5. Complex Numbers

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It is never too late to be what you might have been. —George Eliot

COMPLEX NUMBERS CLASS 11 MATHS

A number of the form \(a + ib\), where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number.

Let \(z_1= a + ib text{ and } z_2 = c + id \). Then
(i) \( z_1 + z_2 = (a + c) + i (b + d) \)
(ii) \(z_1 z_2 = (ac – bd) + i (ad + bc) \)

For any non-zero complex number \(z = a + ib (a \ne 0, b \ne 0)\), there exists the complex number \( \frac{a}{a^2+ b^2} + i \frac{-b}{a^2 + b^2} \) , denoted by \( \frac{1}{z} \) or \( z^{-1}\), called the multiplicative inverse of z such that \( (a + ib)\left ( \frac{a^2}{a^2+b^2}+i \frac{-b}{a^2+b^2} \right ) = 1 + i 0 = 1 \)

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For any integer \( k \), \( i^{4k}=1,\)\( i{4k + 1}=i\), \(i^{4k + 2}= – 1,\) \(i{4k + 3} = – i \)

The conjugate of the complex number \( z = a + ib\), denoted by z , is given by \( z = a – ib\).

The polar form of the complex number \( z = x + iy\) is \(r (\cos θ + i \sin θ)\), where \(r = \sqrt{x^2 + y^2} \) (the modulus of z) and \( \cos θ = \frac{x}{r} \), \(\sin θ = \frac{y}{r}\). (θ is known as the argument of z. The value of θ, such that – π < θ ≤ π, is called the principal argument of z.

A polynomial equation of n degree has n roots.

The solutions of the quadratic equation \(ax^2 + bx + c = 0\), where a, b, c ∈ R, a ≠ 0, \( b^2 – 4ac < 0 \), are given by \( x = \frac{− b \pm \sqrt{4ac – b2} i}{2a} \).

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Complex Numbers Lecture 1
Lecture - 1
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Complex Numbers Lecture 2
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Complex Numbers Lecture 3
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Complex Numbers Lecture 4
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Complex Numbers Lecture 5
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Complex Numbers Lecture 6
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