Complex numbers multiply and divide KUS objectives BAT

Complex numbers: multiply and divide KUS objectives BAT know how multiplying and dividing affects both the modulus and argument of the resulting complex number Starter: Use the trig addition formula to expand simplify

Notes To be able to do this you need to be able to use the addition formulas for sine and cosine

Notes 2 Multiplying a complex number z 1 by another complex number z 2, both in the modulus-argument form Rewrite Now you can expand the double bracket as you would with a quadratic Group terms using the identities to the left You can also factorise the ‘i’ out So when multiplying two complex numbers in the modulusargument form: Multiply the moduli Add the arguments together The form of the answer is the same

Notes 4 Dividing a complex number z 1 by another complex number z 2, both in the modulus-argument form Multiply to cancel terms on the denominator Multiply out Remove i 2 Rewrite (again!) Group real and complex Rewrite terms

WB 5 a) Express the following calculation in the form x + iy: Combine using one of the rules above Multiply the moduli Add the arguments Simplify terms Calculate the cos and sin parts (in terms of i where needed) Multiply out

The cos and sin terms must be added for this to work! Rewrite using the rules you saw in 3 A Combine using a rule from above Simplify Calculate the cos and sin parts Multiply out

WB 5 c) Express the following calculation in the form x + iy: Combine using one of the rules above Divide the moduli Subtract the arguments Simplify You can work out the sin and cos parts Multiply out

Notes 3 Multiplying a complex number z 1 by another complex number z 2, both in the exponential form Rewrite Remember you add the powers in this situation You can factorise the power You can see that in this form the process is essentially the same as for the modulus-argument form: Multiply the moduli together Add the arguments together The answer is in the same form

Notes 5 Dividing a complex number z 1 by another complex number z 2, both in the exponential form Rewrite terms The denominator can be written with a negative power Multiplying so add the powers Factorise the power You can see that in this form the process is essentially the same as for the modulus-argument form: Divide the moduli Subtract the arguments The answer is in the same form

KUS objectives BAT know how multiplying and dividing affects both the modulus and argument of the resulting complex number self-assess One thing learned is – One thing to improve is –

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