The Complex Plane
The complex plane (also known as the Gauss plane or Argand plane) is a geometric method of depicting complex numbers in a complex projective plane. It is usually a modified version of the Cartesian plane, with the real part of a complex number denoted by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.
This provides a way to visually deal with complex numbers, which contain ample advantages. Adding or multiplying complex numbers can be considered as geometric operations on points on the plane if we display complex numbers this way.
Argand Plane
We all know that the pair of numbers (a, b) can be geometrically represented on the AB-plane, where a is ‘referred’ to as abscissa and b are ‘referred’ to as the ordinate. In the same way, we can also represent complex numbers on a plane known as the Argand plane or complex plane. Same as the X-axis and Y-axis in 2D geometry, there are two axes in the Argand plane.
The axis that goes horizontal is referred to as the real axis.
The axis that goes vertical is referred to as the imaginary axis.
The complex number a+ib that corresponds to the ordered pair(a, b) is denoted geometrically as the unique point (a, b) in the AB-plane. Refer to the argand diagram complex numbers to better understand the concept behind the complex plane.
The impulse behind the complex plane emerges from the certainty that a complex number, in its essence, is just an ordered pair of real numbers. Therefore, any complex number can be given a concrete geometric explanation as points on a plane.
The complex number a + bi can simply be interpreted as the point on the Cartesian plane with the coordinates (a, b)(a, b).
Complex Numbers on a Number Line
We are unable to plot complex numbers on a number line as we could do with real numbers. But, we can still depict them graphically. In order to represent a complex number, we would require addressing the two components of the number. We make use of the complex plane grapher, which is a coordinate system where the horizontal axis denotes the real component and the vertical axis denotes the imaginary component. Complex numbers are the points on the plane, written as ordered pairs (a, b), in which a depicts the coordinate for the horizontal axis and b depicts the coordinate for the vertical axis.
Complex numbers do exist and are very much a part of mathematics. The real number line is simply the real axis on the complex plane, but there is so much beyond that single line! You can also refer to complex plane grapher online to plot numbers on the complex plane.
Connections of Complex Plane to the Real Number Line
In Pythagoras's days, the subsistence of irrational numbers had been a surprising discovery! They wondered how something like √2 could exist without an appropriate complete decimal expansion.
The real number line, nevertheless, enables clarifying this dilemma. Why? Because √2 has a particular location on the real number line, depicting that it is indeed a real number. If you take the diagonal of a unit square and place one end on 000, the other end will be in correspondence to the number √2.
Similarly, every complex number does exist since it corresponds to a precise location on the complex plane! Perhaps by being able to envision these numbers, we can understand that calling these numbers "imaginary" was an unsuitable misnomer.
Fun Facts
A complex plane is a modified version of the Cartesian plane where the x-axis and the y-axis have been dubbed as the "real axis" and the "imaginary axis," respectively.
A complex plane is sometimes known as the Argand plane simply because it is used in Argand diagrams.
The complex plane is named after a Paris-based amateur mathematician Jean-Robert Argand (1768 – 1822).
FAQs on Complex Plane
1. What is a complex plane and why is it used?
A complex plane, also known as an Argand plane or Gauss plane, is a two-dimensional coordinate system used to give a geometric representation to complex numbers. It is used because complex numbers, having both a real and an imaginary part, cannot be plotted on a simple one-dimensional number line. The plane provides a way to visualise these numbers as points, making their operations like addition and multiplication easier to understand geometrically.
2. What are the main components of a complex plane?
The complex plane has two main components, which are its axes:
- The horizontal axis is called the real axis. It represents the real part (a) of a complex number (a + bi).
- The vertical axis is called the imaginary axis. It represents the imaginary part (b) of a complex number (a + bi).
A complex number a + bi corresponds to the unique point with coordinates (a, b) on this plane.
3. How do you plot a complex number, for example, z = -3 + 2i, on the Argand plane?
To plot the complex number z = -3 + 2i on the Argand plane, you treat its real and imaginary parts as coordinates.
- First, locate the real part, -3, on the horizontal real axis.
- Next, locate the imaginary part, +2, on the vertical imaginary axis.
- The point where these two values intersect is the geometric representation of the complex number z = -3 + 2i. This corresponds to the coordinate point (-3, 2) in the second quadrant of the plane.
4. What is the difference between an Argand diagram and a complex plane?
There is essentially no difference between an Argand diagram and a complex plane; the terms are often used interchangeably. The complex plane is the conceptual two-dimensional space itself, with its real and imaginary axes. An Argand diagram is the actual graphical representation or plot of one or more complex numbers on that plane. So, you draw an Argand diagram on the complex plane.
5. How are the four quadrants defined in a complex plane?
The quadrants in a complex plane are defined by the signs of the real (a) and imaginary (b) parts of a complex number z = a + bi, similar to a Cartesian plane:
- First Quadrant: Both the real part and the imaginary part are positive (a > 0, b > 0).
- Second Quadrant: The real part is negative, and the imaginary part is positive (a < 0, b > 0).
- Third Quadrant: Both the real part and the imaginary part are negative (a < 0, b < 0).
- Fourth Quadrant: The real part is positive, and the imaginary part is negative (a > 0, b < 0).
6. Why is the complex plane considered two-dimensional (2D) and not one-dimensional (1D) like the real number line?
The complex plane is two-dimensional because every complex number has two independent components: a real part and an imaginary part. A single one-dimensional line can only represent one value at a time (like the real numbers). To uniquely define the position of a complex number, we need two axes—one for the real component and one for the imaginary component—which creates a 2D plane, not a 1D line.
7. How does the polar representation of a complex number relate to its position on the Argand plane?
The polar representation defines a complex number's position using a distance and an angle instead of x-y coordinates. For a complex number z:
- The modulus (|z|) represents the distance of the point z from the origin (0,0) of the plane. It is calculated as √(a² + b²).
- The argument (arg(z)) or amplitude is the angle formed by the line connecting the origin to the point z, measured counter-clockwise from the positive real axis.
This (r, θ) or (|z|, arg(z)) system provides an alternative way to locate the exact same point on the plane as the Cartesian form (a, b).
8. What is the geometric interpretation of adding or subtracting two complex numbers on the complex plane?
Geometrically, adding two complex numbers, z₁ and z₂, on the complex plane can be visualised using the parallelogram law of vector addition. If you represent z₁ and z₂ as position vectors from the origin, their sum, z₁ + z₂, is the vector representing the diagonal of the parallelogram formed by these two vectors. Subtracting z₂ from z₁ (i.e., z₁ - z₂) is equivalent to adding z₁ and (-z₂), where -z₂ is the vector with the same magnitude as z₂ but pointing in the opposite direction.
