We will discuss about the complex numbers formulae.

**1.** Definition of complex number: If an ordered pair (x, y)
of two real numbers x and y is represented by the symbol x + iy, where i = √-1,
then the order pair is called a complex number or an imaginary number. If z = x
+ iy then x is called the real part of the complex number z and y is called its
imaginary part.

**2.** Let z\(_{1}\) = p + iq and z\(_{2}\) = r + is be any two complex
numbers, then their sum z\(_{1}\) + z\(_{2}\) is defined as

z\(_{1}\) + z\(_{2}\) = (p + r) + i(q + s).

**3.** Let z\(_{1}\) = p + iq and z\(_{2}\) = r + is be any two complex
numbers, then the subtraction of z\(_{2}\) from z\(_{1}\) is defined as

z\(_{1}\) - z\(_{2}\) = z\(_{1}\) + (-z\(_{2}\))

= (p + iq) + (-r - is)

= (p - r) + i(q - s)

**4.** Let z\(_{1}\) = p + iq and z\(_{2}\) = r + is be two complex
numbers (p, q, r and s are real), then their product z\(_{1}\)z\(_{2}\) is
defined as

z\(_{1}\)z\(_{2}\) = (pr - qs) + i(ps + qr).

**5.** Division of a complex number z\(_{1}\) = p + iq by z\(_{2}\) = r +
is ≠ 0 is defined as

\(\frac{z_{1}}{z_{2}}\) = \(\frac{pr + qs}{\sqrt{r^{2} + s^{2}}}\) + i\(\frac{qr - ps}{\sqrt{r^{2} + s^{2}}}\)

**6.** In any two complex numbers, if only the sign of the
imaginary part differ then, they are known as complex conjugate of each other. If
x, y are real and i = √-1 then the complex numbers x + iy and x - iy are said
to be conjugate of each other; conjugate of complex number z is denoted by \(\overline{z}\).

**7.** Modulus of a
complex number z = x + iy, denoted by mod(z) or |z| or |x + iy|, is defined as
|z|[or mod z or |x + iy|] = + \(\sqrt{x^{2} + y^{2}}\) ,where a = Re(z), b = Im(z)

If z = x + iy then the positive root of (x\(^{2}\)+ y \(^{2}\)) is called the modulus or absolute value of z and is denoted by |z| or mod z. Thus, if z = x + iy then, |z| = \(\sqrt{x^{2} + y^{2}}\).

Again, if z = x + iy then the unique value of θ satisfying x = |z| cos θ, y = |z| sin θ and - π < θ ≤ π is called the principal value of argument (or amplitude) of z and is denoted by arg z or amp z. If the point p(z) in the Argand Diagram represents the complex number z = (x, y) = x + iy and agr z = θ then

(i) 0 < θ < \(\frac{π}{2}\) when P lies on the first quadrant;

(ii) \(\frac{π}{2}\) < θ < π when P lies on the second quadrant;

(iii) - π < θ < - \(\frac{π}{2}\) when P lies on the third quadrant;

(iv) - \(\frac{π}{2}\) < θ < 0 when P lies on the fourth quadrant.

**8.** z = r(cos θ + i sin θ) where r = |z| and θ = are z, -
π < θ < π, is called the modulus-amplitude form of the complex number z.

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

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

**11.** i = √-1; i\(^{2}\) = - 1; i\(^{3}\) = -i; i\(^{4}\) =
1. Any integral power of i is i or (-i) or 1.

**12.** |z\(_{1}\) + z\(_{2}\) | ≤|z\(_{1}\)| + |z\(_{2}\)|, for two complex numbers z\(_{1}\) and z\(_{2}\).

**13.** |z\(_{1}\)z\(_{2}\)| = |z\(_{1}\)| |z\(_{2}\)|, for two complex numbers z\(_{1}\) and z\(_{2}\).

**14.** |\(\frac{z_{1}}{z_{2}}\)| = \(\frac{|z_{1}|}{|z_{2}|}\), for two complex numbers z\(_{1}\) and z\(_{2}\).

**15. **(a) arg (z\(_{1}\)z\(_{2}\)) = arg z\(_{1}\) - agr z\(_{2}\) + m, for two complex numbers z\(_{1}\) and z\(_{2}\), Where m = 0
or, 2π or, (-2π).

(b) arg (\(\frac{z_{1}}{z_{2}}\)) = arg z\(_{1}\) - agr z\(_{2}\) + m, for two complex numbers z\(_{1}\) and z\(_{2}\), Where m = 0 or, 2π or, (-2π).

**16.** The sum of two
conjugate complex numbers is real.

**17.** The product of two
conjugate complex numbers is real.

**18. **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.

**19.** Cube roots of 1 are 1, ω, ω\(^{2}\) where

ω = \(\frac{-1 + \sqrt{3}i}{2}\) or, \(\frac{-1 - \sqrt{3}i}{2}\);

here ω and ω\(^{2}\) are called the imaginary cube roots of 1.

**20.** The multiplicative inverse of a non-zero complex z is equal
to its reciprocal and is represent as

\(\frac{Re(z)}{|z|^{2}}\) + i\(\frac{(-Im(z))}{|z|^{2}}\)= \(\frac{\overline{z}}{|z|^{2}}\)

**21.** If ω be an
imaginary cube root of unity then ω\(^{3}\) = 1 and 1 + ω + ω\(^{2}\) = 0.

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

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