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Modulus & Argument (DP IB Maths: AI HL)
Revision Note
Modulus & Argument
How do I find the modulus of a complex number?
- The modulus of a complex number is its distance from the origin when plotted on an Argand diagram
- The modulus of is written
- If , then we can use Pythagoras to show…
- A modulus is never negative
What features should I know about the modulus of a complex number?
- the modulus is related to the complex conjugate by…
- This is because
- In general,
- e.g. both and have a modulus of 5, but simplifies to 8i which has a modulus of 8
How do I find the argument of a complex number?
- The argument of a complex number is the angle that it makes on an Argand diagram
- The angle must be taken from the positive real axis
- The angle must be in a counter-clockwise direction
- Arguments are measured in radians
- They can be given exact in terms of
- The argument of is written
- Arguments can be calculated using right-angled trigonometry
- This involves using the tan ratio plus a sketch to decide whether it is positive/negative and acute/obtuse
What features should I know about the argument of a complex number?
- Arguments are usually given in the range
- Negative arguments are for complex numbers in the third and fourth quadrants
- Occasionally you could be asked to give arguments in the range
- The question will make it clear which range to use
- The argument of zero, is undefined (no angle can be drawn)
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
- When two complex numbers, and , are divided to give , their moduli are also divided
- When two complex numbers, and , are multiplied to give , their arguments are added
- When two complex numbers, and , are divided to give , their arguments are subtracted
Examiner Tip
- Always draw a quick sketch to help you see what quadrant the complex number lies in when working out an argument
- Look for the range of values within which you should give your argument
- If it is then you may need to measure it in the negative direction
- If it is then you will always measure in the positive direction (counter - clockwise)
Worked example
a)
Find the modulus and argument of
b)
Find the modulus and argument of
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