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
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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,i√2 (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 + 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√2
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:
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Section 6.1
