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Complex Plane

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The Complex Plane

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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).

FAQ (Frequently Asked Questions)

Q1. What is an Imaginary Unit in a Complex Plane?

Answer: In geometric terms, the Imaginary unit, or i, is the number with the below given equivalent properties:

  • i2 = -1i

  • √-1 = i  (implying that square root of minus 1 equals i)

Having said that, a complex number is any number that can be mathematically expressed as a + bi, where i represents the imaginary unit and ‘a’ and ‘b’ are called real numbers.

A refers to the real part of the number while ‘b’ represents the imaginary part of the number.

Q2. How do we Plot a Complex Number?

Answer: Plotting a complex number is not a difficult task. We can represent every complex number by a point in the complex plane. For example, let’s take into account the number 5 - 7i. This number, also written as 5 + (-7)i, consists of a real part of 5 and an imaginary part of -7.


The position of this number on the complex plane is the point that is in correspondence to 5 on the real axis and -5 on the imaginary axis.


Thus, the number 5 + (-7)i is linked with the point (5, -7). Usually, the complex number a + bi is in correspondence to the point (a, b) in the complex plane.

Q3. Are Complex Numbers Included in Real Numbers? If so, are they Irrational?

Answer: It is the other way around. The Real numbers are a subset of the set which consists of all of the Complex numbers, so are the Imaginary numbers. Just Imagine a large circle with 2 small circles inside it that don't bisect with each other, that would be the set of the Complex number (large circle) and the Real and Imaginary sets (small circles).