Complex Numbers and Quadratic Equations
Class 11 · Mathematics
4 ideas in Class 11 Mathematics, explained in plain words, with games and quizzes to practise.
Complex Numbers and Powers of i
Part of: Complex Numbers and Algebra
Pairs a + ib with real a, b extend reals while i-squared anchors minus-one; equality matches real and imaginary parts; i cycles every four powers.
Why it matters. Imaginary unit solves x-squared-plus-one. Cycling speeds powers.
Formulas
- i^2 = -1
- i^3 = -i and i^4 = 1
- i^(4k + r) = i^r
Key words
- iota (i)
- : Unit with square minus one, building complex numbers.
- Purely imaginary
- : Number with zero real part, such as 0 + 6i.
- Purely real
- : Number with zero imaginary part, an ordinary real.
- Equality rule
- : Matching of real parts and imaginary parts separately.
- Cycle of i
- : i repeats every four powers: i, -1, -i, 1.
Try it in two minutes
- Split real and imaginary.
- Match parts for equality.
- Reduce power modulo four.
Try at home. Compute i to the 50th power.
Algebra, Conjugate and Modulus
Part of: Complex Numbers and Algebra
Addition groups real with real; products expand with i-squared; conjugates flip the imaginary sign; modulus roots sum-of-squares; division rationalises by the conjugate while inverses scale conjugates by modulus-squared.
Why it matters. Conjugate clears imaginary denominators. Modulus sizes complex numbers.
Formulas
- conjugate of a + ib = a - ib
- |a + ib| = sqrt(a^2 + b^2)
- z x conjugate(z) = |z|^2
- 1/z = conjugate(z) / |z|^2
Key words
- Complex number
- : Pair a + ib with real a and b, extending real numbers.
- Conjugate
- : Mirror of a + ib with the imaginary sign flipped.
- Modulus
- : Distance of z from the origin, root of sum of squares.
- Rationalising conjugate
- : Flipped-sign number that clears i from a denominator.
- Multiplicative inverse
- : Conjugate of z scaled down by modulus-squared.
Try it in two minutes
- Group real and imaginary.
- Flip sign for conjugate.
- Multiply by conjugate over modulus-squared.
Try at home. Divide 1 by 2 + 3i.
Argand Plane and Polar Form
Part of: Argand Plane and Quadratics
Points (a, b) plot z with real-x and imaginary-y axes; conjugates mirror across real axis; modulus spans from origin; polar writes r(cos t + i sin t) with argument t.
Why it matters. Geometry visualises operations. Polar form eases multiplication.
Formulas
- z = r(cos t + i sin t)
- r = |z| = sqrt(a^2 + b^2)
- t = angle of the point from the positive real axis
Key words
- Argand plane
- : Graph with real axis x and imaginary axis y for z.
- Modulus r
- : Length r from the origin to the plotted point.
- Argument
- : Angle t of the point from the positive real axis.
- Polar form
- : Radius-angle writing of a complex number.
- Conjugate point
- : Mirror of P(a, b) across the real axis.
Try it in two minutes
- Plot real versus imaginary.
- Mirror for conjugate.
- Read radius and angle.
Try at home. Plot 3 + 4i and its mirror.
Quadratic Equations with Complex Roots
Part of: Argand Plane and Quadratics
Negative discriminants root through i; formula still rules; conjugate pairs always co-occur for real coefficients; sums and products shortcut checks.
Why it matters. Complex roots complete every quadratic. Pairing predicts the twin.
Formulas
- x = (-b +- sqrt(D)) / (2a)
- Sum of roots = -b/a
- Product of roots = c/a
- D < 0 gives a conjugate pair for real coefficients
Key words
- Discriminant
- : b^2 - 4ac, whose sign fixes the root type.
- Conjugate pair
- : Twin roots mirrored across the real axis.
- Sum-product check
- : Shortcut check from sum -b/a and product c/a.
- Complex roots
- : Roots reached by rooting a negative number through i.
Try it in two minutes
- Compute the discriminant.
- Root negatives through i.
- Pair conjugates.
Try at home. Solve x² + 4x + 13 = 0.
What your child will practise
- Algebra, Conjugate and Modulus: Desert Puzzle (Clue Puzzle)
- Algebra, Conjugate and Modulus: Desert Quick Fire (Quick Fire)
- Argand Plane and Polar Form: Space Puzzle (Clue Puzzle)
- Complex Numbers and Powers of i: Ocean Quest (Flip and Match)
- Quadratic Equations with Complex Roots: Jungle Quick Fire (Quick Fire)
- Quiz: Algebra, Conjugate and Modulus (8 questions)
- Quiz: Argand Plane and Polar Form (8 questions)
- Quiz: Complex Numbers and Powers of i (8 questions)
- Quiz: Quadratic Equations with Complex Roots (8 questions)