Download Lecture notes for Section 7.8

Int. Alg. Notes
Section 7.8
Page 1 of 5
Section 7.8: The Complex Number System
Big Idea: When you take the square root of a positive number, you get a positive real number as an answer.
However, you can not take the square root of a negative number and get a real number for an answer, since
there is no real number such that when you square it, you get a negative number for an answer. The imaginary
unit was thus “invented” to allow for taking the square root of negative numbers.
Big Skill: You should be able to write the square root of any negative number using the imaginary unit, and
perform arithmetic with complex numbers.
Nonnegative Property of Real Numbers
For any real number a, a2  0.
What do we do if we want a number that, when we square it, gives a negative number for an answer?
We simply make up such a number!
Definition: The Imaginary Unit
The imaginary unit, denoted by the symbol i, is the number whose square root is –1. That is, i2 = –1.
Taking the square root of both sides, we can see that i  1
Evaluating Square Roots of negative Numbers
If N is a positive real number, then the principal square root of –N is calculated as:
 N  N  1  N
 1 
Ni
Practice:
25 
1.
2.
2 
3.
48 
4. Solve for x: x3 + 16x = 0.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 7.8
Page 2 of 5
Definition: Complex Numbers
Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit.
The number a is called the real part of the complex number, and
the number b is called the imaginary part of the complex number.
Note: Any combination of real and imaginary numbers can always be reduced to a complex number in the
above format (called the standard form).
Note: Any purely real number is really a complex number with 0 imaginary part: x = x + 0i.
Any purely imaginary number is really a complex number with 0 real part: yi = 0 + yi.
Practice:
1. 3  16 
2.
3.
15  75

5
36  22i  9  
Sum of Complex Numbers
To add complex numbers, add reals to reals, and add imaginaries to imaginaries:
(a + bi) + (c + di) = (a + c) + (b + d)i
Difference of Complex Numbers
To subtract complex numbers, subtract reals from reals, and subtract imaginaries from imaginaries:
(a + bi) – (c + di) = (a – c) + (b – d)i
Note: Adding or subtracting complex numbers is just like combining like terms.
Practice:
1.  3  4 j  i   2  2i  
2.
5.6  2.2i    3.2  9.8i  
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 7.8
Page 3 of 5
Product of Complex Numbers
To multiply complex numbers, use the Distributive Property (or the FOIL method), then collect like terms:
(a + bi)(c + di) = (ac – bd) + (ad + bc)i
Practice:
1. 2i  7 18i  
2.
3  5i  7  10i  
2
 2
2 
3. 

i 
2 
 2
4. Show that 1  3i is a solution of x2 + 4 = 2x.
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 7.8
Page 4 of 5
Definition: Complex Conjugate:
For a complex number a + bi, its conjugate is defined as a – bi.
Quotient of Complex Numbers
To divide complex numbers, multiply top and bottom by the complex conjugate of the denominator. Note that
the denominator can be simplified using the difference of squares formula.
a  bi a  bi c  di  ac  bd    bc  ad  i



c  di c  di c  di
c2  d 2
Practice:
1.
4

3  7i
2.
5  2i

3  2i
3.
3 1 i


2i 2  i
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Int. Alg. Notes
Section 7.8
Page 5 of 5
Powers of i: The powers of i follow a pattern that repeats every fourth power:
i0 = 1
i1 = i
i2 = -1
i3 = -i
i4 = 1
i5 = i
i6 = -1
i7 = -i
i8 = 1
i9 = i
To simplify a large power of i, write it as the product of a power that is the largest possible multiple of 4 and the
remaining power. The power that is a multiple of 4 will always be equal to 1…
Example: i9 = i8i1 = i
Practice:
1. i27 =
2. i38 =
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.