Complex Numbers Class 11 Notes for IIT JEE
eSaral Academic and Editorial Team
Summary
Complex Numbers Class 11 Notes cover the imaginary unit $i$, the algebra of complex numbers, conjugate and modulus with their properties, the Argand plane and the argument, polar form and De Moivre’s theorem, the square root of a complex number, the cube roots of unity, simple geometry in the Argand plane, and quadratic equations with complex roots.
In NCERT Class 11 Mathematics, this topic is in the chapter Complex Numbers and Quadratic Equations, which is Chapter 4 of the current book (Chapter 5 in older books). The chapter is the base for the roots of polynomials, for the geometry of the Argand plane and for many JEE algebra problems. These Complex Numbers Class 11 Notes follow the NCERT chapter, and they add the JEE topics that the textbook does not cover in depth.
Complex Numbers Class 11 Notes: Chapter Overview and Syllabus
Topic | CBSE Class 11 | JEE |
|---|---|---|
Need for complex numbers, $i = \sqrt{-1}$, algebraic properties | Yes | Yes |
Argand plane and polar representation | Yes | Yes |
Modulus, argument, conjugate and their properties | Yes | Yes |
Square root of a complex number | Yes | Yes |
Quadratic equations in the complex number system, Fundamental Theorem of Algebra (statement only) | Yes | Yes |
Triangle inequality | Not a separate topic | Yes |
De Moivre’s theorem, cube roots of unity, geometry of complex numbers | Not in the board syllabus | Standard tools in JEE problems |
Check the latest syllabi of CBSE and NTA for the exact lists. De Moivre’s theorem is not in the CBSE Class 11 board syllabus, but it is the quickest way to find powers and roots, so these notes include it.
Tip from the eSaral Maths team: Learn every complex number problem in two forms: algebraic ($a + ib$) for sums and products, and polar ($r\,e^{i\theta}$) for powers, roots and geometry. The right form can turn a long calculation into one line.
Complex Numbers Class 11 Notes
India's Best Exam Preparation for Class 11th - Download Now
India's Best Exam Preparation for Class 11th - Download Now
India's Best Exam Preparation for Class 11th - Download Now
India's Best Exam Preparation for Class 11th - Download Now
India's Best Exam Preparation for Class 11th - Download Now
India's Best Exam Preparation for Class 11th - Download Now
India's Best Exam Preparation for Class 11th - Download Now
India's Best Exam Preparation for Class 11th - Download Now
What Is a Complex Number? in Complex Numbers Class 11 Notes
A complex number is a number of the form $z = a + ib$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, so $i^{2} = -1$. Here $a = \mathrm{Re}(z)$ is the real part, and $b = \mathrm{Im}(z)$ is the imaginary part. Two complex numbers are equal if their real parts and their imaginary parts are equal. If $b = 0$, the number is real, and if $a = 0$ and $b \neq 0$, it is purely imaginary.
Powers of $i$: the values repeat every 4 powers. For an integer $k$:
$$i^{4k} = 1, \qquad i^{4k+1} = i, \qquad i^{4k+2} = -1, \qquad i^{4k+3} = -i$$
A common trap: for positive $a$ and $b$, $\sqrt{-a}\,\sqrt{-b} = i\sqrt{a}\cdot i\sqrt{b} = -\sqrt{ab}$, and not $\sqrt{ab}$. The rule $\sqrt{a}\sqrt{b} = \sqrt{ab}$ holds only when at least one of $a$ and $b$ is non-negative.
Algebra of Complex Numbers in Complex Numbers Class 11 Notes
For $z_1 = a + ib$ and $z_2 = c + id$:
Operation | Result |
|---|---|
Addition | $z_1 + z_2 = (a + c) + i(b + d)$ |
Subtraction | $z_1 - z_2 = (a - c) + i(b - d)$ |
Multiplication | $z_1 z_2 = (ac - bd) + i(ad + bc)$ |
Division | $\dfrac{z_1}{z_2} = \dfrac{z_1\,\bar{z}_2}{\lvert z_2\rvert^{2}} = \dfrac{(ac + bd) + i(bc - ad)}{c^{2} + d^{2}}$ |
Multiplicative inverse | $\dfrac{1}{z} = \dfrac{\bar{z}}{\lvert z\rvert^{2}}$ |
The operations are commutative and associative, multiplication is distributive over addition, $0$ is the additive identity and $1$ is the multiplicative identity.
Conjugate and Modulus in Complex Numbers Class 11 Notes
The conjugate of $z = a + ib$ is $\bar{z} = a - ib$, and the modulus is $\lvert z\rvert = \sqrt{a^{2} + b^{2}}$, the distance of $z$ from the origin.
Properties of the conjugate | Properties of the modulus |
|---|---|
$\overline{\bar{z}} = z$ | $\lvert z\rvert \geq 0$, and $\lvert z\rvert = \lvert\bar{z}\rvert$ |
$\overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2$ | $\lvert z_1 z_2\rvert = \lvert z_1\rvert\,\lvert z_2\rvert$ |
$\overline{z_1 z_2} = \bar{z}_1\,\bar{z}_2$ | $\left\lvert\dfrac{z_1}{z_2}\right\rvert = \dfrac{\lvert z_1\rvert}{\lvert z_2\rvert}$ |
$\overline{\left(\dfrac{z_1}{z_2}\right)} = \dfrac{\bar{z}_1}{\bar{z}_2}$ | $\lvert z^{n}\rvert = \lvert z\rvert^{n}$ |
$z + \bar{z} = 2\,\mathrm{Re}(z)$ and $z - \bar{z} = 2i\,\mathrm{Im}(z)$ | $z\bar{z} = \lvert z\rvert^{2}$ |
$z$ is real if and only if $z = \bar{z}$, and $z$ is purely imaginary if and only if $z = -\bar{z}$.
Triangle inequality and related results (JEE):
$$\lvert z_1 + z_2\rvert \leq \lvert z_1\rvert + \lvert z_2\rvert, \qquad \big\lvert\,\lvert z_1\rvert - \lvert z_2\rvert\,\big\rvert \leq \lvert z_1 - z_2\rvert$$ $$\lvert z_1 + z_2\rvert^{2} + \lvert z_1 - z_2\rvert^{2} = 2\left(\lvert z_1\rvert^{2} + \lvert z_2\rvert^{2}\right)$$
Argand Plane and Argument in Complex Numbers Class 11 Notes
In the Argand plane, the complex number $z = a + ib$ is the point $(a, b)$. The argument $\theta$ is the angle that the line from the origin to $z$ makes with the positive real axis. The principal argument lies in $(-\pi, \pi]$. First find the acute angle $\alpha = \tan^{-1}\left\lvert\dfrac{b}{a}\right\rvert$, and then use the quadrant of $z$:
Quadrant of $z$ | Principal argument $\theta$ |
|---|---|
I ($a > 0$, $b > 0$) | $\alpha$ |
II ($a < 0$, $b > 0$) | $\pi - \alpha$ |
III ($a < 0$, $b < 0$) | $-(\pi - \alpha)$ |
IV ($a > 0$, $b < 0$) | $-\alpha$ |
Do not use $\tan\theta = \dfrac{b}{a}$ alone, because it gives the same value for two opposite points. Properties (up to multiples of $2\pi$): $\arg(z_1 z_2) = \arg z_1 + \arg z_2$, $\arg\dfrac{z_1}{z_2} = \arg z_1 - \arg z_2$, $\arg\bar{z} = -\arg z$ and $\arg(z^{n}) = n\arg z$.
Polar Form and De Moivre’s Theorem in Complex Numbers Class 11 Notes
With $r = \lvert z\rvert$ and $\theta = \arg z$:
$$z = r(\cos\theta + i\sin\theta) = r\,e^{i\theta}$$
In polar form, multiplication multiplies the moduli and adds the arguments:
$$z_1 z_2 = r_1 r_2\big[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\big]$$
De Moivre’s theorem: for an integer $n$,
$$(\cos\theta + i\sin\theta)^{n} = \cos n\theta + i\sin n\theta$$
The $n$-th roots of a complex number $z = r(\cos\theta + i\sin\theta)$ are $n$ distinct numbers:
$$z^{1/n} = r^{1/n}\left[\cos\frac{\theta + 2k\pi}{n} + i\sin\frac{\theta + 2k\pi}{n}\right], \qquad k = 0, 1, \ldots, n - 1$$
On the Argand plane, these roots lie on a circle of radius $r^{1/n}$ and are equally spaced.
Square Root of a Complex Number in Complex Numbers Class 11 Notes
For $z = a + ib$, with $\lvert z\rvert = \sqrt{a^{2} + b^{2}}$:
$$\sqrt{a + ib} = \pm\left[\sqrt{\frac{\lvert z\rvert + a}{2}} + i\,\mathrm{sgn}(b)\sqrt{\frac{\lvert z\rvert - a}{2}}\right]$$
where $\mathrm{sgn}(b)$ is $+1$ if $b > 0$ and $-1$ if $b < 0$. Another method is to put $\sqrt{a + ib} = x + iy$, square both sides, and compare the real and imaginary parts.
Cube Roots of Unity in Complex Numbers Class 11 Notes
The cube roots of $1$ are $1$, $\omega$ and $\omega^{2}$, where:
$$\omega = \frac{-1 + i\sqrt{3}}{2} = e^{2\pi i/3}, \qquad \omega^{2} = \frac{-1 - i\sqrt{3}}{2} = \bar{\omega}$$
Property | Result |
|---|---|
Sum | $1 + \omega + \omega^{2} = 0$ |
Cube | $\omega^{3} = 1$, so $\omega^{3k} = 1$, $\omega^{3k+1} = \omega$ and $\omega^{3k+2} = \omega^{2}$ |
Product | $1\cdot\omega\cdot\omega^{2} = 1$ |
Factorisation | $x^{3} - 1 = (x - 1)(x - \omega)(x - \omega^{2})$ |
Sum of two cubes | $a^{3} + b^{3} = (a + b)(a + \omega b)(a + \omega^{2}b)$ |
The $n$-th roots of unity are $e^{2k\pi i/n}$ for $k = 0, 1, \ldots, n - 1$, and their sum is zero for $n > 1$.
Geometry in the Argand Plane
Condition | Meaning |
|---|---|
$\lvert z_1 - z_2\rvert$ | The distance between the points $z_1$ and $z_2$ |
$\lvert z - z_0\rvert = r$ | A circle with centre $z_0$ and radius $r$ |
$\lvert z - z_1\rvert = \lvert z - z_2\rvert$ | The perpendicular bisector of the line joining $z_1$ and $z_2$ |
$\dfrac{z_3 - z_1}{z_2 - z_1}$ is real | The points $z_1$, $z_2$ and $z_3$ are collinear |
Multiplication by $e^{i\theta}$ | A rotation of the point about the origin through the angle $\theta$ |
Quadratic Equations with Complex Roots in Complex Numbers Class 11 Notes
For $ax^{2} + bx + c = 0$ with real coefficients and a negative discriminant $D = b^{2} - 4ac < 0$:
$$x = \frac{-b \pm i\sqrt{4ac - b^{2}}}{2a}$$
The roots form a conjugate pair. The Fundamental Theorem of Algebra (stated without proof in the chapter) says that every polynomial of degree $n \geq 1$ has at least one root in the complex numbers, so it has exactly $n$ roots, counted with their multiplicity. For the nature and the location of the roots, see eSaral’s Quadratic Equations Class 11 notes.
Key Formulas for Complex Numbers
Concept | Formula |
|---|---|
Powers of $i$ | $i^{4k} = 1$, $i^{4k+1} = i$, $i^{4k+2} = -1$, $i^{4k+3} = -i$ |
Division | $\dfrac{z_1}{z_2} = \dfrac{z_1\bar{z}_2}{\lvert z_2\rvert^{2}}$ |
Modulus | $\lvert z\rvert = \sqrt{a^{2} + b^{2}}$, $z\bar{z} = \lvert z\rvert^{2}$ |
Polar form | $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ |
De Moivre’s theorem | $(\cos\theta + i\sin\theta)^{n} = \cos n\theta + i\sin n\theta$ |
$n$-th roots | $r^{1/n}\left[\cos\dfrac{\theta + 2k\pi}{n} + i\sin\dfrac{\theta + 2k\pi}{n}\right]$ |
Cube roots of unity | $1 + \omega + \omega^{2} = 0$, $\omega^{3} = 1$ |
Triangle inequality | $\lvert z_1 + z_2\rvert \leq \lvert z_1\rvert + \lvert z_2\rvert$ |
Complex roots of a quadratic | $x = \dfrac{-b \pm i\sqrt{4ac - b^{2}}}{2a}$ |
Solved Examples from Complex Numbers Class 11 Notes
Example 1 (powers of $i$): Find $i^{43}$ and $i^{-7}$.
$43 = 4 \times 10 + 3$, so $i^{43} = i^{3} = -i$. $i^{-7} = \dfrac{1}{i^{7}} = \dfrac{1}{-i} = i$.
Example 2 (division): Write $\dfrac{3 + 2i}{1 - i}$ in the form $a + ib$.
$\dfrac{(3 + 2i)(1 + i)}{(1 - i)(1 + i)} = \dfrac{3 + 3i + 2i + 2i^{2}}{2} = \dfrac{1 + 5i}{2} = \dfrac{1}{2} + \dfrac{5}{2}i$.
Example 3 (modulus and argument): Find the modulus, the principal argument and the polar form of $z = -1 + i\sqrt{3}$.
$\lvert z\rvert = \sqrt{1 + 3} = 2$. $\alpha = \tan^{-1}\sqrt{3} = \dfrac{\pi}{3}$, and $z$ is in the second quadrant, so $\theta = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}$. The polar form is $z = 2\left(\cos\dfrac{2\pi}{3} + i\sin\dfrac{2\pi}{3}\right)$.
Example 4 (De Moivre’s theorem): Find $(1 + i)^{8}$.
$1 + i = \sqrt{2}\left(\cos\dfrac{\pi}{4} + i\sin\dfrac{\pi}{4}\right)$, so $(1 + i)^{8} = (\sqrt{2})^{8}\,(\cos 2\pi + i\sin 2\pi) = 16$.
Example 5 (square root): Find $\sqrt{3 + 4i}$.
$\lvert z\rvert = 5$. The real part is $\sqrt{\dfrac{5 + 3}{2}} = 2$ and the imaginary part is $\sqrt{\dfrac{5 - 3}{2}} = 1$, with the same sign as $4$. So $\sqrt{3 + 4i} = \pm(2 + i)$.
Example 6 (cube roots of unity): Find $\omega^{100} + \omega^{200}$.
$100 = 3 \times 33 + 1$, so $\omega^{100} = \omega$. $200 = 3 \times 66 + 2$, so $\omega^{200} = \omega^{2}$. The sum is $\omega + \omega^{2} = -1$.
Example 7 (locus): Find the locus of $z$ if $\lvert z - 1\rvert = \lvert z + i\rvert$.
Put $z = x + iy$. Then $(x - 1)^{2} + y^{2} = x^{2} + (y + 1)^{2}$, which gives $-2x + 1 = 2y + 1$, so $x + y = 0$. The locus is a straight line, the perpendicular bisector of the points $1$ and $-i$.
Example 8 (quadratic equation): Solve $x^{2} - 2x + 5 = 0$.
$D = 4 - 20 = -16$, so $x = \dfrac{2 \pm 4i}{2} = 1 \pm 2i$.
Common Mistakes in Complex Number Problems
Mistake | Correct approach |
|---|---|
Writing $\sqrt{-a}\sqrt{-b} = \sqrt{ab}$ | For $a, b > 0$, it equals $-\sqrt{ab}$ |
Finding the argument from $\tan^{-1}(b/a)$ without the quadrant | Find the acute angle $\alpha$, and then use the quadrant of $z$ |
Writing $\lvert z\rvert^{2} = z^{2}$ | $\lvert z\rvert^{2} = z\bar{z}$, and $z^{2}$ is a complex number |
Forgetting that $\omega^{2} = \bar{\omega}$ | $\omega^{2}$ is the conjugate of $\omega$, so their sum is $-1$ and their product is $1$ |
Using $\lvert z_1 + z_2\rvert = \lvert z_1\rvert + \lvert z_2\rvert$ | This holds only when $z_1$ and $z_2$ have the same argument. In general, it is $\leq$. |
Taking only one $n$-th root | A non-zero complex number has $n$ distinct $n$-th roots |
Using $i^{4k+3} = i$ | $i^{4k+3} = -i$ |
How to Study Complex Numbers Class 11 Notes
- Read the NCERT chapter first, and download the textbook from eSaral’s NCERT books for Class 11 page.
- Practise the algebra until division and conjugates are fast, and learn the quadrant table for the argument.
- Learn the polar form, De Moivre’s theorem and the cube roots of unity. The trigonometry formulas for Class 11 page helps with the angles.
- Then solve the quadratic equations with complex roots. Revise all the chapters with eSaral’s Class 11 Maths notes, and test yourself with eSaral’s JEE Main chapter-wise PYQs.
Frequently Asked Questions
What is a complex number in Class 11?
A complex number is a number of the form $z = a + ib$, where $a$ is the real part, $b$ is the imaginary part and $i = \sqrt{-1}$. Complex numbers extend the real numbers so that equations like $x^{2} + 1 = 0$ have solutions. eSaral’s Complex Numbers Class 11 Notes explain the algebra, the Argand plane and the polar form with examples.
What are the most important formulas in Complex Numbers for JEE Main?
The modulus $\lvert z\rvert = \sqrt{a^{2} + b^{2}}$ and $z\bar{z} = \lvert z\rvert^{2}$, the polar form $z = r(\cos\theta + i\sin\theta)$, De Moivre’s theorem, the $n$-th roots formula, the triangle inequality, and the cube root properties $1 + \omega + \omega^{2} = 0$ and $\omega^{3} = 1$. The number of questions changes every year, so check previous years’ papers.
What is the difference between modulus and argument of a complex number?
The modulus $\lvert z\rvert = \sqrt{a^{2} + b^{2}}$ is the distance of $z$ from the origin, and it is always a non-negative real number. The argument is the angle that the line from the origin to $z$ makes with the positive real axis. The principal argument lies in $(-\pi, \pi]$, and you find it from $\tan^{-1}\left\lvert\dfrac{b}{a}\right\rvert$ and the quadrant of $z$. In Complex Numbers Class 11 Notes, this is the most common area for mistakes.
Is De Moivre's theorem in the CBSE Class 11 syllabus?
No, it is not in the CBSE Class 11 board syllabus, which has the algebra of complex numbers, the Argand plane, the polar representation and the square root. It is a standard tool in JEE problems on powers and roots of complex numbers. The Complex Numbers Class 11 Notes on this page include it, with the $n$-th roots formula.
How do you find the cube roots of unity?
Solve $x^{3} = 1$. The roots are $1$, $\omega = \dfrac{-1 + i\sqrt{3}}{2}$ and $\omega^{2} = \dfrac{-1 - i\sqrt{3}}{2}$. They satisfy $1 + \omega + \omega^{2} = 0$ and $\omega^{3} = 1$, and $\omega^{2}$ is the conjugate of $\omega$. These properties simplify many expressions in algebra problems.
How do you find the square root of a complex number?
Use the formula $\sqrt{a + ib} = \pm\left[\sqrt{\dfrac{\lvert z\rvert + a}{2}} + i\,\mathrm{sgn}(b)\sqrt{\dfrac{\lvert z\rvert - a}{2}}\right]$, or put $\sqrt{a + ib} = x + iy$ and compare the real and imaginary parts. For example, $\sqrt{3 + 4i} = \pm(2 + i)$.
How do you solve a quadratic equation with a negative discriminant?
If $D = b^{2} - 4ac < 0$, the roots are the conjugate pair $x = \dfrac{-b \pm i\sqrt{4ac - b^{2}}}{2a}$. For $x^{2} - 2x + 5 = 0$, the roots are $1 \pm 2i$. Complex Numbers Class 11 Notes link this to the Fundamental Theorem of Algebra.
Which chapter number is Complex Numbers in Class 11 Maths?
In the current NCERT Class 11 Mathematics book, Complex Numbers and Quadratic Equations is Chapter 4. In older books, it was Chapter 5. These Complex Numbers Class 11 Notes follow the NCERT chapter.
Where can I find more Class 11 Maths notes on eSaral?
eSaral has chapter-wise Class 11 Maths notes, NCERT books, a trigonometry formula sheet, notes on Quadratic Equations and JEE Main chapter-wise PYQs, all linked with these Complex Numbers Class 11 Notes on this page.
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eSaral Academic and Editorial Team
Team eSaral is the collective author profile for educational content created by eSaral’s teachers and academic contributors. The team draws on expertise from IIT graduates, doctors, experienced educators and subject specialists to develop resources for JEE, NEET and school students. Our articles aim to explain concepts clearly and help students study with confidence.
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