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The modulus of a complex number is another important attribute of complex numbers. The modulus allows us to disregard the distinction between the real and imaginary parts, and instead use the coefficients, a and b, to measure the overall size of a complex number z.
Complex Modulus
The complex modulus ∣z∣ is a measure of the size of a complex number:
Example
Given that z=3+4i, find ∣z∣.
Discussion
What do you think the complex modulus ∣z∣ represents? Think of formulas you already know that are similar to the one for ∣z∣. Why might we be interested in calculating the value of ∣z∣?
The complex modulus
for z=x+iy is defined by
This has exactly the same form as the distance formula in the Cartesian plane: the distance from (0,0) to (x,y) is
Hence ∣z∣ represents the distance of the point corresponding to z from the origin in the Argand diagram.
We might want to calculate ∣z∣ because it tells us how “large” or “far out” the complex number is.
We can visualize this relationship on the complex plane.
Modulus on the complex plane
On the complex plane, z=a+bi has coordinates (a,b). Therefore
represents the distance of z from the origin:
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Exercise
z is a complex number with z=ki+8, ∣z∣=35k. It is given that k≥0.
Find k.
Select the correct option
Discussion
Let z=1−3i.
Find the values of
zz∗
∣z∣2
What do you notice? Why do you think this is the case?
The conjugate of z=1−3i is
Hence
In fact for any complex number z=a+ib,
while by definition ∣z∣=√a2+b2, so ∣z∣2=a2+b2.
Thus multiplying z by its conjugate always gives the square of its distance from the origin, explaining why zz∗=∣z∣2 in general.
zz*=|z|²
Notice that
The modulus of a complex number has other properties which are similar to those of the conjugate.
Properties of complex modulus
The following properties apply for the complex modulus:
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