Complex numbers There is no number which squares to make -1, so there is no ‘real’ answer! What is ? 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 The imaginary part of z, called Im z is 3 A complex number in the form is said to be in Cartesian form 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

Expand & simplify as usual, remembering that i2 = 1 Manipulation with complex numbers Techniques used with real numbers can still be applied with complex numbers: WB1 z = 5 – 3i, w = 2 + 2i Express in the form a + bi, where a and b are real constants, (a) z2 (b) a) Expand & simplify as usual, remembering that i2 = 1 b) 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 Im Re The complex number can be represented on an Argand diagram by the coordinates Eg Eg Eg Re The modulus of z, Eg The principal argument Eg is the angle from the positive real axis to in the range Eg Remember the definition of arg z

Im Re WB2 The complex numbers z1 and z2 are given by Find, showing your working, (a) in the form a + bi, where a and b are real, (b) the value of The modulus of is (c) the value of , giving your answer in radians to 2 decimal places. Im The principal argument is the angle from the positive real axis to in the range Re

Im Re Im Re WB3 z = 2 – 3i (a) Show that z2 = −5 −12i. Find, showing your working, (b) the value of z2, (c) the value of arg (z2), giving your answer in radians to 2 decimal places. Im Re (d) Show z and z2 on a single Argand diagram. Im Re

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

Im Re Im Re WB4 z1 = − 2 + i (a) Find the modulus of z1 (b) Find, in radians, the argument of z1 , giving your answer to 2 decimal places. Im Re The solutions to the quadratic equation z2 − 10z + 28 = 0 are z2 and z3 (c) Find z2 and z3 , 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 Im Re

WB6 Given that , where a and b are real constants, (a) find the value of a and the value of b. Comparing coefficients of x2 Comparing coefficients of x0 (b) Find the three roots of f(x) = 0. either or (c) Find the sum of the three roots of f (x) = 0. So sum of the three roots is -1

Problem solving with roots If a is a root of f(x) then is a factor In C2 you met the Factor Theorem: 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 z2 + 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

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

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 then Equating real parts: Equating imaginary parts:

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 x4 + 9 = 0 Equating real parts: Equating imaginary parts: Roots are

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

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

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

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

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