Properties of Complex Numbers

We will discuss here about the different properties of complex numbers.

1. When a, b are real numbers and a + ib = 0 then a = 0, b = 0

Proof:

According to the property,

a + ib = 0 = 0 + i  0,

Therefore, from the definition of equality of two complex numbers we conclude that, x = 0 and y = 0.

2. When a, b, c and d are real numbers and a + ib = c + id then a = c and b = d.

Proof:

According to the property,

a + ib = c + id and a, b, c and d are real numbers.

Therefore, from the definition of equality of two complex numbers we conclude that, a = c and b = d.

3. For any three the set complex numbers z$$_{1}$$, z$$_{2}$$ and z$$_{3}$$ satisfies the commutative, associative and distributive laws.

(i) z$$_{1}$$ + z$$_{2}$$ = z$$_{2}$$ + z$$_{1}$$ (Commutative law for addition).

(ii) z$$_{1}$$ z$$_{2}$$ = z$$_{2}$$ z$$_{1}$$ (Commutative law for multiplication).

(iii) (z$$_{1}$$ + z$$_{2}$$) + z$$_{3}$$ = z$$_{1}$$ + (z$$_{2}$$ + z$$_{3}$$) (Associative law for addition)

(iv) (z$$_{1}$$z$$_{2}$$)z$$_{3}$$ = z$$_{1}$$(z$$_{2}$$z$$_{3}$$) (Associative law for multiplication)

(v) z$$_{1}$$(z$$_{1}$$ + z$$_{3}$$) = z$$_{1}$$z$$_{2}$$ + z$$_{1}$$z$$_{3}$$ (Distributive law).

4. The sum of two conjugate complex numbers is real.

Proof:

Let, z = a + ib (a, b are real numbers) be a complex number. Then, conjugate of z is $$\overline{z}$$ = a - ib.

Now, z + $$\overline{z}$$ = a + ib + a - ib = 2a, which is real.

5. The product of two conjugate complex numbers is real.

Proof:

Let, z = a + ib (a, b are real number) be a complex number. Then, conjugate of z is $$\overline{z}$$ = a - ib.

$$\overline{z}$$ = (a + ib)(a - ib) = a$$^{2}$$ - i$$^{2}$$b$$^{2}$$ = a$$^{2}$$ + b$$^{2}$$, (Since i$$^{2}$$ = -1), which is real.

Note: When z = a + ib then |z| = $$\sqrt{a^{2} + b^{2}}$$and, z$$\overline{z}$$ = a$$^{2}$$ + b$$^{2}$$

Hence, $$\sqrt{z\overline{z}}$$ = $$\sqrt{a^{2} + b^{2}}$$

Therefore, |z| = $$\sqrt{z\overline{z}}$$

Thus, modulus of any complex number is equal to the positive square root of the product of the complex number and its conjugate complex number.

6. When 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, z$$_{1}$$ = a + ib and z$$_{2}$$ = c + id be two complex quantities (a, b, c, d and real and b ≠ 0, d ≠0).

According to 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 = $$\overline{z_{1}}$$

Therefore, we conclude that z$$_{1}$$ and z$$_{2}$$ are conjugate to each other.

7. |z$$_{1}$$ + z$$_{2}$$| ≤ |z$$_{1}$$| + |z$$_{2}$$|, for two complex numbers z$$_{1}$$ and z$$_{2}$$.