FP1_Chapter_1 - After completing this chapter you should be...

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1 After completing this chapter you should be able to add, subtract, multiply and divide complex numbers find the modulus and argument of a complex number show complex numbers on an Argand diagram solve equations that have complex roots . The Aurora Borealis (Northern Lights) are part of the Earth’s electromagnetic field. Although complex numbers may seem to have few direct links with real-world quantities, there are areas of application in which the idea of a complex number is extremely useful. For example, the strength of an electromagnetic field, which has both an electric and a magnetic component, can be described by using a complex number. Other areas in which the mathematics of complex numbers is a valuable tool include signal processing, fluid dynamics and quantum mechanics. 1 Complex numbers Complex numbers
CHAPTER 1 2 1.1 You can use real and imaginary numbers. When solving a quadratic equation in Unit C1, you saw how the discriminant of the equation could be used to find out about the type of roots. For the equation a x 2 b x c 0, the discriminant is b 2 4 ac . If b 2 4 ac 0, there are two different real roots. If b 2 4 ac 0, there are two equal real roots. If b 2 4 ac 0, there are no real roots. In the case b 2 4 ac 0, the problem is that you reach a situation where you need to find the square root of a negative number, which is not ‘real’. To solve this problem, another type of number called an ‘imaginary number’ is used. The ‘imaginary number’ _____ ( 1) is called i (or sometimes j in electrical engineering), and sums of real and imaginary numbers, such as 3 2i, are known as complex numbers . A complex number is written in the form a bi . You can add and subtract complex numbers. _____ ( 1) i An imaginary number is a number of the form b i, where b is a real number ( b ). Example 1 Write ______ ( 36) in terms of i. ______ ( 36) _________ (36 1) ___ 36 ____ ( 1) 6i Example 2 Write ______ ( 28) in terms of i. ______ ( 28) _________ (28 1) ___ 28 ____ ( 1) __ 4 __ 7 ____ ( 1) 2 __ 7i or 2i __ 7 or (2 __ 7)i Example 3 Solve the equation x 2 9 0. x 2 9 x _____ ( 9) ________ (9 1) __ 9 ____ ( 1) 3i x 3i ( x 3i, x 3i) Note that just as x 2 9 has two roots 3 and 3, x 2 9 also has two roots 3i and 3i. This can be written as 2i __ 7 or (2 __ 7 )i to avoid confusion with 2 __ 7i .
Complex numbers 3 A complex number is a number of the form a b i, where a and b . For the complex number a b i, a is called the real part and b is called the imaginary part. The complete set of complex numbers is called . Example 4 Solve the equation x 2 6 x 25 0. Method 1 (Completing the square) x 2 6 x ( x 3) 2 9 x 2 6 x 25 ( x 3) 2 9 25 ( x 3) 2 16 ( x 3) 2 16 0 ( x 3) 2 16 x 3 _____ ( 16) 4i x 3 4i x 3 4i, x 3 4i Method 2 (Quadratic formula) x 6 ________________ (6 2 4 1 25) ______________________ 2 6 ______ ( 64) _____________ 2 ______ ( 64) 8i x 6 8i ________ 2 3 4i x 3 4i, x 3 4i In a complex number, the real part and the imaginary part cannot be combined to form a single term.

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