Argand plane is a mathematical concept used to graphically represent complex numbers. It is similar to the Cartesian coordinate system used for real numbers but is specially specified to handle the two-dimensional nature of complex numbers. Argand Plane is also known as the complex plane.
Argand Plane
Table of Contents
Argand Plane is a graph with real numbers on the horizontal axis and imaginary numbers on the vertical. It helps understand complex numbers visually. We have explained in detail about Argand Plane with the properties and examples below.
Argand Plane
Argand plane is used to represent a complex number in a two-dimensional plane.
Argand plane id is named for Jean-Robert Argand, a mathematician. It is a graphical method to represent complex numbers. It’s similar to a two-dimensional plane, having the real numbers on the horizontal axis and the imaginary numbers on the vertical axis.
These numbers are then positioned as the new points on the plane, while each point has the coordinates (a, b) which corresponds to the real and imaginary parts of the number. It is a very good tool for the visualization of abstract arithmetic and geometry.
Diagram of Argand Plane
Polar Representation In Argand Plane
In the Argand plane, which is a graphical representation of complex numbers, polar representation is a way to express complex numbers using their magnitude (distance from the origin) and angle (or argument) with respect to the positive real axis.
Imagine a two-dimensional plane where the horizontal axis represents the real part of a complex number, and the vertical axis represents the imaginary part. In this plane, the origin (0,0) represents the complex number 0.
Now, let’s consider a complex number ? = ? + ??, where a is the real part and b is the imaginary part. The magnitude (or modulus) of this complex number, denoted by ∣z∣, is the distance from the origin to the point representing the complex number in the plane.
It’s calculated using the Pythagorean theorem: ∣?∣ = √a2 + b2
Properties Of Argand Plane
Argand plane shows visually the relationship between complex numbers and their real and imaginary parts.
It brings understanding of number properties, e.g., conjugates, more clearly through geometric visualization.
The points’ distance from the origin equals the modulus of a complex number, the absolute value of its imaginary part.
The angle whose vector points along the positive real axis is the argument of a complex number.
The representation of addition and subtraction of complex numbers by vectors resembles vectorial operations.
Scaling and the rotation of multiplication are implied, while division is its inverse in the Argand plane.
Complex conjugates occur as a mirror image on the real axis.
The notion of roots and powers is illustrated using the division of modulus circle and the scale and the rotation.
Complex numbers can be dealt with geometrically in argand’s plane regarding functions that are complex.
Geometric representation of the solutions is used to help understand equations in their solutions.
Solved Examples on Argand Plane
Example 1: Addition of Complex Numbers on the Argand Plane
Let’s add two complex numbers: z1 = 3 + 4i and z2 = 2 – i.
Solution:
To add them, we simply add their real and imaginary parts separately:
= (3 + 4i) + (2 – i)
= (3 + 2) + (4i – i)
= 5 + 3i
So, the result of adding z1 and z2 is the complex number 5 + 3i.
On the Argand plane, this corresponds to moving 5 units to the right along the real axis and 3 units up along the imaginary axis from the origin.
Example 2: Multiplication of Complex Numbers on the Argand Plane
Let’s multiply two complex numbers: z1 = 2 + 3i and z2 = 1 – 2i.
Solution:
To multiply them, we use the distributive property:
= (2 + 3i) × (1 – 2i)
= 2 × 1 + 2 × (-2i) + 3i × 1 + 3i × (-2i)
= 2 – 4i + 3i – 6i2
Remembering that i2 = -1,
we simplify: = 2 – 4i + 3i + 6 = 8 – i
So, the result of multiplying z1 and z2 is the complex number 8 – i.
On the Argand plane, this corresponds to moving 8 units to the right along the real axis and 1 unit down along the imaginary axis from the origin.
Jean-Robert Argand was a Swiss mathematician after whom the Argand plane is named. He introduced this graphical representation in 1806.
What is polar representation in the Argand plane?
The polar representation of complex numbers in the Argand plane is an alternative to the rectangular form that is based on the distance (magnitude) and angle (or argument) they have with respect to the positive real axis.
What properties of complex numbers are illustrated in the Argand plane?
The Argand plane visually represents properties such as modulus, argument, conjugates, addition, subtraction, multiplication, division, roots, powers, and geometric solutions of equations involving complex numbers.
How does the Argand plane aid in understanding complex arithmetic?
By providing a geometric visualization of complex numbers and their operations, the Argand plane makes it easier to grasp complex arithmetic concepts and relationships.
What are some applications of the Argand plane?
The Argand plane finds applications in various fields, including mathematics, physics, engineering, and signal processing, where complex numbers play a crucial role in modeling and analysis.
Polar representation of complex numbers
Argand plane polar form explanation
Definition of polar representation in Argand plane
Properties of polar representation of complex numbers
Free PDF download polar representation of complex numbers
Polar representation formula and derivation
NCERT solutions polar form of complex numbers
Polar coordinates in Argand diagram
Complex numbers modulus and argument explanation
Conversion of complex numbers to polar form
FAQs on polar representation of complex numbers
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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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