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Some equations, including those deriving from real-world situations, have no real solutions. For example, up to this point, you have been unable to solve equations such as x” =—1. You know that the solutions are x=+J-1, but this is not a real number and is difficult to manipulate, In order to solve this problem, mathematicians denoted the square root of negative one by i. This means that the solutions of the equation x* = —1 can be written as +i. Although this number is known as ‘imaginary, it means that all polynomial equations do indeed have solutions. A good analogy is negative numbers: these can be hard to grasp in isolation. For example, the concept of ‘minus one apple’ is a difficult one, but it’s useful in sums such as 3 apples — 1 apple = 2 apples. In exactly the same way, imaginary numbers are essential in many calculations and have many real-world applications. ‘The imaginary number iis defined as i= /—1 You can solve the equation x* = 9 by square-rooting both sides to give x Then, because /—9 can be written as V9 fT, you can use the fact that j= va to give x=+3i Notice that once you have defined the square root of minus one, the square root of all other negative numbers can be written in terms of i For example, to solve the equation (x—3) =—5, you first square-root both sides to give x-3=+\-5, and since J-5 = V5i, this gives the solutions x= 345i Numbers with both real and imaginary parts are called complex numbers. Complex numbers can be written in the form a+bi | Key point} where a, be R. The set of complex numbers is denoted C Complex numbers can be added, subtracted and multiplied by a constant in the same way as algebraic expressions. Simplify the expression 3(4—7i)—2(3-2i) Multiply real and imaginary parts by the constant. 3(4-7/)-2(3-2i)=(12-21/)-(6-4/), ——_______ =6-17i Simplify real parts: 12 - 6 = 6 Simplify imaginary parts: 20 +4) =-17/ [-] Complex numbers 1 Properties and arithmetic 
