Complex numbers modulus argument KUS objectives BAT find

Complex numbers: modulus argument KUS objectives BAT find the modulus and argument of complex numbers Starter:

The modulus of a complex number is its magnitude – you have already seen how to calculate it by using Pythagoras’ Theorem y (Imaginary) z The argument of a complex number is the angle it makes with the positive real axis θ The argument is usually measured in radians It will be negative if the complex number is plotted below the horizontal axis θ z x (Real)

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

WB 8 a) Find, to two decimal places, the modulus and argument of z = 4 + 5 i Use Pythagoras’ Theorem to find r y (Imaginary) 5 i z 5 Use Trigonometry to find arg z θ -5 4 -5 i 5 x (Real)

WB 8 b) Find, to two decimal places, the modulus and argument of z = -2 + 4 i Use Pythagoras’ Theorem to find r z y (Imaginary) 5 i 4 2. 03 Use Trigonometry to find arg z θ -5 2 5 -5 i Subtract from π to find the required angle (remember π radians = 180°) x (Real)

WB 8 c) Find, to two decimal places, the modulus and argument of z = -3 - 3 i Use Pythagoras’ Theorem to find r y (Imaginary) 5 i Use Trigonometry to find arg z 3 -5 θ 5 3 z -5 i Subtract from π to find the required angle (remember π radians = 180°) As the angle is below the x -axis, its written as negative x (Real)

WB 10 z = 2 – 3 i a) Show that z 2 = − 5 − 12 i. b) Find, showing your working, the value of z 2 , c) Find the value of arg (z 2), giving your answer in radians to 2 decimal places Im Re

b) The modulus of is The principal argument Im is the angle from the positive real axis to in the range Re

WB 12 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. Im Re

modulus and argument Imaginary axis x Real axis A complex number has both modulus and argument Then and

The modulus & argument of a product It can be shown that: It can also be shown that: The modulus & argument of a quotient It can be shown that: It can also be shown that: : Challenge – try constructing a proof of either result

Im Re

From previously,

KUS objectives BAT find the modulus and argument of complex numbers self-assess One thing learned is – One thing to improve is –

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