How to Visualize Complex Numbers on the Argand Plane
The set of complex numbers forms a two-dimensional extension of the real numbers, where each complex number can be associated with a unique point in a plane, enabling a direct geometric interpretation and facilitating algebraic operations such as addition, multiplication, and finding modulus or argument through coordinate geometry.
Coordinates and Representation of Complex Numbers in the Argand Plane
Let $z = a + ib, \, a, b \in \mathbb{R}$ be a complex number. This number may be associated with the ordered pair $(a, b)$, where $a$ is the real part and $b$ is the imaginary part (without the $i$). On a plane called the Argand Diagram, plot the point $P(a, b)$. The horizontal axis is the real axis ($x$-axis), and the vertical axis is the imaginary axis ($y$-axis).
Thus, every complex number determines a unique point in the plane, and conversely, every point $(x, y)$ in this plane corresponds to the complex number $x + iy$. The modulus is the distance from the origin $O(0,0)$ to $P(a,b)$ and is given by $|z| = \sqrt{a^2 + b^2}$, while the argument is the angle $\theta = \arg(z)$ measured from the positive real axis to the line segment $OP$.
The Understanding Complex Numbers page provides further details on set membership and properties.
Conversion of Complex Numbers to Polar Form
The algebraic or Cartesian form $z = a + ib$ can be rewritten in polar form using trigonometric identities. Set $a = r \cos \theta$, $b = r \sin \theta$, where $r = |z| = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}\left(\dfrac{b}{a}\right)$ is the principal value of the argument. Substitution gives $z = r (\cos \theta + i \sin \theta)$.
In this form, $r$ represents the modulus (distance from origin to the point) and $\theta$ the argument (the angle made with positive real axis). The notation $z = r \operatorname{cis} \theta$ is occasionally used, where $\operatorname{cis} \theta = \cos \theta + i \sin \theta$.
The Polar and Euler Form of Complex Numbers page explores conversions in further detail.
Geometric Interpretation of Algebraic Operations
Addition and subtraction of complex numbers in the Argand plane correspond to vector addition and subtraction of their corresponding position vectors. If $z_1 = a_1 + ib_1$ and $z_2 = a_2 + ib_2$, then $z_1 + z_2 = (a_1 + a_2) + i(b_1 + b_2)$; geometrically, this results in the parallelogram law of vector addition.
For multiplication, consider $z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)$. Then, using De Moivre's theorem:
\[ z_1 z_2 = r_1 r_2 [\cos (\theta_1 + \theta_2) + i \sin (\theta_1 + \theta_2)] \]
Multiplication thus produces a new complex number with modulus equal to the product of the moduli and argument equal to the sum of the arguments.
\[ z_1 \div z_2 = \frac{r_1}{r_2} \left[\cos(\theta_1 - \theta_2) + i \sin(\theta_1 - \theta_2)\right] \]
Division yields a complex number whose modulus is the quotient of the moduli and whose argument is the difference of the arguments.
Conjugate of a Complex Number: Geometric Meaning
Given $z = a + ib$, the conjugate is defined as $\overline{z} = a - ib$. Geometrically, if $P(a, b)$ is the point for $z$ in the Argand plane, then $Q(a, -b)$ will be the point for $\overline{z}$. The points $P$ and $Q$ are symmetric with respect to the real axis.
Section Formula in the Geometry of Complex Numbers
Let $z_1$ and $z_2$ represent points $A$ and $B$ in the Argand plane. If $C$ divides $AB$ in the ratio $m:n$ internally, and $z$ is the complex number corresponding to $C$, then:
\[ z = \frac{m z_2 + n z_1}{m + n} \]
This formula directly follows from the section formula in coordinate geometry, since $z$ uniquely corresponds to the point $(x, y)$, and the combination of real and imaginary parts proceeds accordingly.
Equation of a Line in Terms of Complex Numbers
Given two points $A, B$ with affixes $z_1, z_2$, the general parametric point $z$ lying on line $AB$ is given by:
\[ \frac{z - z_1}{z_2 - z_1} \]
is always a real number for every point $z$ on the straight line passing through $z_1$ and $z_2$. This condition provides the equation of a straight line in complex form: $z = z_1 + \lambda(z_2 - z_1), \lambda \in \mathbb{R}$.
The method is further extended in the Geometrical Interpretation of Integrals discussion.
Geometric Interpretation of the Circle in Complex Numbers
A circle with centre at $z_0$ and radius $r$ is defined by the locus:
\[ |z - z_0| = r \]
This follows from the Euclidean distance formula for two points in the plane, as $|z - z_0|$ measures the distance between two complex numbers represented as points.
If the ends of the diameter are $z_1$ and $z_2$, the equation can be derived by setting the diameter as the locus of all $z$ satisfying $\arg\left(\frac{z-z_1}{z-z_2}\right) = \frac{\pi}{2}$, or equivalently $\left|(z-z_1)+(z-z_2)\right|^2=|z-z_1|^2+|z-z_2|^2$ by explicit algebraic expansion.
Worked Example: Length of Line Segment Joining Two Complex Numbers
Given: Find the distance between $z_1 = -1 - i$ and $z_2 = 2 + 3i$.
Subtract: $z_2 - z_1 = (2 + 3i) - (-1 - i) = (2 + 1) + (3i + i) = 3 + 4i$.
Compute modulus: $|z_2 - z_1| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
Result: The line segment between $(-1, -1)$ and $(2, 3)$ has length $5$.
Worked Example: Sum of Squares for Vertices of an Equilateral Triangle
Given: Let $z_1, z_2, z_3$ be vertices of an equilateral triangle, and $z_0$ its circumcentre. Find $z_1^2 + z_2^2 + z_3^2$.
Let $z_1 = z_0 + r e^{i\theta}$, $z_2 = z_0 + r e^{i(\theta + 2\pi/3)}$, $z_3 = z_0 + r e^{i(\theta + 4\pi/3)}$. Compute $z_1^2$:
\[ z_1^2 = (z_0 + r e^{i\theta})^2 = z_0^2 + 2z_0 r e^{i\theta} + r^2 e^{2i\theta} \]
Similarly, $z_2^2 = (z_0 + r e^{i(\theta + 2\pi/3)})^2 = z_0^2 + 2z_0 r e^{i(\theta + 2\pi/3)} + r^2 e^{2i(\theta + 2\pi/3)}$.
Also, $z_3^2 = z_0^2 + 2z_0 r e^{i(\theta + 4\pi/3)} + r^2 e^{2i(\theta + 4\pi/3)}$.
Sum: \[ z_1^2 + z_2^2 + z_3^2 = 3z_0^2 + 2z_0 r \left[e^{i\theta} + e^{i(\theta + 2\pi/3)} + e^{i(\theta + 4\pi/3)}\right] + r^2 \left[e^{2i\theta} + e^{2i(\theta + 4\pi/3)} + e^{2i(\theta + 8\pi/3)}\right] \]
Since $e^{i\theta} + e^{i(\theta + 2\pi/3)} + e^{i(\theta + 4\pi/3)} = 0$ and $e^{2i\theta} + e^{2i(\theta + 4\pi/3)} + e^{2i(\theta + 8\pi/3)} = 0$, only $3z_0^2$ remains.
Result: $z_1^2 + z_2^2 + z_3^2 = 3z_0^2$.
Worked Example: Perimeter of a Regular Hexagon in the Argand Plane
Given: A regular hexagon centre at the origin, one vertex at $z_0 = 1 + 2i$. Compute the perimeter.
The modulus is $|z_0| = \sqrt{1^2 + 2^2} = \sqrt{5}$. The six vertices are $z_k = z_0 e^{i k \frac{2\pi}{6}}$ for $k = 0, 1, \ldots, 5$.
Any side has length $|z_k - z_{k+1}|$. For regular polygons inscribed in a circle of radius $r$, $|z_k - z_{k+1}| = 2r\sin(\pi/6) = 2 |z_0| \cdot 1/2 = |z_0| = \sqrt{5}$.
Perimeter $= 6 \times \sqrt{5}$.
Result: The perimeter is $6\sqrt{5}$.
The Concept of Rotation in Complex Numbers explores implications for regular polygons and their rotations.
FAQs on Understanding the Geometry of Complex Numbers
1. What is the geometry of complex numbers?
The geometry of complex numbers refers to representing complex numbers as points or vectors in the complex plane, also known as the Argand plane. In this representation:
• The horizontal axis (real axis) represents the real part.
• The vertical axis (imaginary axis) represents the imaginary part.
• Each complex number (z = a + bi) corresponds to the point (a, b) in the plane.
This geometric view helps in visualising addition, subtraction, modulus, argument, and multiplication of complex numbers.
2. How do you represent a complex number on the Argand plane?
Complex numbers are represented as points or vectors in the Argand plane.
• The complex number z = a + bi is plotted as the point (a, b).
• The horizontal axis is the real part, and the vertical axis is the imaginary part.
This graphical approach is essential for understanding modulus, argument, and vector operations involving complex numbers.
3. What is the modulus and argument of a complex number?
The modulus is the distance from the origin to the point representing the complex number, while the argument is the angle made with the positive real axis.
• Modulus: |z| = √(a² + b²)
• Argument (arg z): θ = tan-1(b/a)
These concepts are vital for converting complex numbers to polar form and for operations like multiplication and division.
4. State the geometrical interpretation of addition of two complex numbers.
Addition of two complex numbers is interpreted as the vector sum in the complex plane.
• The result is the diagonal of the parallelogram formed by the two complex number vectors.
• If z1 = a + bi and z2 = c + di, their sum z1 + z2 is (a + c) + (b + d)i, which is the point obtained by vector addition.
This makes the Argand diagram helpful in visualising sum and difference operations.
5. What is the polar form of a complex number?
The polar form of a complex number expresses it in terms of its modulus and argument.
• z = r (cos θ + i sin θ), where
- r = modulus = |z|
- θ = argument = arg(z)
The polar form is especially useful for multiplication, division, and for De Moivre's Theorem.
6. How is multiplication of complex numbers represented geometrically?
In geometry, multiplying two complex numbers multiplies their moduli and adds their arguments.
• If z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2):
- |z1z2| = r1 × r2
- arg(z1z2) = θ1 + θ2
This is a key property in the geometry of complex numbers and simplifies various operations.
7. What is De Moivre’s Theorem and how is it applied in geometry of complex numbers?
De Moivre's Theorem states that for any real number θ and integer n, (cos θ + i sin θ)n = cos(nθ) + i sin(nθ).
Applications:
• Finding powers and roots of complex numbers.
• Expressing trigonometric identities.
• Solving equations of the form zn = 1 (roots of unity).
It is widely used for geometric interpretations and calculations involving complex numbers in polar form.
8. What is the geometric meaning of conjugate of a complex number?
The conjugate of a complex number reflects it over the real axis in the Argand plane.
• For z = a + bi, the conjugate z̄ = a – bi.
• Geometrically, the conjugate is the mirror image of the point (a, b) across the real axis.
This concept is important in finding moduli, reciprocals, and simplifying expressions.
9. How do you find the distance between two complex numbers geometrically?
The distance between two complex numbers is the length of the straight line joining their points in the Argand plane.
• For z1 = a + bi and z2 = c + di:
- Distance = |z1 – z2| = √[(a – c)² + (b – d)²]
This formula connects the geometric and algebraic interpretations of complex numbers.
10. Find the modulus and argument of the complex number 1 + i.
The complex number 1 + i has:
• Modulus |z| = √(1² + 1²) = √2
• Argument (θ) = tan⁻¹(1/1) = π/4 radians or 45°
So, 1 + i in polar form is √2 (cos π/4 + i sin π/4). This is a standard example in geometry of complex numbers for conversion between forms.
11. What do you mean by the locus of a complex number?
The locus of a complex number is the set of all points in the Argand plane satisfying a given condition or equation.
• Common loci include lines, circles, and other curves.
• For example, |z – a| = r represents a circle centered at a with radius r.
This concept is frequently tested in exams and is crucial for visualising constraints on complex numbers.
