Download Section 2.2 – Complex Numbers

Chapter 2. Section 2
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Section 2.2 – Complex Numbers
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What does it mean to have 9 ? Why is 9 = 3 ?
The inverse of the square function is the square root function. And 9 = 3 because 32 = 9
In the Real Number system, we know that a negative times a negative is always positive. So how
can we ever achieve squaring a number and getting a negative value?
This translates into the square root function as well, which is why we can't take the square root of a
negative value (like −9 ) in the Real Number system
But sometimes we want to investigate this possibility, so we have what is known as the Complex
Number system
The Number i is defined so that i 2 = −1 or i = −1 , so we can express square roots of negative
numbers according to this value
− p = i2 p = i2
p =i p
Complex Numbers:
• A complex number is a number of the form a +bi, where a and b are real number values. The
number a is said to be the real part and the number b is said to be the imaginary part
• What are the restrictions on a and b?
• Are complex numbers real numbers? Are real numbers complex numbers?
• If b is nonzero, then the complex number a + bi is also called an imaginary number
• Let's take a look at how the entire system maps out in what is known as a Venn diagram
• The real number system we investigated in Chapter R
• We now add complex numbers, imaginary numbers and pure imaginary numbers
C. Bellomo, revised 26-Sep-07
Chapter 2. Section 2
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Operations of Imaginary Numbers:
• The imaginary numbers obey the same laws as the real numbers, like commutative, associative and
distributive
• So we operate on them in much the same way (like we do binomials). We collect the real and
imaginary parts
• The only difference is we always replace i2 with −1
• Example. Simplify ( −6 − 5i ) + (9 + 2i )
( −6 + 9) + (2i − 5i ) = 3 − 3i
• Example. Simplify ( −3 − 4i ) − (8 − i )
( −3 − 4i ) + ( −8 + i ) = ( −3 − 8) + ( −4i + i ) = −11 − 3i
• Example. Simplify 3i (6 + 4i )
3i (6 + 4i ) = 3i (6) + 3i (4i ) = 18i + 12i 2 = −12 + 18i
• Example. Simplify (2 + 3i )(2 + 5i )
(2 + 3i )(2 + 5i ) = 2 ⋅ 2 + 2 ⋅ 5i + 3i ⋅ 2 + 3i ⋅ 5i = 4 + 16i − 15 = −11 + 16i
Complex Conjugates:
• The conjugate of a complex number a + bi is a − bi
• The product of a complex number and its' conjugate is a real number, i.e.
(a + bi )(a − bi ) = a 2 + bi − bi − b 2i 2 = a 2 + b 2
• The conjugate is used to simplify an expression. Somewhere along the way, it was decided that one
shouldn't have a complex number in the denominator of an expression (much the same way of not
having a square root in the denominator)
i
• Example. Simplify
2+i
i ⎛ 2 − i ⎞ 2i − i 2 1
= (1 + 2i )
⎜
⎟=
2 + i ⎝ 2 − i ⎠ 4 − i2 5
1− i
• Example. Simplify
(1 + i ) 2
1− i
1 − i −2i
−2 − 2i −1
=
=
(1 + i )
=
2
(1 + i )
2i −2i
4
2
FG IJ
H K
Powers of i:
• Note the cycle of powers of i: i 1 = i i 2 = −1 i 3 = −i i 4 = 1 i 5 = i...
• We can simplify every power of i into one of the values above
• Example. Simplify i24
i 24 = (i 2 )12 = ( −1)12 = 1
• Example. Simplify (2i)5
(2i ) 5 = 25 i 5 = 32 ⋅ i ⋅ i 4 = 32i (i 2 ) 2 = 32i
C. Bellomo, revised 26-Sep-07