Complex Numbers Complicated Complex does not mean complicated
Complex Numbers
Complicated ? ? Complex does not mean complicated. It means two types of numbers, real and imaginary, which together form a complex, just like a building complex (buildings joined together). 2
REAL NUMBERS & IMAGINARY NUMBERS
Real Numbers are numbers like: Nearly any number you can think of is a Real Number! 1 12. 38 − 0. 8625 3/4 √ 2 1998
Imaginary Numbers when squared give a negative result when we square a positive number , we get a positive result, and when we square a negative number, then also we get a positive result But Imagine, there is such a number, which when squared, gives a negative result Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
The "unit" imaginary number is i, which is the square root of − 1 i 2 = − 1 Examples of Imaginary Numbers: 3 i 1. 04 i − 2. 8 i 3 i/4 (√ 2)i 1998 i 1 12. 38 − 0. 862 5 3/4 √ 2 1998
STANDARD FORM OF A COMPLEX NUMBERA Complex Number is a combination of a Real Number and an Imaginary Number
SQUARE ROOT OF NEGATIVE INTEGERS -
Examples of complex numbers: Real Part a 2 20 Real Numbers: a + 0 i + + – Imaginary Numbers: 0 + bi Simplify: 1. = i 2. = i = 8 i 3. Imaginary Part bi 7 i 3 i = 3 i + = + i a + bi form = + i Simplify using the product property of radicals. = 4 + 5 i
To add or subtract complex numbers: 1. Write each complex number in the form a + bi. 2. Add or subtract the real parts of the complex numbers. 3. Add or subtract the imaginary parts of the complex numbers. (a + bi ) + (c + di ) = (a + c) + (b + d )i (a + bi ) – (c + di ) = (a – c) + (b – d )i
Examples: Add (11 + 5 i) + (8 – 2 i ) = (11 + 8) + (5 i – 2 i ) Group real and imaginary terms. = 19 + 3 i a + bi form Add (10 + ) + (21 – ) = (10 + i ) + (21 – i ) i = = (10 + 21) + (i – i ) Group real and imaginary terms. = 31 a + bi form
Examples: Subtract: (– 21 + 3 i ) – (7 – 9 i) = (– 21 – 7) + [(3 – (– 9)]i = (– 21 – 7) + (3 i + 9 i) = – 28 + 12 i Group real and imaginary terms. a + bi form Subtract: (11 + ) – (6 + ) = (11 + i ) – (6 + i ) Group real and = (11 – 6) + [ – ]i imaginary terms. = (11 – 6) + [ 4 – 3]i = 5 + i a + bi form
The product of two complex numbers is defined as: (a + bi)(c + di ) = (ac – bd ) + (ad + bc)i 1. Use the FOIL ( First. . Outer. . Inner. . Last ) method to find the product. 2. Replace i 2 by – 1. 3. Write the answer in the form a + bi.
Examples: 1. = i i = 5 i 2 = 5 (– 1) = – 5 2. 7 i (11– 5 i) = 77 i – 35 i 2 = 77 i – 35 (– 1) = 35 + 77 i 3. (2 + 3 i)(6 – 7 i ) = 12 – 14 i + 18 i – 21 i 2 = 12 + 4 i – 21(– 1) = 12 + 4 i + 21 = 33 + 4 i
The complex numbers a + bi and a - bi are called conjugates. 15
The product of conjugates is the real number a 2 + b 2. (a + bi) (a – bi) = a 2 – b 2 i 2 = a 2 – b 2(– 1) = a 2 + b 2 Example: (5 + 2 i) (5 – 2 i) = (52 – 4 i 2) = 25 – 4 (– 1) = 29
Dividing Complex Numbers A rational expression, containing one or more complex numbers, is in simplest form when there are no imaginary numbers remaining in the denominator. Example: Multiply the expression by . – 1 Replace i 2 by – 1 and simplify. Write the answer in the form a + bi.
Simplify: Multiply the numerator and denominator by the conjugate of 2 + i. In 2 + i, a = 2 and b = 1. a 2 + b 2 = 22 + 12 – 1 Replace i 2 by – 1 and simplify. Write the answer in the form a + bi.
The Mandelbrot Set The beautiful Mandelbrot Set (pictured here) is based on Complex Numbers. It is a plot of what happens when we take the simple equation z 2+c (both complex numbers) and feed the result back into z time and time again. The color shows how fast z 2+c grows, and black means it stays within a certain range. Here is an image made by zooming into the Mandelbrot set
HOME ASSIGNMENT * Express in the form of a + ib (i) ( 5 – 3 i ) ( 5 + 4 i ) (ii) i ( 8 – 3 i ) ( 5 i ) (iii) 3( 7 + i 7 ) + i (7 + i 7 ) ( iv) (1 – i) – ( – 1 + i 6 ) * Solve each of the following equations: • 2 x²+ x + 1 • 3 x² +1 = 0
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