xSkill: notes, lectures and practice for CBSE and CSE

1.1 The imaginary unit The equation x squared plus 1 equals 0 has no real solution, because a real square cannot be negative. The fix is a new symbol i with i squared equal to minus 1, and a complex number is any expression a plus i b with a and b real. The real number a is the real part and b is the imaginary part. Two complex numbers are equal exactly when their real parts match and their imaginary parts match. The powers of i follow a four step cycle, which makes large powers easy to compute. Power Value i to the power 4k 1 i to the power 4k+1 i i to the power 4k+2 -1 i to the power 4k+3 -i

i^{2} = - 1, i^{4 k} = 1, i^{4 k + 1} = i, i^{4 k + 2} = - 1, i^{4 k + 3} = - i

The defining relation and the four step cycle of powers

1.2 Algebra of complex numbers Addition and subtraction work part by part: add the real parts and add the imaginary parts separately. Multiplication uses the usual distributive law followed by i squared equal to minus 1. Division is done by multiplying the numerator and denominator by the conjugate of the denominator, which clears the imaginary part from the bottom. Operation Result for z1 = a + ib and z2 = c + id Sum (a + c) + i(b + d) Difference (a - c) + i(b - d) Product (ac - bd) + i(ad + bc) Reciprocal conjugate of z divided by modulus squared For a negative real number, the square root is written with i. So the square root of minus 9 is 3i, and the square root of minus a, for positive a, is i times the square root of a. One caution: the rule that the square root of a product splits as the product of square roots fails when both factors are negative. Writing i times i as the square root of minus 1 times minus 1 gives 1, which contradicts i squared equal to minus 1.

Operation	Result for z1 = a + ib and z2 = c + id
Sum	(a + c) + i(b + d)
Difference	(a - c) + i(b - d)
Product	(ac - bd) + i(ad + bc)
Reciprocal	conjugate of z divided by modulus squared

z_{1} z_{2} = (a c - b d) + i (a d + b c), \frac{z _{1}}{z _{2}} = \frac{z _{1} z ˉ _{2}}{z ˉ _{2} z _{2}}

Product and quotient of two complex numbers

1.3 Modulus and conjugate The modulus of a complex number is its distance from the origin in the plane, so the modulus of a plus i b is the square root of a squared plus b squared. The conjugate flips the imaginary part, changing a plus i b into a minus i b. Rule Statement Product of z and its conjugate modulus squared, a real number Conjugate of a sum sum of the conjugates Conjugate of a difference difference of the conjugates Conjugate of a product product of the conjugates Conjugate of a quotient quotient of the conjugates Modulus of a product product of the moduli Modulus of a quotient quotient of the moduli The reciprocal of z equals its conjugate divided by its modulus squared. This single line is the standard way to divide complex numbers and to find multiplicative inverses.

Rule	Statement
Product of z and its conjugate	modulus squared, a real number
Conjugate of a sum	sum of the conjugates
Conjugate of a difference	difference of the conjugates
Conjugate of a product	product of the conjugates
Conjugate of a quotient	quotient of the conjugates
Modulus of a product	product of the moduli
Modulus of a quotient	quotient of the moduli

z = a + ib, ∣ z ∣ = a^{2} + b^{2}, z ˉ = a - ib, z z ˉ = ∣ z ∣^{2}

Modulus, conjugate and their product

1.4 Argand plane and quadratic equations In the Argand plane the complex number a plus i b is the point (a, b), with the real part on the horizontal axis and the imaginary part on the vertical axis. The modulus is the distance from the origin and the conjugate is the mirror image in the real axis. Quadratic equations with real coefficients may have a negative discriminant. When b squared minus 4ac is negative, the square root gains an i, and the two roots are complex conjugates of each other: Discriminant Nature of roots positive two distinct real roots zero two equal real roots negative two conjugate complex roots For the equation a x squared plus b x plus c equals 0 with a not zero, the roots are given by the same quadratic formula, with the negative discriminant written as a positive number times i squared.

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}, when b^{2} - 4 a c < 0, b^{2} - 4 a c = i 4 a c - b^{2}

Quadratic formula with a negative discriminant

Exam

For a division question, multiply top and bottom by the conjugate of the denominator. Do not split the fraction into real and imaginary parts first; the conjugate method is shorter and keeps the working checkable.