What is the magnitude of the complex number 3 – 2i?
In mathematics, a number of the form x + iy where x, y ∈ R (set of real numbers) & i = √(-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| = √(x2 + y2). Since distance is a scalar quantity, |z| ≥ 0 i.e. non-negative. Note that,
- Re(z) ≤ |Re(z)| ≤ |z|
- Im(z) ≤ |Im(z)| ≤ |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 ⇢ |z1 × z2| = |z1| × |z2|
- Magnitude over division ⇢ |z1/z2| = |z1|/|z2| for z2 ≠ 0
- Triangle Inequality ⇢ |z1 + z2| ≤ |z1| + |z2|
- Law of Parallelogram ⇢ |z1 + z2|2 + |z1 – z2|2 = 2 × {|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 – 2i,
The magnitude will be |z| = √(x2 + y2)
= √(32 + (-2)2)
= √13
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| = √(x2 + y2)
= √(52 + 32) = √34
Question 2: What is the magnitude of the complex number -2 + i?
Solution:
For complex number z = -2 + i,
The magnitude will be |z| = √(x2+y2)
= √((-2)2+12)
= √5
Question 3: What is the magnitude of the complex number -5 + 2i?
Solution:
For complex number z = -5 + 2i,
The magnitude will be |z| = √(x2+y2)
= √((-5)2+22)
= √29
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