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.

z **∙** \(\overline{z}\) = (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\(_{2}\).

**11 and 12 Grade Math****From Properties of Complex Numbers** **to HOME PAGE**

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