What is the magnitude of the complex number 3 β 2i?
In mathematics, a number of the form x + iy where x, y β R (set of real numbers) & i = β(-1) (also called iota) is called a complex number. Usually a complex number is denoted by z i.e. z = x + iy where x represents the real part of the complex number z denoted by Re(z) and y represents the imaginary part of complex number z denoted by Im(z).
Representation of z=x+iy on Complex Plane:
We usually use Complex Plane having cartesian co-ordinate system where the x-axis is the Real Axis representing the real part of the complex number and the y-axis is the Imaginary Axis representing the imaginary part of the complex number to study the geometric interpretation of complex numbers.
Magnitude of a Complex Number:
If z = x + iy is a complex number or a point in a complex plane, then the magnitude of a complex number z = x + iy denoted by |z| is the distance of the point z(x, y) from origin O(0, 0) in the complex plane. Magnitude of a complex number z=x+iy is defined as |z| = β(x2 + y2). Since distance is a scalar quantity, |z| β₯ 0 i.e. non-negative. Note that,
- Re(z) β€ |Re(z)| β€ |z|
- Im(z) β€ |Im(z)| β€ |z|
All the complex numbers having the same magnitude will lie on a circle having a center at the origin & radius r = |z|.
Some important properties of the magnitude of complex numbers:
If z1 and z2 are two complex numbers, then
- Magnitude over multiplication β’ |z1 Γ z2| = |z1| Γ |z2|
- Magnitude over division β’ |z1/z2| = |z1|/|z2| for z2 β 0
- Triangle Inequality β’ |z1 + z2| β€ |z1| + |z2|
- Law of Parallelogram β’ |z1 + z2|2 + |z1 β z2|2 = 2 Γ {|z1|2 + |z2|2}
Representation of |z| on Complex Plane:
What is the magnitude of the complex number 3-2i
Solution:
For complex number z = 3 β 2i,
The magnitude will be |z| = β(x2 + y2)
= β(32 + (-2)2)
= β13
Similar Problems
Question 1: What is the magnitude of the complex number 5 + 3i?
Solution:
For complex number z = 5 + 3i,
The magnitude will be |z| = β(x2 + y2)
= β(52 + 32) = β34
Question 2: What is the magnitude of the complex number -2 + i?
Solution:
For complex number z = -2 + i,
The magnitude will be |z| = β(x2+y2)
= β((-2)2+12)
= β5
Question 3: What is the magnitude of the complex number -5 + 2i?
Solution:
For complex number z = -5 + 2i,
The magnitude will be |z| = β(x2+y2)
= β((-5)2+22)
= β29