Complex Number is a combination of both Real and Imaginary Numbers. In other words, Complex Numbers are defined as the numbers that are in the form of x+iy where x, y are real numbers and i =√-1.

z = x+iy here x is the real part of the Complex Number and is denoted by Re Z and y is called the Imaginary Part and is denoted as Im Z. In the later sections, you will find What is a Complex Number and Properties of Complex Numbers. We tried explaining each and every Property of Complex Number in detail with Proofs.

## What are Complex Numbers?

If x, y ∈ R, then ordered pair (x, y) = x + iy is called a complex number. It is denoted by z. Where x is the real part and is denoted as Re(z) and y is the imaginary part of the complex number and represented as Im(z).

(i) If Re(z) = x = 0, then the number z is a purely imaginary number

(ii) If Im(z) = y = 0 then the number z is a purely real number.

### Properties of Complex Numbers

**1. If x, y are two real numbers and x+iy =0 then x = 0 and y = 0**

Proof:

Since, x + iy = 0 = 0 + i0, thus by the definition of equality of two complex numbers we can say that, x = 0 and y = 0.

**2. If x, y, p, q are real and x + iy = p + iq then x = p and y = q**

Proof:

Given x + iy = p + iq

rearranging the equation we get x − p = -i(y − q)

⇒ (x − p)^{2} = i^{2} (y − q)^{2}

⇒ (x − p)^{2} + (y − q)^{2} = 0 (We know i^{2} = -1)……..(1)

Since x, y, p, q are real, and (x − p)^{2} and (y − q)^{2} are both non-negative. Equation (1) is satisfied if each square is separately zero.

Thus we can write the equation as follows

(x − p)^{2} = 0 or x = p and (y − q)^{2} = 0 or y = q.

**3. Similar to real numbers, the set of complex numbers also satisfy the commutative, associative, and distributive laws**

Proof:

If z_{1}, z_{2} and z_{3} be three complex numbers then,

z_{1} + z_{2} = z_{2} + z_{1} (commutative law of addition) and z_{1}. z_{2} = z_{2}. z_{1} (commutative law of multiplication)

(z_{1} + z_{2}) + z_{3} = z_{1} + (z_{2} + z_{3}) (associative law of addition) and (z_{1}. z_{2}) z3 = z_{1} (z_{2}. z_{3}) (associative law of multiplication)

z_{1}(z_{2} + z_{3}) = z_{1} z_{2} + z_{1} z_{3} (distributive law)

**4. Sum and Product of Two Conjugate Complex Quantities are both Real.**

Proof:

Consider z = x + iy is a complex number where x, y are real.

Then, the conjugate of z is = x − iy.

Now, z + \(\overline {z}\)= x + iy + x − iy = 2x, is real.

and z. \(\overline {z}\) = (x + iy)(x − iy) = x^{2} − i^{2}y^{2} = x^{2} + y^{2} is also real.

**5. For two complex quantities z _{1} and z_{2}, |z_{1}+ z_{2}| ≤ |z_{1} | + |z_{2} |**

Proof:

Let z_{1} = r_{1}(cosθ_{1} + isinθ_{1} ) and z_{2} = r_{2}(cosθ_{2} + isinθ_{2} ).

Hence |z_{1} | = r_{1} and |z_{2} | = r_{2}

Now

z_{1} + z_{2} = r_{1}(cosθ_{1}isinθ_{1}) + r_{2}(cosθ_{2} + isinθ_{2})

= (r_{1}(cosθ_{1}+ r_{2}cosθ_{2} )+ i(r_{1}sinθ_{1}+ r_{2}sinθ_{2})

Hence |z_{1}+ z_{2} | = √(r_{1}cosθ_{1}+ r_{2}cosθ_{2})_{2} + (r_{1}sinθ_{1}+ r_{2}sinθ_{2})_{2}

= √r_{1}2(cos_{2}θ_{1}+ sin2_{1}) + r_{2}2(cos2θ_{2}+ sin2θ_{2}) + 2r_{1}r_{2} (cosθ_{1} cosθ_{2}+ sinθ_{1} sinθ_{2})

= √r1_{2} + r2_{2} + 2r_{1}r_{2}cos (θ_{1}– θ_{2})

Now, |cos(θ_{1}– θ_{2})| ≤ 1

Hence |z_{1}+ z_{2}| ≤ √r1_{2} + r2_{2} + 2r_{1}r_{2} or |z_{1}+ z_{2} | ≤ |z_{1}| + |z_{2} |

**6. If the sum of two complex numbers is real and the product of two complex numbers is also real then the complex numbers are conjugate to each other.**

Proof:

Let us consider z_{1} = a + ib and z_{2} = c + id are two complex quantities (a, b, c, d and real and b ≠ 0, d ≠0).

As per the property,

z_{1} + z_{2} = a+ ib + c + id = (a + c) + i(b + d) is real.

Therefore, b + d = 0

⇒ d = -b

and,

z_{1}.z_{2} = (a + ib)(c + id) = (a + ib)(c +id) = (ac – bd) + i(ad + bc) is real.

Therefore, ad + bc = 0

⇒ -ab + bc = 0, (Since, d = -b)

⇒ b(c – a) = 0

⇒ c = a (Since, b ≠ 0)

Hence, z_{2} = c + id = a + i(-b) = a – ib

Thus, we can say that z_{1} and z_{2} are conjugate to each other.