Learning these TS Inter 2nd Year Maths 2A Formulas Chapter 1 Complex Numbers will help students to solve mathematical problems quickly.

## TS Inter 2nd Year Maths 2A Complex Numbers Formulas

→ A complex number is an ordered pair of real numbers. It is denoted by (a, b); a ∈ R, b ∈ R.

z = a + ib

Re(z) = a ; Im(z) = b

→ Two complex numbers z_{1} = a + ib and z_{2} = c + id are said to be equal if a = c, b = d.

→ Algebra of complex numbers

(a) z = z_{1} + z_{2} = (a + c) + i (b + d)

(b) z = z_{1} – z_{2} = (a – c) + i (b – d)

(c) z = z_{1}/z_{2} = \(\frac{a c+b d}{c^2+d^2}+\frac{i(b c-a d)}{c^2+d^2}\)

(d) z = z_{1} . z_{2} = (ac – bd) + i (ad + be)

→ If z = a + ib, then conjugate of complex number is z̅ = a – ib

z . z̅ = a^{2} + b^{2}

→ If z = a + ib, then modulus of z is represented by |z| = \(\sqrt{a^2+b^2}\)

→ Any real number θ satisfy the equation x = r cos θ; y = r sin θ.

→ Arg z = tan^{-1}\(\frac{{Im}(z)}{{Re}(z)}\) = tan^{-1}\(\frac{y}{x}\), -π < Arg z < π

(a) Arg (z_{1}. z_{2}) = Arg z_{1} + Arg z_{2}+ nπ for some π ∈ {-1,0,1}.

(b) Arg (z_{1}/z_{2}) = Arg z_{1} – Arg z_{2} + nπ, π ∈ (-1,0,1}.