Complex Numbers — Class 11 Maharashtra HSC Mathematics revision notes
A 4-page visual revision sheet for Complex Numbers, written for Class 11 Maharashtra HSC 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 in one look
no real number squares to -1, so the number system was widened until one did
The imaginary unit
- x^2+1=0 has no solution among the real numbers: no real number squares to -1
- define i (iota) by i=√-1, so i^2=-1. This is the imaginary unit
- a number of the form bi, bin R, b≠0, is called an imaginary number, e.g. 5i, 2/7i
- a complex number is z=a+ib, a,bin R
- a is the real part Re(z), b is the imaginary part Im(z)
- the set of all such z is C, and Rsubset C
Equality and conjugate
- equality
a+ib=c+idLeftrightarrow a=c and b=d - conjugate of z=a+ib
z=a-ib
Solving by equality of complex numbers
Q. if 7a+i(3a-b)=21-3i, find a and b
- match real parts: 7a=21, so a=3
- match imaginary parts: 3a-b=-3, so 3(3)-b=-3
Answer: a=3, b=12
Properties of the conjugate
- Double conjugate: overlinez=z
- z real: z=zLeftrightarrow z is purely real
- z imaginary: z=-zLeftrightarrow z is purely imaginary
- Modulus link: zcdotz=a^2+b^2
The four operations
- 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) - division, multiply top and bottom by z_2
z_1/z_2=fracz_1z_2z_2z_2
Multiplying two complex numbers
Q. find z_1z_2 if z_1=2+3i and z_2=3-2i
- z_1z_2=(2+3i)(3-2i)=2(3-2i)+3i(3-2i)
- =6-4i+9i-6i^2
- i^2=-1, so -6i^2=+6: =6-4i+9i+6
Answer: 12+5i
Page 1 of 4. The full sheet, all 4 pages, opens free in the app.
What this Complex Numbers sheet covers
- The imaginary unit
- Equality and conjugate
- equality
- conjugate of z=a+ib
- Solving by equality of complex numbers
- Properties of the conjugate
- Double conjugate
- z real
- z imaginary
- Modulus link
- The four operations
- addition
- subtraction
- multiplication
- division, multiply top and bottom by z_2
- Multiplying two complex numbers
- Dividing by rationalising
- Powers of i, by remainder
- Reducing a large power of i
- Square root of a complex number
- set up
- match parts
- Finding a square root
- Roots of ax squared + bx + c = 0, a,b,c real
- Complex roots of a quadratic
- The Argand plane
- Argument by quadrant
- Modulus and argument
- Polar and exponential form
- polar form
- exponential form
- Converting to polar form
- De Moivre's theorem
- for any integer n
- multiplying two polar forms
- A power by De Moivre's theorem
- Cube roots of unity
- solving x^3=1
- the three roots, with ω
- the defining relations
- Properties of 1, omega, omega squared
- Cycle
- Shift by 1
- Shift by 2
- Reciprocal
- 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 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.