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Math 202
Complex Numbers Name:
The goal of this worksheet is to familiarize you with the basic ideas surrounding complex numbers. In your math classes you have dealt with several different kinds of numbers: integers (positive and negative whole numbers), rational numbers (fractions), and real numbers. The complex numbers are a new extension of the idea “number”.
I. Why complex numbers?
Solve the quadratic equation x^2 + x − 1 = 0. How many solutions are there?
Solve the quadratic equation x^2 + x + 1 = 0. How many solutions are there?
One way to understand the complex numbers is as an effort to allow for solutions to this (and all) quadratic equations (as well as all other polynomial equations). In particular, we make the following definition.
Definition 1. Let
−1 be the number i.
What is i^2?
Write the solutions to (2) in the form a + bi.
Definition 2. We define a complex number to be a number of the form z = a + bi where a and b are real numbers.
The real part of z is a, and the imaginary part pf z is b
The complex conjugate of z is defined to be z¯ = a − bi
- Find all solutions to the equation x^4 + 1 = 0.
Notice that
i appears in your solutions. A very important theorem tells us that when we use the complex numbers all polynomials of all degrees have solutions that can be written in the form a + bi. In a little bit we will see how to write
i in the form a + bi.
II. Arithmetic with the complex numbers
Let z = a + bi and w = u + vi. Addition and subtraction works like you would expect it to. z ± w = (a + bi) ± (u + vi) = (a ± u) + (b ± v)i 1
2
Multiplication requires a bit more work. zw = (a + bi)(u + vi) = au + avi + bui + bvi^2 = au − bv + avi + bui = (au − bv) + (av + bu)i For division we need a nifty little identity.
- (The nifty identity) Compute z z¯.
- Use (1) to write
z
in a way that has a real denominator.
- Write
z
in the form c + di.
- Use (3) to compute
w z
and write it in the form c + di.
- Exercises: (a) Write (3 + πi) − (5 − i) in the form a + bi. (b) Write (1 + 4i)(3 + 2i) in the form a + bi. (c) Write (1 + i)^2 + (1 + i) in the form a + bi. (d) Write
2 − i 5 + 4i
in the form a + bi. (e) Show that for complex numbers z and w, the following equality holds zw¯ = ¯z w¯. (f) Compute in^ for n = ± 1 , ± 2 , ± 3 , ±4. Do you see a pattern? What is i^45? What about i−^37?
III. Visualizing complex numbers
When we visualize real numbers, we generally think of them as lying on the “number line”. To specify a complex number z = a + bi we need to specify two real number, namely a and b. Thus, to visualize the complex numbers we need two real dimensions, that is we need a plane. We call this plane the “complex plane”, and to plot z = a + bi we plot the point (a, b) on the standard Cartesian plane.
- Sketch a complex plane and plot the following points, 1 + i, (1 + i)^2 , (1 + i)−^1 , 1 − i, −1 + i, − 1 − i.
One of the most useful properties requires polar coordinates.
Write the following points which are given in rectangular coordinates in polar coordinates, (a) (0, 3) (b) (− 1 , 1) (c) (− 2 , −4)
Write the following points which are given in polar coordinates in rectangular coordinates, (a) (2, 0) (b) (8, π/2) (c) (e, 7 π/4)
Sketch a graph of the region described by the following expressions given in polar coordi- nates. (a) r = 3
(^4) Isn’t it nice to see the things we learned this term at work proving interesting things?
- Exercises: (a) What is eπi?
(b) Use Euler’s formula to write the following complex numbers in polar form (reıθ). (i) 1 + i (ii) − 3 − 4 i (iii) −1 + 3i
(c) Use Euler’s formula to write the following complex numbers in rectangular form (a+bi). (i)
−i (ii) (1 + i)^100 (iii) (3, −3)^3 /^4 (iv) (
5 + 2i)
√ 2
(d) Use Euler’s formula to prove cos^2 (2θ) = cos^2 (θ) − sin^2 (θ).
