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Complex Numbers for Polynomials | Part 1

(What you need to know about complex numbers for polynomials)

In this section we learn about complex numbers for polynomials. The purpose of this section is to learn the must-knows about complex numbers to be able to move-on to more advanced theorems and techniques for polynomials. In particular, the Fundamental Theorem of Algebra (FTA) and the Complex Conjugate Zeros Theorem.

Definition: Imaginary Numbers

An imaginary number is any number that can be written: \[a.i\] Where:

  • \(a \in \mathbb{R}\), \(a\) can be any real number
  • \(i = \sqrt{-1}\).

Since \(i = \sqrt{-1}\) an immediate and important result is that: \[i^2 = -1\] So, for instance, given the two imaginary numbers \(3i\) and \(4i\), we can calculate their product as follows: \[\begin{aligned} 3i \times 4i & = 3\times 4 \times i^2 \\ & = 12 \times (-1) \\ 3i \times 4i & = -12 \end{aligned}\]

Definition: Complex Numbers

A complex number is made of two parts:

  • a Real Part
  • an Imaginary Part
We often refer to complex numbers with the variable \(z\) and write: \[z = a + ib\] Where \(a\) and \(b\) are both real numbers and are the respective real and imaginary parts of the complex number \(z\) and \(i\) is \(i = \sqrt{-1}\).

Example

Each of the following are complex numbers:

  • The number defined as: \[z = 2 + 3i\] is a complex number. Its real part is \(2\), its imaginary part is \(3\).
  • The number defined as: \[z = 3 - 5i\] is a complex number. Its real part is \(3\), its imaginary part is \(-5\).
  • The number defined as: \[z = 6i\] is a complex numbers. Its real part is \(0\), its imaginary part is \(6\).
  • The number defined as: \[z = -7\] is a complex numbers. Its real part is \(-7\), its imaginary part is \(0\).

Notation

Given a complex number, \(z = a+ib\), to refer to its real and imaginary parts, \(a\) and \(b\), we write:

  • \(Re(z) = a\)
  • \(Im(z) = b\)

Example

Consider the complex number \(z = 2 + 4i\). This number has real and imaginary: \[Re(z) = 2 \quad Im(z) = 4\]

Addition/Subtraction with Complex Numbers

Given two complex numbers, \(z_1 = a+ib\) and \(z_2 = c + id\), we add/subtract by adding/subtracting their respective real and imaginary parts: \[\begin{aligned} z_1 \pm z_2 & = \begin{pmatrix} a + ib \end{pmatrix} \pm \begin{pmatrix} c + id \end{pmatrix} \\ & = a + ib + c + id \\ z_1 \pm z_2 & = \begin{pmatrix} a \pm c \end{pmatrix} + i \begin{pmatrix} b \pm d \end{pmatrix} \end{aligned}\]

Tutorial: Adding & Subtracting Complex Numbers

In this tutorial we learn the rule for adding and subtracting with complex numbers and work through the following example:

Given \(z_1 = 2+3i\), \(z_2 = 6+4i\), \(z_3=5i\) and \(z_4 = 7\), find:

  1. \(z_1 + z_2\)
  2. \(z_1 - z_2\)
  3. \(z_2 + z_3\)
  4. \(z_4 - z_1\)

Example 1

Given the two complex numbers \(z_1 = 2+3i\) and \(z_2 = 3 - 5i\) we can add them as follows: \[\begin{aligned} z_1 + z_2 & = \underbrace{2+3i}_{z_1} + \underbrace{3-5i}_{z_2} \\ & = \underbrace{2+3}_{\text{real parts together}} + \underbrace{3i-5i}_{\text{imaginary parts together}} \\ z_1 + z_2 & = 5 - 2i \end{aligned}\]

Example 2

Given the two complex numbers \(z_1 = 8+5i\) and \(z_2 = 3 + 2i\) we can subtract \(z_2\) from \(z_1\) as follows: \[\begin{aligned} z_1 - z_2 & = \underbrace{8+5i}_{z_1} - \underbrace{3+2i}_{z_2} \\ & = \underbrace{8-3}_{\text{real parts}} + \underbrace{5i-2i}_{\text{imaginary parts}} \\ z_1 - z_2 & = 5 + 3i \end{aligned}\]

Multiplication by a Scalar

Given a complex number \(z = a+ib\) we can multiply it by a scalar \(k \in \mathbb{R}\) as follows: \[\begin{aligned} k.z & = k\begin{pmatrix} a + ib \end{pmatrix} \\ & = k.a + ik.b \end{aligned}\] put "simply" we multiply both the real and the imaginary part by the number \(k\).

Tutorial: Multiplying a Complex Number by a Scalar (a Real Number)

In this tutorial we learn the rule for complex numbers by scalars and work through the following example:

Given \(z = 2+3i\), find:

  1. \(4z\)
  2. \(-5z\)

Example 3

Given the complex number \(z = 3 + 4i\) we can find \(5z\) as follows: \[\begin{aligned} 5z & = 5\begin{pmatrix} 3 + 4i\end{pmatrix}\\ & = 5\times 3 + 5\times 4i \\ & = 15 + 20i \end{aligned}\]

Complex Conjugates

Given a complex number \(z = a+ib\), we define its complex conjugate

, typically written \(z^*\), as: \[z^* = a - ib\] We say that two complex numbers are complex conjugates of each other if they have the same real part and opposite imaginary parts.

Example

Given the complex number \(z = 3+2i\), its complex conjugate is: \[z^*=3 - 2i\]

Addition & Subtraction with Conjugates

Given a complex number \(z = a + ib\) and its conjugate \(z^* = a - ib\), their sum and difference leads to the following results:

  • addition \(z+z^*\): \[z + z^* = 2.Re(z)\]
  • subtraction \(z-z^*\): \[z - z^* = 2i.Im(z)\]

Example

Given \(z = 3 + 5i\) and its complex conjugate \(z^* = 3 - 5i\):

  • \(z + z^* = 2\times 3 = 6\)
  • \(z - z^* = 2i \times 5 = 10i\)

Exercise 1

For each of the following questions, express your answer in its simplest form:

  1. Find \(z_1+z_2\) where \(z_1 = 4 + i\) and \(z_2 = -2 + 3i\).
  2. Find \(z_1 - z_2\) where \(z_1 = 6 + 3i\) and \(z_2 = 4 + i\).
  3. Find \(5z\) where \(z = -3 + 5i\).
  4. Find \(4z_1 + 2z_2\) where \(z_1 = 1 + 4i\) and \(z_2 = 3 - i\).
  5. Find \(z_1 + z^*_1 - z_2\) where \(z_1 = -4 + 2i\) and \(z_2 = -6i\)
  6. Find \(z_1 - 3z_2\) where \(z_1 = 3-2i\) and \(z_2 = 1+ 5i\).
  7. Find \(z_2 + \begin{pmatrix} z_1 - z^*_1 \end{pmatrix}\), where \(z_1 = 3 + 2i\) and \(z_2 = 4 + 5i\).
  8. Find \(4z_1 + 3z_2 - 2z_3\) where \(z_1 = -2 + i\), \(z_2 = 3 + 4i\) and \(z_3 = 4\).
  9. Find \(z_1 + z_2 +z^*_1 - z^*_2\) where \(z_1 = 1 - 3i\) and \(z_2 = 6 +i\).
  10. Find \(5z_1 - 4z_2 + z_3\) where \(z_1 = 3+4i\), \(z_2 = 1 - 3i\) and \(z_3 = 4 + 2i\).

Note: this exercise can be downloaded as a worksheet to practice with: Worksheet 1

Solution Without Working

Multiplying Two, or more, Complex Numbers

Given two complex numers \(z_1 = a+ib\) and \(z_2 = c + id\), we find the product: \[z_1 . z_2 = \begin{pmatrix}a + ib \end{pmatrix}.\begin{pmatrix}c + id \end{pmatrix}\] using the "usual" rules of distributivity, keeping in mind that \(i^2 = -1\).

Example 4

Given the two complex numbers \(z_1 = 4+5i\) and \(z_2 = 3 + 2i\) we multiply these two complex numbers together as follows: \[\begin{aligned} z_1.z_2 & = \begin{pmatrix} 4 + 5i \end{pmatrix}.\begin{pmatrix} 3 + 2i \end{pmatrix} \\ & = 4\times 3 + 4\times 2i + 5i \times 3 + 5i \times 2i \\ & = 12 + 8i + 15i + 10i^2 \\ & = 12 + 8i + 15i - 10 \\ z_1.z_2 & = 2 + 23i \end{aligned}\]

Tutorial: Multiplying a Complex Numbers Together

In this tutorial we learn the how to multiply complex numbers together. We work through the following example:

Given \(z_1 = 3 - 2i\), \(z_2 = 4 + i\) and \(z_3 = 6i\), find:

  1. \(z_1.z_2\)
  2. \(z_2.z_3\)

Important Result: Product of Complex Conjugates

Given a complex number \(z = a+ib\) and its complex conjugate \(z^* = a - ib\), their product will always equal to the sum of the squares of the real and the imaginary part of \(z\): \[z.z^* = a^2+b^2\]

Proof

This result can be shown as follows: \[\begin{aligned} z.z^* & = \begin{pmatrix}a + ib \end{pmatrix}.\begin{pmatrix} a - ib \end{pmatrix} \\ & = a^2 - iab + iab - b^2i^2 \\ z.z^* & = a^2 + b^2 \end{aligned}\]

Example

Given \(z = 2 + 5i\) and its conjugate \(z^* = 2 - 5i\), their product is: \[\begin{aligned} z.z^* &= \begin{pmatrix} 2+5i \end{pmatrix}.\begin{pmatrix} 2-5i \end{pmatrix} \\ & = 2^2 + 5^2 \\ z.z^* & = 29 \end{aligned}\]

Exercise 2

For each of the following questions, express your answer in its simplest form:

  1. Expand and simplify: \(\begin{pmatrix}1 + 3i \end{pmatrix}.\begin{pmatrix}5 + 2i \end{pmatrix}\)
  2. Expand and simplify: \(\begin{pmatrix}3 - 4i \end{pmatrix}.\begin{pmatrix}2 + 6i \end{pmatrix}\)
  3. Given \(z_1 = 3 - 4i\) and \(z_2 = 1 + 5i\), find: \[2.z_2 - z_1.z^*_1\]
  4. Expand and simplify: \(\begin{pmatrix}2 - i \end{pmatrix}.\begin{pmatrix}6 - 3i \end{pmatrix}\)
  5. Given \(z_1 = 3i\) and \(z_2 = 7+i\), find: \[3.z_1.z^*_1 + z_2.z^*_2\]
  6. Expand and simplify: \(\begin{pmatrix}5 + 3i \end{pmatrix}.\begin{pmatrix}7 + i \end{pmatrix}\)
  7. Given \(z_1 = 5 +2i\) and \(z_2 = 1+i\), find: \[4.z_1 \begin{pmatrix}z^*_1 - z_2 \end{pmatrix}\]
  8. Expand and simplify: \(\begin{pmatrix}-2 + 5i \end{pmatrix}.\begin{pmatrix}3 - 2i \end{pmatrix}\)
  9. Given \(z_1 = -4 +i\) and \(z_2 = 6 + 7i\), find: \[\begin{pmatrix}z_1+z_2\end{pmatrix}.\begin{pmatrix}z^*_1 - z^*_2 \end{pmatrix}\]
  10. Find \(2z_1 + z_2.z_3\), where \(z_1 = 1+2i\), \(z_2 = 2 + 4i\) and \(z_3 = 3 - i\).
  11. Find \(z_1.z_2 - 4z_3\), where \(z_1 = 2 + i\), \(z_2 = 3 - i\) and \(z_3 = 3 + 2i\).
  12. Find \(3z_1.z_2 - 2z_3\), where \(z_1 = 5 + 2i\), \(z_2 = 2 - 3i\) and \(z_3 = 6 + 4i\).

Note: this exercise can be downloaded as a worksheet to practice with: Worksheet 2

Division with Complex Numbers

Given two complex numbers \(z_1 = a + ib\) and \(z_2 = c + id\), we can divide \(z_1\) by \(z_2\) using the complex conjugate of \(z_2\). Given \(z_2 = c + id\) its complex conjugate is \(z^*_2 = c - id\). Using the fact that \(z_2 \times z^*_2 = c^2 + d^2\), we can divide \(z_1\) by \(z_2\) as explained in the following tutorial.

Tutorial: Division with Complex Numbers

In this tutorial we learn the how to divide one complex number by another. We work through the following examples:

  1. \(\frac{3 - 4i}{1+2i}\)
  2. \( \frac{3+i}{-2i}\)

Example

Given \(z_1 = 8 -2i\) and \(z_2 = 2 + 5i\) we find \(z_1 \div z_2\) as follows: \[\begin{aligned} z_1 \div z_2 & = \frac{z_1}{z_2} \\ & = \frac{8 - 2i}{2 + 5i} \\ & = \frac{8 - 2i}{2 + 5i} \times \frac{2 - 5i}{2 - 5i} \\ & = \frac{\begin{pmatrix} 8 - 2i \end{pmatrix}. \begin{pmatrix} 2 - 5i \end{pmatrix}}{\begin{pmatrix} 2 + 5i \end{pmatrix}. \begin{pmatrix} 2 - 5i \end{pmatrix}} \\ & = \frac{16 - 40i - 4i +10i^2}{4 + 25} \\ & = \frac{16 - 44i -10}{29} \\ & = \frac{6 - 44i}{29}\\ z_1 \div z_2 & = \frac{6}{29} - \frac{44}{29}i \end{aligned}\]

Exercise 3

  1. Given \(z_1 = 3+4i\) and \(z_2 = 1+2i\), find:
    1. \(z_1 \div z_2\)
    2. \(z_2 \div z_1\)
  2. Given \(z_3 = -2 + 3i\) and \(z_4 = 5+i\), find:
    1. \(z_3 \div z_4\)
    2. \(z_4 \div z_3\)
  3. Given \(z_5 = 6 - 2i\) and \(z_6 = 8i\), find:
    1. \(z_5 \div z_6\)
    2. \(z_6 \div z_5\)
  4. Given \(z_1 = 1 + 3i\), \(z_2 = 4-i\) and \(z_3 = 5 + 7i\), find: \[z_1+z_2 \div z_3\]
  5. Given \(z_1 = 5-3i\), \(z_2 = 4i\) and \(z_3 = 1 + 2i\), find: \[3z_1 - \frac{z_2}{2z_3}\]

Note: this exercise can be downloaded as a worksheet to practice with: Worksheet 3

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