1After completing this chapter you should be able to •add, subtract, multiply and divide complex numbers•find the modulusand argumentof 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.1Complex numbersComplex numbers
CHAPTER121.1 You can use real and imaginary numbers. When solving a quadratic equation in Unit C1, you saw how the discriminantof the equation could be used to find out about the type of roots.For the equation ax2bxc0, the discriminant is b24ac.If b24ac0, there are two different real roots.If b24ac0, there are two equal real roots.If b24ac0, there are no real roots.In the case b24ac0, 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 abi.You can add and subtract complex numbers.√_____(1)iAn imaginary number is a number of the form bi, where bis a real number (b ).Example 1Write √______(36)in terms of i.√______(36) √_________(36 1) √___36 √____(1) 6iExample 2Write √______(28)in terms of i.√______(28) √_________(28 1) √___28 √____(1) √__4 √__7 √____(1) 2 √__7i or 2i √__7 or (2 √__7)iExample 3Solve the equation x29 0.x29x√_____(9) √________(9 1) √__9 √____(1) 3ix3i (x3i, x3i)Note that just as x29has two roots 3 and −3, x29 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 numbers3A complex number is a number of the form abi, where a and b .For the complex number abi, ais called the realpartand bis called the imaginary part.The complete set of complex numbers is called .Example 4Solve the equation x26x25 0.Method 1 (Completing the square)x2 6x(x3)29x2 6x25 (x3)29 25 (x3)216(x3)216 0(x3)216x3√_____(16) 4ix3 4ix3 4i, x3 4iMethod 2(Quadratic formula)x6 √________________(624 1 25) ______________________26 √______(64) _____________2√______(64) 8ix6 8i________23 4ix3 4i, x3 4iIn a complex number, the real part and the imaginary part cannot be combined to form a single term.