A complex number in the form is said to be in Cartesian form Complex numbers What is ? There is no number which squares to make -1, so there is no ‘real’ answer! Mathematicians have realised that by defining the imaginary number, many previously unsolvable problems could be understood and explored. If, what is: A number with both a real part and an imaginary one is called a complex number Eg Complex numbers are often referred to as z, whereas real numbers are often referred to as x The real part of z, called Re z is 2 The imaginary part of z, called Im z is 3

Manipulation with complex numbers WB1 z = 5 – 3i, w = 2 + 2i Express in the form a + bi, where a and b are real constants, (a) z 2 (b) a) b) Techniques used with real numbers can still be applied with complex numbers: Expand & simplify as usual, remembering that i 2 = 1 An equivalent complex number with a real denominator can be found by multiplying by the complex conjugate of the denominator If then its complex conjugate is

Modulus and argument Re Im The complex number can be represented on an Argand diagram by the coordinates Eg is the angle from the positive real axis to in the range The principal argument The modulus of z, Eg Remember the definition of arg z

WB2 The complex numbers z 1 and z 2 are given by Find, showing your working, (b) the value of (c) the value of, giving your answer in radians to 2 decimal places. (a)(a) in the form a + bi, where a and b are real, The modulus of is Re Im is the angle from the positive real axis to in the range The principal argument

WB3z = 2 – 3i Find, showing your working, (b) the value of  z 2 , (c) the value of arg (z 2 ), giving your answer in radians to 2 decimal places. (d) Show z and z 2 on a single Argand diagram. (a) Show that z 2 = −5 −12i. Re Im Re Im

If then We get no real answers because the discriminant is less than zero This tells us the curve will have no intersections with the x-axis Complex roots In C1, you saw quadratic equations that had no roots. Eg Quadratic formula We can obtain complex roots though We could also obtain these roots by completing the square:

WB4 z 1 = − 2 + i (a) Find the modulus of z 1 (b) Find, in radians, the argument of z 1, giving your answer to 2 decimal places. The solutions to the quadratic equation z 2 − 10z + 28 = 0 are z 2 and z 3 (c) Find z 2 and z 3, giving your answers in the form p  i  q, where p and q are integers. (d) Show, on an Argand diagram, the points representing your complex numbers Re Im Re Im

WB6 Given that, where a and b are real constants, (c) Find the sum of the three roots of f (x) = 0. (b) Find the three roots of f(x) = 0. (a) find the value of a and the value of b. Comparing coefficients of x 2 Comparing coefficients of x 0 either or So sum of the three roots is -1

Problem solving with roots In C2 you met the Factor Theorem: If a is a root of f ( x ) then is a factor Eg Given that x = 3 is a root of the equation, (a) write down a factor of the equation, (b) Given that x = -2 is the other root, find the values of a and b is the other factor is the equation factorised expanding In FP1 you apply this method to complex roots…

Problem solving with complex roots We have seen that complex roots come in pairs: Eg This leads to the logical conclusion that if a complex number is a root of an equation, then so is its conjugate We can use this fact to find real quadratic factors of equations: WB5 Given that 2 – 4i is a root of the equation z 2 + pz + q = 0, where p and q are real constants, (a) write down the other root of the equation, (b) find the value of p and the value of q. Factor theorem: If a is a root of f ( x ) then is a factor

WB7 Given that 2 and 5 + 2i are roots of the equation (c) Show the three roots of this equation on a single Argand diagram. (a) write down the other complex root of the equation. (b) Find the value of c and the value of d. Re Im

Problem solving by equating real & imaginary parts Eg Given that where a and b are real, find their values Equating real parts: Equating imaginary parts:

WB8 Given that z = x + iy, find the value of x and the value of y such that where z* is the complex conjugate of z. z + 3iz* = −1 + 13i Equating real parts: Equating imaginary parts: then

Eg Find the square roots of 3 – 4i in the form a + ib, where a and b are real Equating real parts: Equating imaginary parts: as b real Square roots are -2 + i and 2 - i

Eg Find the roots of x = 0 Equating real parts: Equating imaginary parts: Roots are

Re Im Modulus-argument form of a complex number If and then known as the modulus-argument form of a complex number Eg express in the form From previously, so Eg express in the form and

The modulus & argument of a product Eg if and It can be shown that: This is easier than evaluating and then finding the modulus… It can also be shown that: Eg if and Re Im

It can be shown that: Eg if and The modulus & argument of a quotient This is much easier than evaluating and then finding the modulus… It can also be shown that: Eg if and From previously,

WB9 z = – 24 – 7i (a) Show z on an Argand diagram. (c) find the values of a and b (d) find the value of It is given that w = a + bi, a  ℝ, b  ℝ. (b) Calculate arg z, giving your answer in radians to 2 decimal places. Given also that and Re Im where and Modulus-argument form given

w is a root of. Find the values of a and b Manipulation with complex numbers Re Im Modulus and argument Complex roots Equating real & imaginary parts Find the values of p and q Equating real parts: Equating imaginary parts: Complex numbers Using: Also