# 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| = √(x^{2 }+ y^{2}). 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 z_{1} and z_{2 }are two complex numbers, then

**Magnitude over multiplication ⇢**|z_{1}× z_{2}| = |z_{1}| × |z_{2}|**Magnitude over division ⇢**|z_{1}/z_{2}| = |z_{1}|/|z_{2}| for z_{2}≠ 0**Triangle Inequality ⇢**|z_{1}+ z_{2}| ≤ |z_{1}| + |z_{2}|**Law of Parallelogram ⇢**|z_{1}+ z_{2}|^{2 }+ |z_{1}– z_{2}|^{2 }= 2 × {|z_{1}|^{2 }+ |z_{2}|^{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| = √(x

^{2 }+ y^{2})= √(3

^{2 }+ (-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| = √(x

^{2 }+ y^{2})= √(5

^{2 }+ 3^{2}) = √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| = √(x

^{2}+y^{2})= √((-2)

^{2}+1^{2})= √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| = √(x

^{2}+y^{2})= √((-5)

^{2}+2^{2})= √29