Carson Elementary And Intermediate Algebra 3e

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Carson Elementary and Intermediate Algebra 3e
Section 10.7
Objectives
1.
Write imaginary numbers using i.
2.
Perform arithmetic operations with complex numbers.
3.
Raise i to powers.
Vocabulary
Imaginary unit
2
The number represented by i, where i =
1
and i
= -1.
Imaginary number
A number that can be expressed in the form bi, where b is a real number and i is the
imaginary unit.
Complex number
A number that can be expressed in the form a + bi, where a and b are real number and i
is the imaginary unit.
The complex conjugate of a complex number a + bi is a – bi.
Complex
conjugates
Prior Knowledge
Simplifying radicals.
New Concepts
1. Write imaginary numbers using i.
A whole group of problems exists that cannot be solved in the real number system. These problems all
involve the square root of -1. If you look at the history of the development of mathematics, each time a new
situation arose, a new number system was developed to meet the need. The natural numbers were sufficient
to count animals in herds but then some way was needed to represent no animals and a loss of animals.
Hence the introduction of whole numbers and integers. Then some way to represent part of a whole was
needed so the rational numbers were developed. Now, in your learning, there is a need to work with
problems that involve the square root of -1, so another new number system is introduced - the complex
numbers.
By definition we create a group of numbers called imaginary numbers such that
i
1
2
i
1
The second equation is consistent with the work we did earlier with radicals: when a radical is squared, the
result is the radicand.
We can now represent square roots of negative numbers using imaginary numbers. All of the other radical
skills that you have learned still apply.
Example 1:
49
1
49
i
7
i 7
Example 2:
11
1
11
i
11
Example 3:
20
1
20
i
20
i
4
5
i 2
5
Once the negative is removed, the remaining radical can be simplified as we did earlier.
V. Zabrocki 2011
page 1

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