Introduction to Complex Numbers
[Pages:35]Basic Mathematics
Introduction to Complex Numbers
Martin Lavelle
The aim of this package is to provide a short study and self assessment programme for students who wish to become more familiar with complex numbers.
Copyright c 2001 mlavelle@plymouth.ac.uk Last Revision Date: June 11, 2004
Version 1.1
Table of Contents
1. The Square Root of Minus One! 2. Real, Imaginary and Complex Numbers 3. Adding and Subtracting Complex Numbers 4. Multiplying Complex Numbers 5. Complex Conjugation 6. Dividing Complex Numbers 7. Quiz on Complex Numbers
Solutions to Exercises Solutions to Quizzes
The full range of these packages and some instructions, should they be required, can be obtained from our web page Mathematics Support Materials.
Section 1: The Square Root of Minus One!
3
1. The Square Root of Minus One!
If we want to calculate the square root of a negative number, it rapidly
becomes clear that neither a positive or a negative number can do it.
E.g.,
-1 = ?1, since
12 = (-1)2 = +1 .
To find -1 we introduce a new quantity, i, defined to be such that
i2 = -1. (Note that engineers often use the notation j.)
Example 1
(a)
-25 = 5i
Since (5i)2 = 52 ? i2
= 25 ? (-1)
= -25 .
Section 2: Real, Imaginary and Complex Numbers
4
16
4
(b)
- =i
9
3
Since
4 (
i)2
=
16 ? (i2)
3
9
16 =- .
9
2. Real, Imaginary and Complex Numbers
Real numbers are the usual positive and negative numbers. If we multiply a real number by i, we call the result an imaginary number. Examples of imaginary numbers are: i, 3i and -i/2. If we add or subtract a real number and an imaginary number, the result is a complex number. We write a complex number as
z = a + ib where a and b are real numbers.
Section 3: Adding and Subtracting Complex Numbers
5
3. Adding and Subtracting Complex Numbers
If we want to add or subtract two complex numbers, z1 = a + ib and z2 = c + id, the rule is to add the real and imaginary parts separately:
z1 + z2 = a + ib + c + id = a + c + i(b + d) z1 - z2 = a + ib - c - id = a - c + i(b - d) Example 2
(a) (1 + i) + (3 + i) = 1 + 3 + i(1 + 1) = 4 + 2i
(b) (2 + 5i) - (1 - 4i) = 2 + 5i - 1 + 4i = 1 + 9i
Exercise 1. Add or subtract the following complex numbers. (Click on the green letters for the solutions.)
(a) (3 + 2i) + (3 + i)
(c)
(-1
+
3i)
+
1 2
(2
+
2i)
(b) (4 - 2i) - (3 - 2i)
(d)
1 3
(2
-
5i)
-
1 6
(8
-
2i)
Section 3: Adding and Subtracting Complex Numbers
6
Quiz To which of the following does the expression (4 - 3i) + (2 + 5i)
simplify?
(a) 6 - 8i (c) 1 + 7i
(b) 6 + 2i (d) 9 - i
Quiz To which of the following does the expression (3 - i) - (2 - 6i)
simplify?
(a) 3 - 9i (c) 1 - 5i
(b) 2 + 4i (d) 1 + 5i
Section 4: Multiplying Complex Numbers
7
4. Multiplying Complex Numbers
We multiply two complex numbers just as we would multiply expressions of the form (x + y) together (see the package on Brackets)
(a + ib)(c + id) = ac + a(id) + (ib)c + (ib)(id) = ac + iad + ibc - bd = ac - bd + i(ad + bc)
Example 3
(2 + 3i)(3 + 2i) = 2 ? 3 + 2 ? 2i + 3i ? 3 + 3i ? 2i = 6 + 4i + 9i - 6 = 13i
Section 4: Multiplying Complex Numbers
8
Exercise 2. Multiply the following complex numbers. (Click on the green letters for the solutions.)
(a) (3 + 2i)(3 + i) (c) (-1 + 3i)(2 + 2i)
(b) (4 - 2i)(3 - 2i) (d) (2 - 5i)(8 - 3i)
Quiz To which of the following does the expression (2 - i)(3 + 4i)
simplify?
(a) 5 + 4i (c) 10 + 5i
(b) 6 + 11i (d) 6 + i
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- list of mathematical symbols basic knowledge 101
- oxidation numbers rules
- a level mathematics p complex numbers
- complex numbers for high school students
- real number chart lamar state college orange
- understanding swr by example arrl home
- list of mathematical symbols by subject
- rational and irrational numbers wpmu dev
- chapter 5 complex numbers
- introduction to complex numbers
Related searches
- complex numbers khan academy
- complex numbers operations
- complex numbers calculator
- simplify complex numbers calculator
- adding complex numbers calculator
- multiplying complex numbers calculator
- complex numbers to polar form
- solve complex numbers calculator
- synthetic division complex numbers calculator
- power of complex numbers calculator
- complex numbers calculator with steps
- divide complex numbers calculator