“i cycles in fours”
Powers of i
i, −1, −i, 1, then it repeats. Divide the power by 4 and use the remainder.
Mathematics, Class 11 and 12 · Gujarat Board, CBSE, GUJCET and JEE
Complex Numbers and Quadratic Equations formulas
Complex numbers turn every quadratic into a solvable one. The modulus-argument form is the part that carries into physics and engineering, and De Moivre's theorem is the one result worth learning by heart.
18 formulas, 4 memory hooks, and the 5 mistakes that cost the most marks.
Basics
| What | Formula | Symbols and notes |
|---|---|---|
| Imaginary unit | i = √(−1), i² = −1 | i³ = −i, i⁴ = 1, then it repeats |
| Standard form | z = a + ib | a is the real part, b the imaginary part |
| Conjugate | z̄ = a − ib | reflect across the real axis |
| Modulus | |z| = √(a² + b²) | the distance from the origin |
| Argument | θ = tan−1(b/a) | adjusted for the quadrant of the point |
| Polar form | z = r(cosθ + i sinθ) | r = |z| |
Operations
| What | Formula | Symbols and notes |
|---|---|---|
| Multiplication of moduli | |z₁z₂| = |z₁||z₂| | arguments add |
| Division | |z₁/z₂| = |z₁|/|z₂| | arguments subtract |
| Conjugate product | z·z̄ = |z|² | always real and non-negative |
| De Moivre's theorem | (cosθ + i sinθ)n = cos nθ + i sin nθ | for integer n |
| Triangle inequality | |z₁ + z₂| ≤ |z₁| + |z₂| | equality when the arguments are equal |
| Cube roots of unity | 1, ω, ω², with 1 + ω + ω² = 0 and ω³ = 1 | ω = (−1 + i√3)/2 |
Quadratic equations
| What | Formula | Symbols and notes |
|---|---|---|
| Roots | x = (−b ± √(b² − 4ac))/2a | for ax² + bx + c = 0 |
| Discriminant | D = b² − 4ac | D > 0 real distinct, D = 0 real equal, D < 0 complex |
| Sum of roots | α + β = −b/a | faster than solving |
| Product of roots | αβ = c/a | together these rebuild the equation |
| Forming an equation | x² − (sum)x + product = 0 | when the roots are known |
| Complex roots | occur in conjugate pairs | when the coefficients are real |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
Darshit Sir has taught this in Paldi since 2006, in the classroom and live online.
Memory hooks
How to remember this chapter.
“Multiply lengths, add angles”
Multiplying complex numbers
Moduli multiply and arguments add. Division does the reverse. Thinking of it as a rotation plus a stretch makes the algebra unnecessary.
“Check D first”
Any quadratic question
The discriminant tells you the nature of the roots before you solve anything, and that is often all the question wanted.
“Omega sums to zero”
Cube roots of unity
1 + ω + ω² = 0 and ω³ = 1. Nearly every ω question collapses using one of those two.
Where marks go
The mistakes that cost the most.
- Writing √(−4) × √(−9) as √36. The rule √a × √b = √(ab) fails for negative numbers; the answer is −6.
- Reading the argument straight off tan−1(b/a) without checking the quadrant.
- Forgetting that complex roots come in conjugate pairs when the coefficients are real.
- Using D to find the roots when only their nature was asked, and losing time.
- Treating |z| as z. The modulus is a real, non-negative number.
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