Class 11 Maths • Chapter 04 • Comprehensive Interactive Notes
A complex number is defined as \( z = a + ib \), where \( a, b \in R \). Here, \( a \) is the real part and \( b \) is the imaginary part. The unit \( i \) (iota) is defined as \( \sqrt{-1} \).
Calculate \( i^n \) for any integer n.
Pattern: \( i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1 \)
Two complex numbers are equal if and only if their real parts and imaginary parts are equal.
If \( a + ib = c + id \)
Then:
Exam Tip: Always equate real parts and imaginary parts separately.
A complex number \( z = x + iy \) corresponds to the point \( P(x, y) \) in the Argand plane.
Enter Real (a) and Imaginary (b) parts.
The argument of \( z = a + ib \) is the angle \( \theta \) made by the line joining origin to point z with the positive real axis.
\( \tan \theta = \frac{b}{a} \)
\( \arg(z) = \theta \)
Principal Argument: Value of argument lying between \( (-\pi, \pi] \).
Note: Full polar form is studied in Class 12.
Operations on complex numbers behave like polynomials with \( i \).
For \( z_1 = a + ib \) and \( z_2 = c + id \):
\( z_1 z_2 = (ac - bd) + i(ad + bc) \)
Note: \( i^2 = -1 \) is used here.
Modulus: \( |z| = \sqrt{a^2 + b^2} \) (Distance from origin)
Conjugate: \( \bar{z} = a - ib \) (Reflection in Real axis)
Property: \( z\bar{z} = |z|^2 \)
Solving \( ax^2 + bx + c = 0 \) when discriminant \( D < 0 \).
The solution is given by: \( x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-b \pm \sqrt{|D|}i}{2a} \)
| Discriminant (D) | Nature of Roots |
|---|---|
| D > 0 | Two distinct real roots |
| D = 0 | Equal real roots |
| D < 0 | Complex conjugate roots |
Solve \( ax^2 + bx + c = 0 \)
| Identity | Formula |
|---|---|
| Modulus Properties | \( |z_1 z_2| = |z_1| |z_2| \) \( |\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|} \) |
| Conjugate Properties | \( \overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2} \) \( \overline{z_1 z_2} = \bar{z_1} \bar{z_2} \) |
| Multiplicative Inverse | \( z^{-1} = \frac{\bar{z}}{|z|^2} \) |
For any complex numbers \( z_1, z_2 \):
\( |z_1 + z_2| \le |z_1| + |z_2| \)
\( ||z_1| - |z_2|| \le |z_1 - z_2| \)
Meaning: Distance in complex plane follows triangle rules.
Exam Tip: Often asked as “prove that” or MCQ.
1. The value of \( i^{4k+3} \) is:
2. The conjugate of \( 3 - 4i \) is:
3. If \( z = x + iy \), then \( |z| \) represents:
4. Solution of \( x^2 + 2 = 0 \) is:
5. Multiplicative inverse of \( i \) is: