The complex number ?is said to be in Cartesian form. There are, however, other ways to write a complex number, such as in modulus-argument (polar) form.

How do I write a complex number in modulus-argument (polar) form?

• The?Cartesian form?of a complex number, , is written in terms of its real part, , and its imaginary part,
• If we let ?and , then it is possible to write a complex number in terms of its modulus, , and its argument, , called the?modulus-argument (polar) form, given by...
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• It is usual to give arguments in the range
  • Negative arguments should be shown clearly, e.g. without simplifying ? to either
  • Occasionally you could be asked to give arguments in the range
• If a complex number is given in the form , then it is not currently in modulus-argument (polar) form due to the minus sign, but can be converted as follows…
  • By considering transformations of trigonometric functions, we see that ?and
  • Therefore ?can be written as , now in the correct form and indicating an argument of -
• To convert from modulus-argument (polar) form back to Cartesian form,?evaluate the real and imaginary parts
  • E.g. ?becomes

What are the rules for moduli and arguments under multiplication and division?

• When two complex numbers, ?and , are multiplied to give , their?moduli?are also?multiplied
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• When two complex numbers, ?and , are divided to give , their?moduli?are also?divided
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• When two complex numbers, ?and , are multiplied to give , their?arguments?are?added
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• When two complex numbers, and , are divided to give , their?arguments?are?subtracted
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How do I multiply complex numbers in modulus-argument (polar) form?

How do I divide complex numbers in modulus-argument (polar) form?