Use these Inter 2nd Year Maths 2A Formulas PDF Chapter 1 Complex Numbers to solve questions creatively.

## Intermediate 2nd Year Maths 2A Complex Numbers Formulas

**Definition of a complex number:**

→ A number of the type z = x + yi, where x, y ∈ R and i = √- 1 i.e., i^{2} = – 1 is called a complex number ‘x’ is called real part of z, and ‘y’ is called imaginary part of z. We write x = Re(z) and y = Im(z). A number z = x + yi is said to be purely real iff y = 0 (x ≠ 0) and purely imaginary iff x = 0 (y ≠ 0)

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

→ Two complex numbers z_{1} = (a, b), z_{2} = (c, d) are said to be equal iff a = c and b = d.

→ If z_{1} = (a, b), z_{2} = (c, d) then

- z
_{1}+ z_{1}= (a + c, b + d) - z
_{1}– z_{2}= (a – c, b – d) - z
_{1}. Z_{2}= (ac – bd, ad + bc) and - \(\frac{z_{1}}{z_{2}}=\left(\frac{a c+b d}{c^{2}+d^{2}}, \frac{b c-a d}{c^{2}+d^{2}}\right)\)

**Modulus and Amplitude:**

→ The modulus of a complex number z = x + iy is defined as a non-negative real number r = \(\sqrt{x^{2}+y^{2}}\). It is denoted by |z|.

→ Any real number θ satisfying the equation cos θ = \(\frac{x}{r}\), sin θ = \(\frac{y}{r}\) is called an amplitude or argument of z. The unique argument θ of z satisfying – π < θ ≤ π is called the principal argument of z and is denoted by Arg z.

→ The polar form or modulus amplitude form of the complex number

z = x + iy is r(cos θ + i sin θ)

**Conjugate of a complex number:**

→ If z = x + iy i.e., x, y ∈ R, then the complex number x – iy is called conjugate of z and is written as z̅. The sum and product of a complex number and its Conjugate is always purely real.

**Some properties of modulus, amplitude and conjugate:**

- (z̅) = z
- z + z̅ = 2 Re (z) and z – z̅ = 2 Im (z)
- \(\overline{Z_{1} Z_{2}}\) = \(\overline{Z_{1}}\) \(\overline{Z_{2}}\)
- \(\) and \(\) (z
_{2}≠ 0) and |z_{1}z_{2}| = |z_{1}| |z_{2}| - zz̅ = |z|
^{2} - |z| = |z̅|;
- |z
_{1}+ z_{2}| ≤ |z_{1}| + |z_{2}|, |z_{1}– z_{2}| ≤ |z_{1}| + |z_{2}| - |z
_{1}– z_{2}| > | |z_{1}| – |z_{2}| |

In (vii) and (viii) equality holds iff amp (z_{1}) – amp (z_{2}) is an integral multiple of 2π. - amp (z
_{1}z_{2}) = amp (z_{1}) + amp (z_{2}) + nπ, for some n ∈ {- 1, 0, 1} - amp \(\left(\frac{Z_{1}}{Z_{2}}\right)\) = amp (z
_{1}) – amp (z_{2}) + nπ, for some n ∈ {- 1, 0, 1} - \(\frac{1}{{cis} \alpha}\) = cis(- α)
- cis α cis β = cis (α + β)
- \(\frac{{cis} \alpha}{{cis} \beta}\) = cis (α – β)

**De-Moivre’s theorem:**

- If n is any integer, then (cos θ + i sin θ)
^{n}= cos nθ + i sin nθ - If n = \(\frac{p}{q}\), where p and q are integers having no common factor and q > 1, then cos nθ + i sin nθ is one of the q values of (cos θ + i sin θ)
^{n} - If z
_{0}= r_{0}cis θ_{0}≠ 0, then the n^{th}roots of z_{0}are α_{k}= r_{0}^{1/n}cis \(\left(\frac{2 k \pi+\theta_{0}}{n}\right)\),

k = 0, 1, 2, 3 … (n-1)

**Cube roots of unity:**

- The cube roots of unity are 1, ω = \(\frac{-1+\sqrt{3 i}}{2}\) and ω
^{2}= \(\frac{-1-\sqrt{3 i}}{2}\) - 1 + ω + ω
^{2}= 0 and w^{3}= 1; 1 + ω = – ω^{2}, 1 + ω^{2}= – ω, ω + ω^{2}= – 1 - Either of the two non-real cube roots of unity is square of the other.
- For either of the two non – real cube roots a.and of unity α + β = – 1, αβ = – 1, α
^{2}= β, β^{2}= α and α^{3}= β^{3}= 1 - (-1)
^{1/3}= – 1, – ω – ω^{2} - The n
^{th}roots of unity are cis\(\left(\frac{2 k \pi}{n}\right)\), k = 0, 1, 2, ….. (n – 1)

**Formulae:**

→ Modulus of Z = \(\sqrt{x^{2}+y^{2}}\)

→ If \(\sqrt{a+i b}\) = (x + iy), then x = \(\sqrt{\frac{\sqrt{a^{2}+b^{2}}+a}{2}}\) and y = \(\sqrt{\frac{\sqrt{a^{2}+b^{2}}-a}{2}}\)

→ Conjugate of a + ib = a – ib

→ Conjugate of a – ib = a + ib

→ Any number of the form x + iy where x, y ∈ R and i^{2} = -1 is called a Complex Number.

→ In the complex number x + iy, x is called the real part and y is called the imaginary part of the complex number.

→ A complex number is said to be purely imaginary if its real part is zero and is said to be purely real if its imaginary part is zero.

(a) Two complex numbers are said to be equal if their real parts are equal and their imaginary parts are equal.

(b) In the set of complex numbers, there is no meaning to the phrases one complex is greater than or less than another i.e. If two complex numbers are not equal, we say they are unequal.

(c) a+ ib > c + id is meaningful only when b = 0, d = 0.

→ Two complex numbers are conjugate if their sum and product are both real. They are of the form a + ib, a – ib.

→ cisθ_{1} cisθ_{2} = cis(θ_{1} + θ_{2}), \(\frac{\operatorname{cis} \theta_{1}}{\operatorname{cis} \theta_{2}}\) = cis(θ_{1} – θ_{2}), \(\frac{1}{\cos \theta+i \sin \theta}\) = cosθ – isinθ

→ \(\frac{a_{1}+i b_{1}}{a_{2}+i b_{2}}=\frac{\left(a_{1} a_{2}+b_{1} b_{2}\right)+i\left(a_{2} b_{1}-a_{1} b_{2}\right)}{a_{2}^{2}+b_{2}^{2}}\)

→ \(\frac{1+i}{1-i}\) = i, \(\frac{1-i}{1+i}\) = i

→ \(\sqrt{x^{2}+y^{2}}\) is called the modulus of the complex number x + iy and is denoted by r or |x + iy|

→ Any value of 0 obtained from the equations cos θ = \(\frac{x}{r}\), sin θ = \(\frac{y}{r}\) is called an amplitude of the complex number.

→ The amplitude lying between -π and π is called the principal amplitude of the complex number. Rule for choosing the principal amplitude.

→ If θ is the principal amplitude, then -π < 0 < π

→If α is the principle amplitude of a complex number, general amplitude = 2nπ + α where n ∈ Z.

- Amp (Z
_{1}Z_{2}) = Amp Z_{1}+ AmpZ_{2} - Amp z + Amp z̅ = 2π (when z is a negative real number) = 0 (otherwise)

→ r(cos θ + i sin θ) is the modulus amplitude form of x + iy.

→ If the amplitude of a complex number is \(\frac{\pi}{2}\), its real part is zero.

→ If the amplitude of a complex number is \(\frac{\pi}{4}\), its real part is equal to its imaginary part.