Complex Numbers and Quadratic Equations — Class 11 CBSE Mathematics revision notes
A 3-page visual revision sheet for Complex Numbers and Quadratic Equations, written for Class 11 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.
Page 1 as PDF
Complex Numbers and Quadratic Equations in one look
a second axis for numbers, so every quadratic finally has a full set of roots
Why i exists
- x^2+1=0 has no solution among real numbers, since a real square is never negative
- define i so that i^2=-1: a new kind of number, built to fill exactly this gap
- z=a+bi, with a and b real: a the real part, b the imaginary part
- when b=0, z is purely real. when a=0 and b≠ 0, z is called purely imaginary
- two complex numbers are equal only when both parts match: a=c and b=d
Algebra of complex numbers
| operation | rule |
|---|---|
| addition | (a+bi)+(c+di)=(a+c)+(b+d)i |
| subtraction | (a+bi)-(c+di)=(a-c)+(b-d)i |
| multiplication | (a+bi)(c+di)=(ac-bd)+(ad+bc)i |
| conjugate | z=a-bi |
| modulus | |z|=√a^2+b^2 |
| division | a+bi/c+di=(a+bi)(c-di)/c^2+d^2 |
Conjugate and modulus
- conjugate of z=a+bi: flip the sign of the imaginary part
z=a-bi - a number times its own conjugate is always real
zz=a^2+b^2 - modulus: distance of z from the origin
|z|=√a^2+b^2
Powers of i cycle every 4
- n leaves remainder 1: i¹ = i
- n leaves remainder 2: i² = -1
- n leaves remainder 3: i³ = -i
- n leaves remainder 0: i⁴ = 1
Page 1 of 3. The full sheet, all 3 pages, opens free in the app.
What this Complex Numbers and Quadratic Equations sheet covers
- Why i exists
- Algebra of complex numbers
- Conjugate and modulus
- conjugate of z=a+bi: flip the sign of the imaginary part
- a number times its own conjugate is always real
- modulus: distance of z from the origin
- Powers of i cycle every 4
- n leaves remainder 1
- n leaves remainder 2
- n leaves remainder 3
- n leaves remainder 0
- Square root of a negative real number
- Polar form, reading z off the Argand plane
- Finding the argument
- A quadratic equation with complex roots
- A quadratic equation with real roots
- Nature of the roots of ax²+bx+c=0
- The quadratic formula, restated for any real a, b, c
- the two roots
- sum of the roots
- product of the roots
- Symbols
- Traps that cost marks
- Tricks that save time
How CapCompass uses this sheet
These notes are not a standalone PDF. CapCompass plans Mathematics chapter by chapter against your real exam dates and free hours, and when the plan reaches Complex Numbers and Quadratic Equations the sheet opens inside the study block itself — at the page covering the sub-topic that block is on, with the timer still running. Revision arrives at the moment the plan says revise, instead of having to be gone looking for.