Transcript MCR3U - Santowski OCF.1.7 - Operations With Complex Numbers 1

OCF.1.7 - Operations With Complex Numbers
MCR3U - Santowski
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(A) Review
• A complex number is a number that has two components: a real part and
an imaginary part (ex = 2 + 3i )
• In general, we can write any complex number in the form of a + bi. We
designate any complex number by the letter z. As such z = a + bi.
• Complex numbers have conjugates which means that the two complex
numbers have the same real component and the imaginary components are
“negative opposites”. If we designate z = a + bi, then we designate the
conjugate as z = a - bi
• The same algebra rules that we learned for polynomials apply for complex
numbers - the concepts of like terms and the distributive rule apply. For
example, the terms 7i and i2 are like terms; the terms 6 and 6i are unlike
terms
• Remember that i2 = -1 and that i = (-1)
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(B) Operations With Complex Numbers
• (1) Addition and Subtracting
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ex 1.
ex 2.
ex 3.
ex 4.
Add (5 - 2i) and (-3 + 7i)
Find the sum of (-4 - 3i) and its conjugate
Simplify (7 - 4i) - (2 + 3i)
Subtract 3 - 5i from its conjugate
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(B) Operations With Complex Numbers
• (2) Multiplying
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ex 1. Simplify (i) i7 (ii) (i2)5
(iii) i75
ex 2. Multiply 4 + 3i by 2 - i
ex 3. Expand and simplify (-3 - 2i)2
ex 4. Find the product of 5 - 2i and its conjugate
ex 5. Find the product of every complex number and its
conjugate
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(B) Operations With Complex Numbers
• (3) Dividing
• A prerequisite skill is the idea of “rationalizing the denominator”
- in other words, we have something in the denominator that we
can algebraically “remove” or change.
• Specifically, we do not want a term containing i in the
denominator, so we must “remove it” using algebraic concepts
(recall i2 = -1 and recall the product of conjugates)
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(B) Operations With Complex Numbers – Examples with Division
• ex 1. Simplify 5/2i
• 5/2i = 5/2i * i/i multiply the fraction by i/i  why?
• 5i/2i2 = 5i/(2(-1)) = 5i/-2 = -2.5i or -2 ½ i
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ex 2. Simplify (2 + 3i)/(1 – 2i)
Multiply the “fraction” by the conjugate of its denominator  why??
= (2 + 3i) / (1 – 2i) * (1 + 2i)/(1 + 2i)
= (2 + 3i)(1 + 2i) / (1 + 2i)(1 – 2i)
= (2 + 3i + 4i + 6i2) / (1 + 2i – 2i – 4i2)
= (2 + 7i – 6) / (1 + 4)
= (-4 + 7i) / 5
Or -4/5 + 7i/5
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(B) Operations With Complex Numbers – Examples with
Substitution
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Evaluate f(2 - i) if f(x) = 2x2 - 8x - 2
f(2 - i) = 2(2 - i)(2 - i) - 8(2 - i) - 2
f(2 - i) = 2(4 - 4i + i2) - 16 + 8i - 2
f(2 - i) = 8 - 8i + 2(-1) - 18 + 8i
f(2 - i) = -12
• which means that 2 - i is not a factor of f(x)
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(C) Web Links
• Link #1 :
http://www.uncwil.edu/courses/mat111hb/Izs
/complex/complex.html#Content then
connect to Complex arithmetic
• Link #2 :
http://www.purplemath.com/modules/complex
2.htm from PurpleMath
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(D) Homework
• Handout from MHR, page 150, Q1-7 eol
• Nelson text, p336, Q1,3,5,6-10,13
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