Introduction to Complex Numbers
Imaginary numbers were introduced to solve equations like x^2 + 1 = 0. The imaginary unit i is defined as i = sqrt(-1), so i^2 = -1.
A complex number is of the form a + ib, where a, b in R. Here a is the real part (Re(z)) and b is the imaginary part (Im(z)).
The set of complex numbers is denoted by C.
Powers of i
i^1 = ii^2 = -1i^3 = -ii^4 = 1- In general,
i^(4k + r) = i^rfor integer k and r = 0, 1, 2, 3.
Algebra of Complex Numbers
Equality
a + ib = c + id if and only if a = c and b = d.
Addition
(a + ib) + (c + id) = (a + c) + i(b + d)
Subtraction
(a + ib) - (c + id) = (a - c) + i(b - d)
Multiplication
(a + ib)(c + id) = (ac - bd) + i(ad + bc)
Conjugate
Conjugate of z = a + ib is bar(z) = a - ib.
z bar(z) = a^2 + b^2bar(z_1 + z_2) = bar(z_1) + bar(z_2)bar(z_1 z_2) = bar(z_1) bar(z_2)
Division
(a + ib)/(c + id) = ((a + ib)(c - id))/(c^2 + d^2) (using rationalisation)
Modulus and Argument
Modulus (Absolute Value)
|z| = sqrt(a^2 + b^2) for z = a + ib.
Properties:
|z_1 z_2| = |z_1||z_2||z_1/z_2| = |z_1|/|z_2|(forz_2 != 0)|z_1 + z_2| <= |z_1| + |z_2|(Triangle inequality)|z_1 - z_2| >= ||z_1| - |z_2||
Argument (Amplitude)
arg(z) = theta where tan theta = b/a, measured in the appropriate quadrant.
theta = tan^(-1)(b/a)adjusted for quadrant.
Polar Form
z = r(cos theta + i sin theta), where r = |z| and theta = arg(z).
Euler's Form
z = r e^(i theta) where e^(i theta) = cos theta + i sin theta.
Square Roots of Complex Numbers
To find sqrt(a + ib), let sqrt(x + iy) = pm [sqrt((|z| + a)/2) + i sqrt((|z| - a)/2)] (with appropriate sign for y).
Quadratic Equations with Complex Roots
The quadratic equation ax^2 + bx + c = 0 with discriminant D = b^2 - 4ac:
- If
D > 0: Two distinct real roots. - If
D = 0: Two equal real roots. - If
D < 0: Two distinct complex conjugate roots.
Roots: x = (-b pm sqrt(D))/(2a)
For D < 0, roots are (-b)/(2a) pm i sqrt(|D|)/(2a).
Nature of Complex Roots
Complex roots always occur in conjugate pairs for quadratic equations with real coefficients.
Worked Examples
Example 1: Express (2 + 3i)/(1 - i) in a + ib form.
Solution: Multiply numerator and denominator by (1 + i):
((2+3i)(1+i))/((1-i)(1+i)) = (2+2i+3i+3i^2)/(1+1) = (2+5i-3)/2 = (-1+5i)/2 = -1/2 + (5/2)i
Example 2: Find the modulus and argument of z = -1 + i*sqrt(3).
Solution: r = sqrt((-1)^2 + 3) = sqrt(4) = 2. tan theta = sqrt(3)/(-1) = -sqrt(3). Since z is in QII, theta = pi - pi/3 = 2pi/3.
Example 3: Solve x^2 + 4x + 5 = 0.
Solution: D = 16 - 20 = -4. x = (-4 pm sqrt(-4))/2 = (-4 pm 2i)/2 = -2 pm i.
Common Mistakes
sqrt(a) sqrt(b) = sqrt(ab)fails for negative numbers:sqrt(-4) sqrt(-9) = 2i * 3i = -6, notsqrt(36) = 6.- Argument quadrant: Always locate the quadrant before finding
theta = tan^(-1)(b/a). - Conjugate of sum:
bar(z_1 + z_2) = bar(z_1) + bar(z_2), NOTbar(z_1) + z_2. - Equality condition:
a + ib = c + idrequires both real and imaginary parts equal.
ISC Exam Focus
- Theory (70%): Properties of conjugates, modulus, argument, polar form conversion.
- Application (30%): Solving quadratic equations, finding square roots of complex numbers.
- ISC frequently asks: express in
a + ibform, find modulus and argument, solve quadratic with complex roots. - 4-6 mark questions involving these concepts appear regularly.
Self-Test Questions
Q1: Express (1 + i)/(1 - i) in a + ib form.
Answer: (1 + i)/(1 - i) = ((1+i)^2)/(1+1) = (1 + 2i -1)/2 = i = 0 + 1i.
Q2: Find the modulus and argument of z = 1 + i*sqrt(3).
Answer: |z| = sqrt(1 + 3) = 2. theta = tan^(-1)(sqrt(3)) = pi/3 (Q1).
Q3: Solve 2x^2 + x + 1 = 0.
Answer: D = 1 - 8 = -7. x = (-1 pm i sqrt(7))/4.
Q4: If z = 2 + 3i, find z bar(z).
Answer: z bar(z) = (2+3i)(2-3i) = 4 + 9 = 13.
Q5: Express z = sqrt(3) + i in polar form.
Answer: r = sqrt(3 + 1) = 2, theta = tan^(-1)(1/sqrt(3)) = pi/6. Polar form: z = 2(cos(pi/6) + i sin(pi/6)).
Q6: Find the square root of 3 + 4i.
Answer: Let sqrt(z) = a + ib. Then a^2 - b^2 = 3 and 2ab = 4. Also a^2 + b^2 = sqrt(3^2 + 4^2) = 5. Solving: a^2 = 4, b^2 = 1. So a = pm 2, b = pm 1. Since ab > 0, roots are 2 + i and -2 - i.
