10.7 - 10.7 Complex Numbers Imaginary Numbers Previously,...

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Unformatted text preview: 10.7 Complex Numbers Imaginary Numbers Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”. Imaginary Unit The imaginary unit 1', is the number whose square is — 1. That is i2=—l and i=\/——1 Write the following with the inotation. V-32= _25= *‘V—121= Complex Numbers Real numbers and imaginary numbers are both subsets of a new set of numbers. Complex Numbers A complex number is a number that can be written in the form a + bi, where a and b are real numbers. Standard Form of Complex Numbers Complex numbers can be written in the form a + bi (called standard form), with both a and b as real numbers. a is a real number and bi would be an imaginary number. Ifb = 0, a + bi is a real number. If a = 0, a + bi is an imaginary number. Write each of the following in the form of a complex number in standard form a + bi. Examples: 6 4:.) P- P 4724: H3; :3 awe, :> .41 6+\/—25= 64—19% :) CwmlaLLX‘ WIDM Sum or Difference of Com lex Numbers If a + bi and c + dz' are complex numbers, then their sum is (a + bi) + (c + di) = (a + c) + (b + d)i Their difference is (a+bi)—(c+di)=(a—c)+(b-d)i Add or subtract the followin com lex numbers. Write the answer in standard form a + bi. 1. (1+6i)+(;—2;j ‘ {Hep—L6 4% 2. (8 + 2:) -_.(4i) geai Multiplying Complex Numbers The technique for multiplying complex numbers varies depending on whether the numbers are written as single term (either the real or imaginary component is missing) or two terms. Note that the produot rule for radicals does NOT apply for imaginary numbers. 1/_16. _16' Lii 5E 30 L1 Jig; J: Multiply the following complex numbers. 81' - 7i . 2. ’36 L 6t60 Multiply the following complex numbers. Write the answer in standard form a + bi. 5:14 — 7:) 55‘ + 2,0 I 20E #3 b” E L Lo ‘1 - 36‘ C'" D 2.0 i + 3 '9' Multiply the following complex numbers. Write the answer in standard form a + bi. (6W0 Lllr+LLtEwStIL~lilia ug.+3ifim;0 l+a+3i+wa 13444;; \l/ ‘L' Rr 3* Complex Coniugates The complex numbers (a + bi) and (a — bi) are complex conjugates of each other, and w+wmpwa=&+fi The conjugate of a + bi is a — bi. The conjugate ofa — bi is a + bi. The product of (a + bi) and (a — bi) is (a + bi)(a —- bi) a2 — abi + abi # b2? E—Een a2 + b2, which is a real number. Use complex conjugates ' ' the following complex numbers. Write the answer in standard form. + E , __i;__l- an +1ei-F%L +632 #14311 I M ?_ DiéidEthselfollowing complex numbers. I G "— q LL P fiP 1 123;" two—bi j by; 63%: M “305 QHEELLEE‘LJ 247 27 Patterns ofi TEE El 3g E... «I 35. i The powers recycle through each multiple of 4. -4k 1 = 1 Simplify each of the following powers. F 6 2— __F 1.; . 5‘1 ' 2, .53 L L t"? .25" l 2 Lara 50”" i ...
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This note was uploaded on 01/16/2012 for the course MATH 096 taught by Professor Jimcotter during the Fall '11 term at Truckee Meadows Community College.

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10.7 - 10.7 Complex Numbers Imaginary Numbers Previously,...

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