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.
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