Complex numbers are the basic concept in mathematics, extending the concept of numbers to include solutions for equations that don’t have real solutions. Complex numbers have applications in many scientific research, signal processing, electromagnetism, fluid dynamics, quantum mechanics, engineering and vibration analysis.
Complex Numbers are numbers that can be written in the form (a + ib), where a represents the real part and ib represents the imaginary part and a & b are the real numbers and i is an imaginary unit called “iota” that represent √-1 and i2 = -1.
Example: 2 + 3i, is a complex number in which 2 is a real number and 3i is an imaginary number. They can be written as a + ib where a and b are rational numbers that can be represented on a number line extending to infinity.
What is iota (i) ?
The iota is referred to by the alphabet ‘i’. iota is helpful to represent the imaginary part of the complex number. It is also very helpful to find the square root of negative numbers as the value of i2 = -1, which is used to find the value of the square root of negative numbers,
Example: √-9 = √i232 = ±3i
Power of iota (i)
The value of i2 = -1 is the fundamental aspect of a complex number.
i = √-1
i2 = -1
i3 = i.i2 = i(-1) = -i
i4 = (i2)2 = (-1)2 = 1
i4n = (i2)2n = (-1)2n = (-12)n = (1)n = 1
i4n + 1 = i
i4n + 2 = -1
i4n + 3 = -i
Classification of Complex Numbers
As we know the standard form of a complex number is z = (a + ib) where a, b ∈ R, and “i” is iota (an imaginary unit). So depending on the values of “a” (called real part) and “b” (called imaginary part), they are classified into four types.
Zero Complex Number:
If a = 0 & b = 0, then it is called a zero complex number.
The only example of this is 0.
Purely Real Numbers:
If a ≠ 0 & b = 0, then it is called a purely real number i.e., a number with no imaginary part.
If a = 0 & b ≠ 0, then it is called a purely imaginary number i.e., a number with no real part.
All numbers with no real parts are examples of this type of number i.e., -7i, -5i, -i, i, 5i, 7i, etc.
Imaginary Numbers:
If a ≠ 0 & b ≠ 0, then it is called an imaginary number.
For example, (-1 – i), (1 + i), (1 – i), (2 + 3i), etc.
Different Forms of Complex Numbers
There are various forms of complex numbers that are,
Rectangular FormForm: Itisalso called Standard Form and it is represented by(a + ib), where a and b are the real numbers. Example: (5 + 5i), (-7i), (-3 – 4i), etc.
Polar Form: It is the representation of a complex number where coordinates are represented as (r, θ), where r is the distance from the origin and θ is the angle between the line joining the point and origin and the positive x-axis. Any complex number is represented as r [cos θ + i sin θ]. Example: [cos π/2 + i sin π/2], 5[cos π/6 + i sin π/6], etc.
Exponential Form: The Exponential Form, is the representation of complex numbers using Euler’s Formula and in this form, the complex number is represented by reiθ, where r is the distance of a point from the origin and θ is the angle between the positive x-axis and radius vector. Examples: ei(0), ei(π/2), 5.ei(π/6), etc.
Note: All three forms of the complex numbers discussed above are interconvertible i.e., these can be converted from one form to another very easily.
Geometrical Representation of Complex Numbers
As we know is complex number (z = a + i b) is represented by a unique point p(a, b) on the complex plane and every point on the complex plane represents a unique complex number.
Complex Plane
The plane on which the complex numbers are uniquely represented is called the Complex plane or Argand plane or Gaussian plane.
The Complex plane has two axes:
X-axis or Real Axis
Y-axis or Imaginary Axis
X-axis or Real Axis
All the purely real complex numbers are uniquely represented by a point on it.
The real part Re(z) of all complex numbers is plotted concerning it.
That’s why theX-axis is also called the Real axis.
Y-axis or Imaginary Axis
All the purely imaginary complex numbers are uniquely represented by a point on it.
Imaginary part Im(z) of all complex numbers is plotted concerning it.
That’s why the Y-axis is also called the Imaginary axis.
Steps to represent Complex Numbers on a Complex plane
To represent any complex number z = (a + i b) on the complex plane follow these conventions:
The real part of z (Re(z) = a) becomes the X-coordinate of the point p
The imaginary part of z (Im(z) = b) becomes the Y-coordinate of the point p
And finally z (a + i b) ⇒ p (a, b) which is a point on the complex plane.
Modulus and Argument of Complex Number
In the complex plane, a complex number z = a + ib can be represented as a point (a, b) where a is the real part and b is the imaginary part.
The modulus of a complex number represents its absolute value and is defined as the distance between the origin and the given point in the complex plane. It is also referred to as the magnitude of the complex number. For a complex number z = a + ib, the modulus is given by:
The argument of a complex number is the angle θ\thetaθ between its radius vector and the positive real (x) axis. Mathematically, for z = a + ib, it is expressed as:
Operations on Complex Numbers
The following operations can be performed on Complex Numbers:
Addition: We can add two complex numbers, by simply adding their real and imaginary parts separately.
For example, (3 + 2i) + (1 + 4i) = 4 + 6i.
Subtraction: We can subtract two complex numbers, by simply subtracting their real and their imaginary parts separately.
For example, (3 + 2i) – (1 + 4i) = 2 – 2i.
Multiplication: We can multiply two complex numbers using the distributive property and the fact that i2 = -1.
Division: We can divide one complex number by another, by simply multiplying both the numerator and the denominator by the complex conjugate of the denominator and further simplifying the expression.
We can easily find the conjugate of a complex number, by simply changing the sign of its imaginary part. Conjugate of a complex number is often denoted with a bar above the number, such as z̄.
For example, the conjugate of 3 + 2i is 3 – 2i.
Identities for Complex Numbers
For any two complex numbers z1 and z2 the following algebraic identities can be given:
(z1 + z2)2 = (z1)2 + (z2)2 + 2 z1 × z2
(z1 – z2)2 = (z1)2 + (z2)2 – 2 z1 × z2
(z1)2 – (z2)2 = (z1 + z2)(z1 – z2)
(z1 + z2)3 = (z1)3 + 3(z1)2 z2 +3(z2)2 z1 + (z2)3
(z1 – z2)3 = (z1)3 – 3(z1)2 z2 +3(z2)2 z1 – (z2)3
Some other formulas related to complex numbers are as follows:
Euler’s Formula: Euler’s Formula shows the relation between the imaginary power of exponent and trigonometric ratio sin and cos and is given by:
eix = cos x + i sin x
De Moivre’s Formula: The Moivre’s Formula expresses the nth power of a complex number in polar form and is given by:
(cos x + i sin x)n = cos (nx) + i sin (nx)
Properties of Complex Numbers
There are various properties of complex numbers, some of which are as follows:
For any complex number z = a + ib, if z = 0 then a = 0 as well as b = 0.
For 4 real numbers a, b, c, and d such that z1 =a + ib and z2 = c + id. If z1 = z2 then a = c, and b=d.
Addition of a complex number with its conjugate results in a purely real number i.e., z + z̄ = Real Number.
Let z = a + ib, z + z̄ = a + ib + a – ib ⇒ z + z̄ = 2a (which is purely real)
The product of a complex number with its conjugate results is a purely real number as well i.e., z × z̄ = Real Number.
Let z = a + ib, then z × z̄ = (a + ib) × (a – ib) ⇒ z × z̄= a2 – i2b2 ⇒ z × z̄ = a2 + b2 (which is purely real)
Complex Numbers are commutative under the operation of addition and multiplication. Let’s consider two complex numbers z1 and z2, and then.
z1+z2 = z2+z1 z1 × z2 = z2 × z1
Complex Numbers are associative with the operation of addition and multiplication. Let’s consider three complex numbers z1, z2, and z3 then
Complex Numbers hold the distributive property of multiplication over addition as well. Let’s consider three complex numbers z1, z2, and z3 then
z1 × (z2+z3) = z1 × z2 + z1 × z3
Importance of Complex Numbers
Complex numbers extend the real number system to include solutions to equations that have no real solutions and also provide a complete framework for solving a wide range of mathematical problems and have numerous applications in various fields.
Solving Polynomial Equations: Some polynomial equations do not have real solutions. For instance, the equation x2 + 1 = 0 has no real solution because the square of any real number is non-negative. However, using complex numbers, the solutions are x = i and x = −i.
Mathematical Completeness: The set of complex numbers is algebraically closed, meaning every non-constant polynomial equation with complex coefficients has a solution in the complex numbers. This is a significant advantage over the real numbers, which do not have this property.
Simplifying Calculations: In some cases, working with complex numbers can simplify mathematical expressions and calculations, especially in trigonometry and calculus.
Applications in Various Fields: Complex numbers are used extensively in engineering, physics, computer science, and other fields to model and solve real-world problems.
Solved Examples of Complex Numbers
Example 1: Plot these complex numbers z= 3 + 2i on the Complex plane. Solution:
Given: z = 3 + 2 i
So, the point is z(3, 2). Now we plot this point on the below graph, here in this graph x-axis represents the real part and y-axis represents the imaginary part.
Example 2: Plot these complex numbers z1 = (2 + 2 i), z2 = (-2 + 3 i), z3 = (-3 – i), z4 = (1 – i) on the Complex plane. Solution:
So, the points are z1 (2, 2), z2(-2, 3), z3(-3, -1), and z4(1, -1). Now we plot these points on the below graph, here in this graph x-axis represents the real part and y-axis represents the imaginary part.
FAQs on Complex Numbers
Define Complex Numbers.
Numbers of the form a + ib is called complex number, where a and b are the real number and i is the imaginary unit which represents square root of -1.
What is a Complex Number example?
A complex number is in the form a + bi, where i is the imaginary unit. Example: 3 + 2i.
Is 7 a Complex Number?
Yes, 7 is a complex number because it can be written as 7+0i7 + 0i7+0i.
What is the difference between a real number and a complex number?
Difference between real and complex numbers is that we only need one number to represent any real number but need two real numbers to represent any complex number.
What is the real part and imaginary part of a complex number?
In a complex number a + ib, a is the real part of the complex number, and b is called the imaginary part of the complex number.
What is the complex conjugate of a complex number?
For a complex number a + ib its complex conjugate will be a – ib . Complex conjugates can be found by simply changing the sign of the imaginary part.
What is the modulus of a complex number?
Distance between the origin and the point represented by a complex number in the argand plane is called the modulus of that complete number and for z = a + ib, it is mathematically given by: |z| = √(a2 + b2)
What is the argument of a complex number?
Angle between the radius vector of a complex number and the positive x-axis is called the argument of a complex number and for z = a + ib, it is mathematically given by: θ = tan-1(b/a)
What is the polar form of a complex number?
For any complex number, z = a + ib, the polar form of this is given by: r [cos θ + i sin θ]
What is Euler’s formula?
Euler’s Formula shows the relation between the imaginary power of exponent and trigonometric ratio sin and cos and is given by: eix = cos x + i sin x
Case Study: Complex Numbers in Electrical Engineering
Introduction:
Complex numbers, often represented in the form a + bi (where a and b are real numbers, and i is the imaginary unit with i² = -1), might seem abstract. However, they are indispensable tools in various fields, especially electrical engineering. This case study explores how complex numbers simplify the analysis of alternating current (AC) circuits.
Background:
Electrical circuits can be categorized as direct current (DC) or alternating current (AC). DC circuits involve current flowing in one direction, while AC circuits involve current that periodically reverses direction. Analyzing DC circuits is relatively straightforward using Ohm’s Law (V = IR). However, AC circuits introduce complexities due to the presence of reactive components like inductors and capacitors. These components impede current flow differently depending on the frequency of the AC signal.
The Problem:
Consider a simple AC circuit containing a resistor (R), an inductor (L), and a capacitor (C) connected in series. We want to determine the impedance (Z) of this circuit, which is the total opposition to current flow in an AC circuit. Impedance is analogous to resistance in DC circuits, but it also considers the frequency-dependent effects of inductors and capacitors.
Traditional Approach (Without Complex Numbers):
Calculating impedance without complex numbers involves cumbersome trigonometric calculations and phase angle considerations. It becomes difficult to represent the phase relationship between voltage and current, which is crucial in AC circuit analysis.
The Complex Number Solution:
Complex numbers provide an elegant and efficient way to represent and manipulate AC circuit quantities.
Representation: We represent impedance (Z), resistance (R), inductive reactance (Xʟ), and capacitive reactance (Xc) as complex numbers:
Resistance: R (a real number)
Inductive Reactance: jωL (where ω is the angular frequency and L is the inductance)
Capacitive Reactance: -j/(ωC) (where C is the capacitance)
Impedance: Z = R + j(ωL – 1/(ωC))
Notice how the imaginary part of the impedance accounts for the reactance of the inductor and capacitor. The ‘j’ represents the imaginary unit i.
Calculation: The impedance of the series RLC circuit is simply the sum of the complex representations of the individual components, as shown above. This is a significant simplification compared to the trigonometric approach.
Magnitude and Phase: The magnitude of the complex impedance (|Z|) represents the total opposition to current flow. The argument (angle) of the complex impedance (φ) represents the phase difference between the voltage and current. These are easily calculated:
|Z| = sqrt(R² + (ωL – 1/(ωC))²)
φ = arctan((ωL – 1/(ωC))/R)
Ohm’s Law for AC Circuits: Ohm’s Law can be extended to AC circuits using complex numbers:
V = IZ (where V and I are also complex representations of voltage and current, respectively).
Phase Information: Complex numbers naturally represent the phase relationship between voltage and current, a critical aspect of AC circuit analysis.
Concise Representation: Complex notation provides a compact and efficient way to represent circuit elements and their interactions.
Powerful Analysis Tools: Complex numbers enable the use of powerful mathematical tools like phasors and Fourier analysis for analyzing complex AC circuits.
Conclusion:
This case study demonstrates the power of complex numbers in simplifying the analysis of AC circuits. By representing circuit quantities as complex numbers, electrical engineers can easily calculate impedance, phase angles, and other crucial parameters, leading to more efficient circuit design and analysis. This example highlights how abstract mathematical concepts can have practical and profound applications in real-world engineering problems. Sources and related content
Er. Neeraj K.Anand is a freelance mentor and writer who specializes in Engineering & Science subjects. Neeraj Anand received a B.Tech degree in Electronics and Communication Engineering from N.I.T Warangal & M.Tech Post Graduation from IETE, New Delhi. He has over 30 years of teaching experience and serves as the Head of Department of ANAND CLASSES. He concentrated all his energy and experiences in academics and subsequently grew up as one of the best mentors in the country for students aspiring for success in competitive examinations.
In parallel, he started a Technical Publication "ANAND TECHNICAL PUBLISHERS" in 2002 and Educational Newspaper "NATIONAL EDUCATION NEWS" in 2014 at Jalandhar. Now he is a Director of leading publication "ANAND TECHNICAL PUBLISHERS", "ANAND CLASSES" and "NATIONAL EDUCATION NEWS".
He has published more than hundred books in the field of Physics, Mathematics, Computers and Information Technology. Besides this he has written many books to help students prepare for IIT-JEE and AIPMT entrance exams. He is an executive member of the IEEE (Institute of Electrical & Electronics Engineers. USA) and honorary member of many Indian scientific societies such as Institution of Electronics & Telecommunication Engineers, Aeronautical Society of India, Bioinformatics Institute of India, Institution of Engineers. He has got award from American Biographical Institute Board of International Research in the year 2005.
CBSE Class 11 Maths Syllabus for 2023-24 with Marking Scheme
CBSE syllabus for class 11 Maths is divided into 5 units. The table below shows the units, number of periods and marks allocated for maths subject. The maths theory paper is of 80 marks and the internal assessment is of 20 marks.
No.
Units
Marks
I.
Sets and Functions
23
II.
Algebra
25
III.
Coordinate Geometry
12
IV.
Calculus
08
V.
Statistics and Probability
12
Total Theory
80
Internal Assessment
20
Grand Total
100
2025-26 CBSE Class 11 Maths Syllabus
Below you will find the CBSE Class Maths Syllabus for students.
Unit-I: Sets and Functions
1. Sets
Sets and their representations, empty sets, finite and infinite sets, equal sets, subsets, and subsets of a set of real numbers, especially intervals (with notations), universal set, Venn diagrams, union and intersection of sets, difference of sets, complement of a set and properties of complement.
2. Relations & Functions
Ordered pairs, Cartesian product of sets, number of elements in the Cartesian product of two finite sets, Cartesian product of the set of reals with itself (upto R x R x R), definition of relation, pictorial diagrams, domain, co-domain and range of a relation. Function as a special type of relation. Pictorial representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum, exponential, logarithmic and greatest integer functions, with their graphs. Sum, difference, product and quotients of functions.
3. Trigonometric Functions
Positive and negative angles, measuring angles in radians and in degrees and conversion from one measure to another, definition of trigonometric functions with the help of unit circle, truth of the identity, signs of trigonometric functions, domain and range of trigonometric functions and their graphs, expressing sin (x±y) and cos (x±y) in terms of sinx, siny, cosx & cosy and their simple applications.
Unit-II: Algebra
1. Complex Numbers and Quadratic Equations
Need for complex numbers, especially√−1, to be motivated by the inability to solve some of the quadratic equations. Algebraic properties of complex numbers, Argand plane.
2. Linear Inequalities
Linear inequalities, algebraic solutions of linear inequalities in one variable and their representation on the number line.
3. Permutations and Combinations
The fundamental principle of counting. Factorial n. (n!) Permutations and combinations, derivation of Formulae for nPr and nCr and their connections, simple applications.
4. Binomial Theorem
Historical perspective, statement and proof of the binomial theorem for positive integral indices, Pascal’s triangle, simple applications.
5. Sequence and Series
Sequence and series, arithmetic progression (A. P.), arithmetic mean (A.M.), geometric progression (G.P.), general term of a G.P., sum of n terms of a G.P., infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M.
Unit-III: Coordinate Geometry
1. Straight Lines
Brief recall of two-dimensional geometry from earlier classes. Slope of a line and angle between two lines. Various forms of equations of a line: parallel to axis, point-slope form, slope-intercept form, two-point form, intercept form and normal form. General equation of a line. Distance of a point from a line.
2. Conic Sections
Sections of a cone: circles, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle.
3. Introduction to Three-Dimensional Geometry
Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points.
Unit-IV: Calculus
1. Limits and Derivatives
Derivative introduced as rate of change both as that of distance function and geometrically, intuitive idea of limit, limits of polynomials and rational functions trigonometric, exponential and logarithmic functions, definition of derivative relate it to the slope of the tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions.
Unit-V: Statistics and Probability
1. Statistics
Measures of Dispersion: Range, mean deviation, variance and standard deviation of ungrouped/grouped data.
2. Probability
Events; occurrence of events, ‘not’, ‘and’ and ‘or’ events, exhaustive events, mutually exclusive events, Axiomatic (set theoretic) probability, connections with other theories of earlier classes. Probability of an event, probability of ‘not’, ‘and’ and ‘or’ events.
Students can also get the syllabus of all the subjects by visiting CBSE Class 11 Syllabus page. Learn Maths & Science in an interactive & fun-loving way with Anand Classes App/Tablet.
Frequently Asked Questions on CBSE Class 11 Maths Syllabus 2025-26
Q1
What is the marks distribution for internals and theory exams according to the CBSE Maths Syllabus for Class 11?
The marks distribution for internals is 20 marks and the theory exam is 80 marks based on the CBSE Class 11 Maths Syllabus.
Q2
Which is the most important chapter in the CBSE Class 11 Maths Syllabus?
The important chapter in the CBSE Class 11 Maths Syllabus is Algebra which is for 25 marks in the overall weightage.
Q3
What are the chapters covered in Unit III of the CBSE Class 11 Maths Syllabus?
The chapters covered in Unit III of the CBSE Class 11 Maths Syllabus are straight lines, conic sections and an introduction to three-dimensional geometry.
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