# Properties of Complex Numbers | Basic Algebraic Properties of Complex Numbers

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 =âˆš-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 âˆˆ 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 âˆ’ p = -i(y âˆ’ q)

â‡’ (x âˆ’ p)2 = i2 (y âˆ’ q)2

â‡’ (x âˆ’ p)2 + (y âˆ’ q)2 = 0 (We know i2 = -1)……..(1)

Since x, y, p, q are real, and (x âˆ’ p)2 and (y âˆ’ 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 âˆ’ p)2 = 0 or x = p and (y âˆ’ 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 z1, z2 and z3 be three complex numbers then,

z1 + z2 = z2 + z1 (commutative law of addition) and z1. z2 = z2. z1 (commutative law of multiplication)

(z1 + z2)Â + z3 = z1 + (z2 + z3) (associative law of addition) and (z1. z2) z3 = z1 (z2. z3) (associative law of multiplication)

z1(z2 + z3) = z1 z2 + z1 z3 (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 âˆ’ iy.

Now, z + $$\overline {z}$$= x + iy + x âˆ’ iy = 2x, is real.

and z. $$\overline {z}$$ = (x + iy)(x âˆ’ iy) = x2 âˆ’ i2y2 = x2 + y2 is also real.

5. For two complex quantities z1 and z2,Â  |z1+ z2| â‰¤ |z1 | + |z2 |

Proof:

Let z1 = r1(cosÎ¸1 + isinÎ¸1 ) and z2 = r2(cosÎ¸2 + isinÎ¸2 ).

Hence |z1 | = r1 and |z2 | = r2

Now

z1 + z2 = r1(cosÎ¸1isinÎ¸1) + r2(cosÎ¸2 + isinÎ¸2)

= (r1(cosÎ¸1+ r2cosÎ¸2 )+ i(r1sinÎ¸1+ r2sinÎ¸2)

Hence |z1+ z2 | = âˆš(r1cosÎ¸1+ r2cosÎ¸2)2 + (r1sinÎ¸1+ r2sinÎ¸2)2

= âˆšr12(cos2Î¸1+ sin21) + r22(cos2Î¸2+ sin2Î¸2) + 2r1r2 (cosÎ¸1 cosÎ¸2+ sinÎ¸1 sinÎ¸2)

= âˆšr12 + r22 + 2r1r2cos (Î¸1– Î¸2)

Now, |cos(Î¸1– Î¸2)| â‰¤ 1

Hence |z1+ z2| â‰¤ âˆšr12 + r22 + 2r1r2 or |z1+ z2 | â‰¤ |z1| + |z2 |

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 z1 = a + ib and z2 = c + id are two complex quantities (a, b, c, d and real and b â‰  0, d â‰ 0).

As per the property,

z1 + z2 = a+ ib + c + id = (a + c) + i(b + d) is real.

Therefore, b + d = 0

â‡’ d = -b

and,

z1.z2 = (a + ib)(c + id) = (a + ib)(c +id) = (ac â€“ bd) + i(ad + bc) is real.

Therefore, ad + bc = 0

â‡’ -ab + bc = 0, (Since, d = -b)

â‡’ b(c – a) = 0

â‡’ c = a (Since, b â‰  0)

Hence, z2 = c + id = a + i(-b) = a – ib

Thus, we can say that z1 and z2 are conjugate to each other.

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