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Complex Numbers: Understanding Real and Imaginary Numbers - Prof. Errin E. White, Study notes of Pre-Calculus

Complex numbers, explaining the difference between real and imaginary numbers. Real numbers include rational and irrational numbers, while complex numbers cannot be expressed as the ratio of two integers. Complex numbers are represented as a + bi, where a and b are real numbers. Adding and subtracting complex numbers, and the importance of remembering restrictions when dealing with radicals.

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COMPLEX NUMBERS 7.1
WHAT IS REAL?: Real numbers are the numbers that you have grown up with. You started
with counting or whole numbers: 1, 2, 3, 4, .... Then you probably learned about fractions 1
2,3
7, etc.
These are called rational numbers which comes from the word ratio because they can be constructed from
the ratio of two integers. Next, you probably learned about negative numbers (Rational numbers and
Integers can be positive or negative.) Finally, you found out about some irrational numbers. These num-
bers, like 2, and πare not rational ... they can NOT be written as the ratio of 2 integers. They can
be distinguished by the fact that their decimal expansions never repeat a pattern whereas a fraction like
1
3=.333333333333333333333333333333333333333333333333333333333333333 (forever) is rational.
NUMBER LINE LINK: One way of thinking of Real Numbers is that they are all the possible, tiny,
infinitesimal points on a number line. (No matter how close together you pick two points on the number
line, there is always at least one more in between them.)
WHAT ISN’T REAL: The thing that you were probably told was impossible (and rightly so in the
Real Numbers!) was to take the square root of a negative number. However, there is another whole sys-
tem of numbers, that contains the Real Numbers. This system is the COMPLEX NUMBERS, and in the
system of complex numbers square roots of negative numbers DO EXIST! But since they are NOT REAL
NUMBERS, the basic one 1 is defined as ifor imaginary. (Yes, you can claim that mathematicians are
now ”making up” things to make trouble for you. However, there are actually many applications of Complex
Numbers.)
COMPLEX NUMBER BASICS: The basic imaginary unit is i:
1 = iand i2=1
i
i= 1 This is natural and useful in Simplification and in Rationalization.
WARNING: idoes NOT equal 1
A complex number is any number that can be expressed in the form:
a+bi where aand bare Real Numbers
Examples: 2 3i,3
4+ 2i,i2 (Here, a= 0, and we write the iin front of the radical so that i
does not appear to be under the radical, which would be incorrect.)
ADDING & SUBTRACTING COMPLEX NUMBERS is a very natural thing: You just combine like terms.
Example: (2 + 4i)(5 7i) = 2 + 4i5+7i=3 + 11i
WHAT WE WILL USE LATER: In solving certain equations later in the course, we will get answers
that have square roots of negative numbers. We have to be able to correctly interpret these as Complex
Numbers.
Example: 18 = 3i2
Now keep in mind, in Notes #11 on Radicals, I mentioned to remember the restriction put on multiplying
radicals together when we got to Complex Numbers.... Now is the time:
1
Section 6.1
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COMPLEX NUMBERS 7.

WHAT IS REAL?: Real numbers are the numbers that you have grown up with. You started

with counting or whole numbers: 1, 2, 3, 4, .... Then you probably learned about fractions^1 2

,^3

, etc.

These are called rational numbers which comes from the word ratio because they can be constructed from the ratio of two integers. Next, you probably learned about negative numbers (Rational numbers and Integers can be positive or negative.) Finally, you found out about some irrational numbers. These num- bers, like

2, and π are not rational ... they can NOT be written as the ratio of 2 integers. They can be distinguished by the fact that their decimal expansions never repeat a pattern whereas a fraction like 1 3 = .333333333333333333333333333333333333333333333333333333333333333 (forever) is rational.

NUMBER LINE LINK: One way of thinking of Real Numbers is that they are all the possible, tiny, inf initesimal points on a number line. (No matter how close together you pick two points on the number line, there is always at least one more in between them.)

WHAT ISN’T REAL: The thing that you were probably told was impossible (and rightly so in the Real Numbers!) was to take the square root of a negative number. However, there is another whole sys- tem of numbers, that contains the Real Numbers. This system is the COMPLEX NUMBERS, and in the system of complex numbers square roots of negative numbers DO EXIST! But since they are NOT REAL NUMBERS, the basic one

−1 is defined as i for imaginary. (Yes, you can claim that mathematicians are now ”making up” things to make trouble for you. However, there are actually many applications of Complex Numbers.)

COMPLEX NUMBER BASICS: The basic imaginary unit is i:

√ −1 = i and i^2 = − 1

i i = 1 This is natural and useful in Simplification and in Rationalization.

WARNING: i does NOT equal − 1

A complex number is any number that can be expressed in the form: a + bi where a and b are Real Numbers

Examples: 2 − 3 i,

  • 2i, i

2 (Here, a = 0, and we write the i in front of the radical so that i

does not appear to be under the radical, which would be incorrect.)

ADDING & SUBTRACTING COMPLEX NUMBERS is a very natural thing: You just combine like terms. Example: (2 + 4i) − (5 − 7 i) = 2 + 4i − 5 + 7i = −3 + 11i

WHAT WE WILL USE LATER: In solving certain equations later in the course, we will get answers that have square roots of negative numbers. We have to be able to correctly interpret these as Complex Numbers. Example:

−18 = 3i

Now keep in mind, in Notes #11 on Radicals, I mentioned to remember the restriction put on multiplying radicals together when we got to Complex Numbers.... Now is the time:

1

Section 6.

REMEMBER RESTRICTIONS:

A

B =

AB ONLY if A ≥ 0 and B ≥ 0. So what do we do when A and/or B is negative? You use Order of Operations, which guides you to do what you can inside parentheses first. Oh, you don’t see the parentheses? These are some of those ”understood parentheses” .... Everything underneath a radical is ”held together” by that radical just like parentheses do... So we would do what is under the radical first. Follow the Examples: √ − 6

−8 = (i

6)(i

  1. = i^2

3 CORRECT!

BUT if we forgot the RESTRICTIONS above: √ − 6

3 WRONG!!!!!!!!!!.

WAY TO REMEMBER: Get the i′s out f irst, then you can deal with the other part of the radical problem in a normal manner... Just remember to simplify i^2 = −1. Also, I don’t mind if you ”skip steps”, as long as your work is correct and logical, so that I can follow it. However, do not do everything on scratch paper and just write down an answer. If you skip a step, that is for something you did ”in your head”; so I should be able to do the same.

MULTIPLYING WITH i: These problems are really done in a very natural manner. You just Dis- tribute and FOIL as needed, and combine like terms. Just remember that (i)(i) = i^2 = −1, so that every term that is multiplied by i^2 has to be rewritten with the proper new sign and combined with other terms accordingly. Example: (3 − 2 i)(4 + 5i) = 12 + 15i − 8 i − 10 i^2 = 12 + 7i − 10(−1) = 22 + 7i

”RATIONALIZING DENOMINATORS”: The book does not actually use the term ”rational- ize” here, but the idea is certainly fitting. The i represents the square root of −1 and since we are used to thinking of ”getting rid of” radicals from the denominator, it is the same idea. Technically, we will be putting the number into that standard form (a + bi). There are two types, just like there are two types of square roots that we have had to rationalize, and the methods are identical to the corresponding process in square roots. I will show an Example of each: 4 5 i

5 i

i i

4 i 5 i^2

4 i − 5

4 i 5 2 i 5 − 3 i

2 i 5 − 3 i

5 + 3i 5 + 3i

(2i)(5 + 3i) 52 − (3i)^2

10 i + 6i^2 25 − 9 i^2

10 i − 6 25 + 9

−6 + 10i 34

10 i 34

i

Notice that I made use of the CONJUGATE as I did in rationalizing binomial radicals. I made the fi- nal steps here to get the number simplified and into the proper f orm of a + bi.

Section 6.