 15 degrees. Set up the equation for solving for the other angle of the triangle.

5. Set up the equation for solving for the principal amount of a loan, if the amount of interest owed in one year is \$700 and the interest rate is 5% per year.

# Week 2: Equations – Day 4

Predicting Solutions

Without solving these equations, predict the maximum and minimum number of solutions:

1. 3x + 54 = 8x – 6

2. 4x – 7 = 4x + 20

3. 5 + 4x = 3x2 + 6x

4. 10x5 – 47x4 + 3x3 – x2 + 8 = 0

5. 1 = x2 + 3

# Week 2: Equations – Day 3

Equations Involving Exponential Expressions

Exponential expressions have two basic parts: exponent and base. The simplest equations involve one or the other.

1. If 2300 x 2n = 2501 what is n?

2. If 2100 x 3100 = a100 what is a?

3. If 100n x 53 = 20n x 563 what is n?

4. If 249 x 2n = 1/4, what is n?

5. If (2n)10 = 326 what is n?

# Week 2: Equations – Day 2

Review of easy equation-solving techniques.

1. How many solutions are there to:

1/(x2 + 1) = 0

2. What are the solutions of:

(x – 2)/(x2 + 1) = 0

3. What are the solutions of:

x2 – 28 = 3x

4. What are the solutions of:

(x + 3)2 = 25

5. Solve:

# Week 2: Equations – Day 1

Solve for x:

1.   3x – 3 = 5 – x

2.   1/2 – 1/x = 1/x

3. x4 – 1 = 0

4. 5x2 – 3 = 2x

5. (x2 – 2x – 3)/(x + 1) = 0