Welcome, future engineers! In this deep dive, we're going to thoroughly explore the fundamental representation of complex numbers and their elegant geometric interpretation using the Argand diagram. This forms the backbone for understanding advanced concepts in complex numbers, especially for JEE Main & Advanced.
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Understanding the Standard Form: $z = a + ib$
At its core, a
complex number is an extension of the real number system, introduced to provide solutions to equations that have no real roots, such as $x^2 + 1 = 0$. The key to this extension is the introduction of the imaginary unit, denoted by
$i$, which is defined as:
$mathbf{i = sqrt{-1}}$
From this definition, it follows that $mathbf{i^2 = -1}$.
A complex number, in its standard or
algebraic form, is generally expressed as:
$mathbf{z = a + ib}$
Here, 'a' and 'b' are
real numbers. Let's break down the components:
* The term
'a' is called the
real part of the complex number $z$, denoted as
Re(z) = a.
* The term
'b' is called the
imaginary part of the complex number $z$, denoted as
Im(z) = b.
Important Note for JEE: It's crucial to remember that both 'a' and 'b' *must be real numbers*. The 'i' is just a marker indicating the imaginary component. For instance, if you have an expression like $z = 3 + isqrt{2}$, then $Re(z) = 3$ and $Im(z) = sqrt{2}$.
Let's look at some special cases:
1.
Purely Real Numbers: If $b = 0$, then $z = a + i(0) = a$. This means every real number is a complex number with its imaginary part equal to zero.
* Example: $z = 5 implies Re(z)=5, Im(z)=0$.
2.
Purely Imaginary Numbers: If $a = 0$, then $z = 0 + ib = ib$. These are complex numbers whose real part is zero.
* Example: $z = -3i implies Re(z)=0, Im(z)=-3$.
3.
Zero Complex Number: A complex number $z = a+ib$ is zero if and only if both its real and imaginary parts are zero, i.e., $a=0$ and $b=0$.
####
Equality of Complex Numbers
Two complex numbers, $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$, are said to be equal if and only if their respective real parts are equal and their respective imaginary parts are equal.
$mathbf{a_1 + ib_1 = a_2 + ib_2 iff a_1 = a_2 ext{ and } b_1 = b_2}$
This property is extremely useful for solving equations involving complex numbers, as it allows us to equate the real and imaginary parts separately, often leading to a system of two real equations.
Example 1: Find the real numbers $x$ and $y$ such that $(x-1) + i(y+2) = 3 + 4i$.
Solution:
Using the equality of complex numbers, we equate the real parts and the imaginary parts:
Real parts: $x-1 = 3 implies x = 4$
Imaginary parts: $y+2 = 4 implies y = 2$
So, $x=4$ and $y=2$.
JEE Focus: Algebraic Simplification to $a+ib$ Form
Often, complex numbers are not directly given in the $a+ib$ form. You'll need to perform algebraic operations (addition, subtraction, multiplication, division, powers) to convert them into this standard form before identifying real and imaginary parts. Division typically involves multiplying the numerator and denominator by the conjugate of the denominator.
Example 2: Express $z = frac{1+i}{1-i}$ in the form $a+ib$ and find its real and imaginary parts.
Solution:
To express $z$ in $a+ib$ form, we multiply the numerator and denominator by the conjugate of the denominator, which is $1+i$.
$z = frac{1+i}{1-i} imes frac{1+i}{1+i}$
$z = frac{(1+i)^2}{(1)^2 - (i)^2}$
$z = frac{1^2 + 2(1)(i) + i^2}{1 - (-1)}$
$z = frac{1 + 2i - 1}{1 + 1}$
$z = frac{2i}{2}$
$z = i$
So, $z = 0 + 1i$.
Therefore, $Re(z) = 0$ and $Im(z) = 1$.
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The Argand Diagram: Visualizing Complex Numbers
While the algebraic form $z = a + ib$ is powerful for calculations, it doesn't immediately give us a visual sense of the complex number. This is where the
Argand Diagram (also known as the
Complex Plane or
Gaussian Plane) comes into play. It provides a geometric representation, allowing us to plot and visualize complex numbers as points or vectors in a 2D plane.
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Mapping Complex Numbers to Points
The core idea is simple: just as a point $(x, y)$ in the Cartesian coordinate system represents an ordered pair of real numbers, a complex number $z = a + ib$ can be uniquely represented by an ordered pair of real numbers $(a, b)$. We can then plot this ordered pair on a two-dimensional plane.
In the Argand Diagram:
* The
horizontal axis is called the
Real Axis. It represents the real part 'a' of the complex number.
* The
vertical axis is called the
Imaginary Axis. It represents the imaginary part 'b' of the complex number.
So, a complex number $z = a + ib$ is represented by the point
$P(a, b)$ in the complex plane. Alternatively, it can be represented by a position vector $vec{OP}$ from the origin $O(0,0)$ to the point $P(a,b)$.
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Plotting Complex Numbers – Examples
Let's plot a few complex numbers on the Argand diagram:
1. $z_1 = 3 + 2i implies$ Point $(3, 2)$
2. $z_2 = -2 + 4i implies$ Point $(-2, 4)$
3. $z_3 = -4 - 3i implies$ Point $(-4, -3)$
4. $z_4 = 5 - i implies$ Point $(5, -1)$
5. $z_5 = 3$ (purely real) $implies$ Point $(3, 0)$ – Lies on the Real Axis.
6. $z_6 = -2i$ (purely imaginary) $implies$ Point $(0, -2)$ – Lies on the Imaginary Axis.
```html
Visual Representation (Imagine this as a standard XY plane, but X is Real Axis and Y is Imaginary Axis):
^ Imaginary Axis (Im(z))
|
z2(-2,4) .
| . z1(3,2)
|
<----(-4,0)---+---(0,0)----+---(3,0)-----> Real Axis (Re(z))
| O z5
|
. z6(0,-2)
|
z3(-4,-3) .
. z4(5,-1)
```
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Geometric Interpretation of Related Complex Numbers
The Argand diagram makes it easy to visualize relationships between complex numbers:
*
Conjugate: The conjugate of $z = a+ib$ is $ar{z} = a-ib$. Geometrically, $ar{z}$ is the reflection of $z$ across the
Real Axis. If $z$ is at $(a,b)$, $ar{z}$ is at $(a,-b)$.
*
Negative: The negative of $z = a+ib$ is $-z = -a-ib$. Geometrically, $-z$ is the reflection of $z$ through the
Origin. If $z$ is at $(a,b)$, $-z$ is at $(-a,-b)$.
Example 3: Plot $z=2+3i$, $ar{z}$, and $-z$ on the Argand plane.
Solution:
* $z = 2+3i implies$ Point $P(2,3)$
* $ar{z} = 2-3i implies$ Point $Q(2,-3)$ (reflection of $P$ across Real Axis)
* $-z = -2-3i implies$ Point $R(-2,-3)$ (reflection of $P$ through Origin)
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Modulus of a Complex Number: Distance from the Origin
One of the most significant geometric interpretations from the Argand diagram is the concept of the
modulus of a complex number.
The
modulus of a complex number $z = a+ib$, denoted by
$|z|$, represents the
distance of the point $P(a,b)$ from the origin $O(0,0)$ in the Argand plane.
Using the distance formula from coordinate geometry, the distance between $(a,b)$ and $(0,0)$ is:
$|z| = sqrt{(a-0)^2 + (b-0)^2}$
$mathbf{|z| = sqrt{a^2 + b^2}}$
This is a fundamental formula. Note that $|z|$ is always a non-negative real number. It's akin to the magnitude of a vector.
Example 4: Find the modulus of $z = 3 - 4i$.
Solution:
Here, $a=3$ and $b=-4$.
$|z| = sqrt{3^2 + (-4)^2}$
$|z| = sqrt{9 + 16}$
$|z| = sqrt{25}$
$|z| = 5$
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Distance Between Two Complex Numbers
The Argand diagram also allows us to interpret the distance between any two complex numbers.
If $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$ are two complex numbers, then the expression
$|z_1 - z_2|$ represents the
distance between the points corresponding to $z_1$ and $z_2$ in the Argand plane.
Let's derive this:
$z_1 - z_2 = (a_1 + ib_1) - (a_2 + ib_2)$
$z_1 - z_2 = (a_1 - a_2) + i(b_1 - b_2)$
Now, taking the modulus of this difference:
$|z_1 - z_2| = sqrt{(a_1 - a_2)^2 + (b_1 - b_2)^2}$
This is precisely the distance formula between two points $(a_1, b_1)$ and $(a_2, b_2)$ in a Cartesian plane.
JEE Application: Locus Problems
This interpretation is incredibly powerful for solving locus problems in complex numbers.
* The equation
$|z - z_0| = r$ represents all complex numbers $z$ whose distance from a fixed complex number $z_0$ is constant and equal to $r$. Geometrically, this describes a
circle with center $z_0$ and radius $r$.
* The equation
$|z - z_1| = |z - z_2|$ represents all complex numbers $z$ that are equidistant from two fixed complex numbers $z_1$ and $z_2$. Geometrically, this describes the
perpendicular bisector of the line segment joining $z_1$ and $z_2$.
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Advanced JEE Example: Manipulating the $a+ib$ Form
This example demonstrates how the fundamental concept of equating real and imaginary parts (from the $a+ib$ form) is used in more complex algebraic problems.
Example 5: If $(x+iy)^3 = u+iv$, show that $frac{u}{x} + frac{v}{y} = 4(x^2-y^2)$.
Solution:
Given $(x+iy)^3 = u+iv$.
First, expand the left side using the binomial theorem or $(A+B)^3 = A^3 + 3A^2B + 3AB^2 + B^3$:
$(x+iy)^3 = x^3 + 3x^2(iy) + 3x(iy)^2 + (iy)^3$
$(x+iy)^3 = x^3 + 3ix^2y + 3x(i^2y^2) + i^3y^3$
Recall $i^2 = -1$ and $i^3 = i^2 cdot i = -i$.
$(x+iy)^3 = x^3 + 3ix^2y + 3x(-1)y^2 + (-i)y^3$
$(x+iy)^3 = x^3 + 3ix^2y - 3xy^2 - iy^3$
Now, group the real and imaginary parts:
$(x+iy)^3 = (x^3 - 3xy^2) + i(3x^2y - y^3)$
We are given that this is equal to $u+iv$.
By the equality of complex numbers, we equate the real and imaginary parts:
$u = x^3 - 3xy^2$ --- (1)
$v = 3x^2y - y^3$ --- (2)
Now, we need to show $frac{u}{x} + frac{v}{y} = 4(x^2-y^2)$.
From (1), divide by $x$ (assuming $x
eq 0$):
$frac{u}{x} = frac{x^3 - 3xy^2}{x} = x^2 - 3y^2$ --- (3)
From (2), divide by $y$ (assuming $y
eq 0$):
$frac{v}{y} = frac{3x^2y - y^3}{y} = 3x^2 - y^2$ --- (4)
Now, add equations (3) and (4):
$frac{u}{x} + frac{v}{y} = (x^2 - 3y^2) + (3x^2 - y^2)$
$frac{u}{x} + frac{v}{y} = x^2 + 3x^2 - 3y^2 - y^2$
$frac{u}{x} + frac{v}{y} = 4x^2 - 4y^2$
$mathbf{frac{u}{x} + frac{v}{y} = 4(x^2 - y^2)}$
This completes the proof. This problem showcases how fundamental algebraic manipulation and the correct identification of real and imaginary parts are key to solving JEE-level problems.
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Summary and Key Takeaways
The representation of complex numbers in the $a+ib$ form and their visualization on the Argand diagram are foundational concepts.
Concept |
Description |
Key Formula/Property |
|---|
Algebraic Form ($a+ib$) |
Standard way to write a complex number. 'a' is real part, 'b' is imaginary part. Both 'a' and 'b' are real numbers. |
$z = a+ib$, $Re(z)=a$, $Im(z)=b$ |
Equality |
Two complex numbers are equal if their real and imaginary parts are individually equal. |
$a_1+ib_1 = a_2+ib_2 iff a_1=a_2 ext{ and } b_1=b_2$ |
Argand Diagram |
Geometric representation of complex numbers as points $(a,b)$ or position vectors in a 2D plane (Complex Plane). |
Real Axis (horizontal), Imaginary Axis (vertical) |
Modulus |
Distance of the complex number $z=a+ib$ from the origin $(0,0)$ in the Argand plane. Always non-negative. |
$|z| = sqrt{a^2+b^2}$ |
Distance Between Two Points |
The distance between $z_1$ and $z_2$ in the Argand plane. Crucial for locus problems. |
$|z_1-z_2| = sqrt{(a_1-a_2)^2 + (b_1-b_2)^2}$ |
Mastering these concepts is crucial for building a strong foundation in complex numbers for both board exams and competitive exams like JEE. Practice converting complex expressions to $a+ib$ form and interpreting geometric properties on the Argand diagram.