Download Section10.7

10.7 Complex Numbers
Up to this point, we have not been able to take the square root of a negative
number. If the radicand is negative, we have to stop and say the expression is
“not a real number” or it is “undefined.” This problem with the set of real
numbers can be overcome by coming up with (inventing) a new number that
does allow us to take the square root of negative numbers.
We define the new number, called the imaginary unit, 𝒊, to be the principal
square root of −1.
Definition of 𝒊
Thus, 𝑖 2 = (√−1)
𝑖 2 = −1
𝑖 = √−1
2
With this definition and the product rule for square roots, we can find the
square root of any negative number in terms of 𝑖.
√−5 = √−1 ∙ 5 = √−1 ∙ √5 = 𝑖√5
√−36 = √−1 ∙ 36 = √−1 ∙ √36 = 𝑖 √36 = 6𝑖
Square Root of Any Negative Number
√−𝑏 = √−1 ∙ 𝑏 = 𝑖√𝑏 where 𝑏 is any positive real number
Examples:
√−9 = 𝑖√9 = 3𝑖
√−
9
16
= 𝑖√
9
16
3
= 𝑖
4
√−6 = 𝑖 √6
√−8 = 𝑖 √8 = 2𝑖√2
√−80 = 𝑖 √80 = 𝑖√16 ∙ 5 = 4𝑖√5
Complex Numbers
The numbers in our examples are not real numbers. They are a new type of
number called imaginary numbers or just imaginaries. We now form a new
set of numbers, the set of complex numbers, that consists of two mutually
exclusive sets: the set of real numbers and the set of imaginary numbers.
See the diagram at the top of page 720.
Standard Form of a Complex Number
A complex number is written in the standard form of
𝒂 + 𝒃𝒊 where 𝑎 and 𝑏 are real numbers
𝑎 is the 𝐫𝐞𝐚𝐥 𝐩𝐚𝐫𝐭 and 𝑏 is the 𝐢𝐦𝐚𝐠𝐢𝐧𝐚𝐫𝐲 𝐩𝐚𝐫𝐭 of the complex number
 if 𝑏 = 0 then 𝑎 + 0𝑖 = 𝑎 is a 𝐫𝐞𝐚𝐥 𝐧𝐮𝐦𝐛𝐞𝐫
2
2
6 + 0𝑖 = 6
+ 0𝑖 =
√5 + 0𝑖 = √5
3
3
 if 𝑎 ≠ 0 and 𝑏 ≠ 0 then 𝑎 + 𝑏𝑖 is an 𝐢𝐦𝐚𝐠𝐢𝐧𝐚𝐫𝐲 𝐧𝐮𝐦𝐛𝐞𝐫
1
3
3 + 2𝑖
4 − 7𝑖
+ 𝑖
2
4
 if 𝑎 = 0 and 𝑏 ≠ 0 then 0 + 𝑏𝑖 = 𝑏𝑖 is an imaginary number
called a 𝐩𝐮𝐫𝐞 𝐢𝐦𝐚𝐠𝐢𝐧𝐚𝐫𝐲
0 + 2𝑖 = 2𝑖
0 − 8𝑖 = −8𝑖
0 + 0.7𝑖 = 0.7𝑖
--------------------------------------------------------------------------------------------Operations with Complex Numbers
Adding and Subtracting
We add and subtract complex numbers by combining the real parts and
combining the imaginary parts.
(2 + 3𝑖) + (−4 + 5𝑖) = (2 + (−4)) + (3𝑖 + 5𝑖) = −2 + 8𝑖
--------------------------------------------------------------------------------------------(5 − 11𝑖) + (7 + 4𝑖) = (5 + 7) + (−11𝑖 + 4𝑖) = 12 − 7𝑖
--------------------------------------------------------------------------------------------(−5 + 7𝑖)— (11 − 6𝑖) = −5 + 7𝑖 − 11 + 6𝑖 =
(−5 − 11) + (7𝑖 + 6𝑖) = −16 + 13𝑖
--------------------------------------------------------------------------------------------6 − (2 + 𝑖) = 6 − 2 − 𝑖 = (6 − 2) − 𝑖 = 4 − 𝑖
--------------------------------------------------------------------------------------------−2𝑖 − (3 − 9𝑖) = −2𝑖 − 3 + 9𝑖 = −3 + (−2𝑖 + 9𝑖) = −3 + 7𝑖
Multiplying
When multiplying complex numbers, you may get a factor of 𝑖 2 .
When this occurs, remember to replace 𝑖 2 with − 1 and continue
simplifying.
2(3 − 4𝑖) = 2 ∙ 3 − 2 ∙ 4𝑖 = 6 − 8𝑖
--------------------------------------------------------------------------------------------3𝑖(6 + 𝑖) = 3𝑖 ∙ 6 + 3𝑖 ∙ 𝑖 = 18𝑖 + 3𝑖 2 = 18𝑖 + 3(−1) =
18𝑖 − 3 = −3 + 18𝑖
--------------------------------------------------------------------------------------------(−5 + 2𝑖)(7 − 6𝑖) = −5 ∙ 7 + 5 ∙ 6𝑖 + 2𝑖 ∙ 7 − 2𝑖 ∙ 6𝑖 =
−35 + 30𝑖 + 14𝑖 − 12𝑖 2 = −35 + 44𝑖 − 12(−1) =
−35 + 44𝑖 + 12 = −23 + 44𝑖
Dividing
Divisor is a Monomial and Real
Use the distributive property and simplify the division in each term.
3+9𝑖
3
3
=3+
9𝑖
3
= 1 + 3𝑖
Divisor is a Monomial and Imaginary
Multiply the numerator and denominator by i then use the distributive
property and simplify the division in each term.
−2+4𝑖
5𝑖
=
(−2+4𝑖)𝑖
5𝑖∙𝑖
=
−2𝑖+4𝑖 2
5𝑖 2
=
−4−2𝑖
−5
=
−4
−5
+
−2
−5
4
2
5
5
𝑖= + 𝑖
Divisor is a Binomial of the form a + bi
Multiply the numerator and denominator by the complex conjugate of the
denominator.
𝒂 + 𝒃𝒊 and 𝒂 − 𝒃𝒊 are complex conjugates.
When complex conjugates are multiplied together, the result is a real number.
(𝑎 + 𝑏𝑖)(𝑎 − 𝑏𝑖) = 𝑎2 − (𝑏𝑖)2 = 𝑎2 − 𝑏 2 𝑖 2 = 𝑎2 − 𝑏 2 (−1) = 𝑎2 + 𝑏 2
2−3𝑖
4+𝑖
=
(2−3𝑖)(4−𝑖)
(4+𝑖)(4−𝑖)
=
8−2𝑖−12𝑖+3𝑖 2
4 2 +12
=
5−14𝑖
17
=
5
17
−
14
17
𝑖
--------------------------------------------------------------------------------------------Powers of i
The powers of i are cyclical. They repeat in a cycle of 4.
𝑖1 = 𝑖
𝑖 2 = −1
𝑖 3 = 𝑖 2 ∙ 𝑖 = −1 ∙ 𝑖 = −𝑖
𝑖 4 = 𝑖 2 ∙ 𝑖 2 = (−1)(−1) = 1
𝑖 5 = 𝑖 4 ∙ 𝑖 = 1𝑖 = 𝑖
𝑖 7 = 𝑖 4 ∙ 𝑖 3 = 1 ∙ −𝑖 = −𝑖
𝑖 6 = 𝑖 4 ∙ 𝑖 2 = 1 ∙ −1 = −1
𝑖8 = 𝑖4 ∙ 𝑖4 = 1 ∙ 1 = 1
To simplify powers of i, divide the exponent by 4.
 If the remainder is 1, the value of the expression is 𝑖.
 If the remainder is 2, the value of the expression is −1.
 If the remainder is 3, the value of the expression is −𝑖.
 If the remainder is 0, the value of the expression is 1.
𝑖 23
23 ÷ 4 = 5 remainder 3
𝑖 23 = −𝑖
𝑖 81
81 ÷ 4 = 20 remainder 1
𝑖 81 = 𝑖
NOTE: On your calculator,
1⁄
a remainder of 1 will show as the decimal .25
4
2⁄ = 1⁄
a remainder of 2 will show as the decimal .5
4
2
3
a remainder of 3 will show as the decimal .75
⁄4
a remainder of 0 will show a whole number answer