Definition formula and how to plot complex numbers on 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 in Mathematics and Argand Diagram
1. What is the complex plane in mathematics?
The complex plane is a two-dimensional plane used to represent complex numbers, where the horizontal axis is the real part and the vertical axis is the imaginary part. It is also called the Argand plane.
- A complex number is written as z = a + bi.
- The point (a, b) represents the complex number on the plane.
- The x-axis is the real axis and the y-axis is the imaginary axis.
2. How do you plot a complex number on the complex plane?
To plot a complex number on the complex plane, represent z = a + bi as the point (a, b).
- Identify the real part a on the horizontal axis.
- Identify the imaginary part b on the vertical axis.
- Mark the point where these coordinates meet.
3. What is the modulus of a complex number in the complex plane?
The modulus of a complex number is its distance from the origin in the complex plane and is given by |z| = √(a² + b²).
- For z = a + bi, use the distance formula.
- It represents the length of the vector from (0, 0) to (a, b).
4. What is the argument of a complex number?
The argument of a complex number is the angle θ it makes with the positive real axis, calculated using tan θ = b/a.
- For z = a + bi, θ = tan⁻¹(b/a).
- Adjust θ based on the quadrant of the point (a, b).
5. What is the difference between the real axis and imaginary axis?
The real axis represents real numbers while the imaginary axis represents imaginary numbers in the complex plane.
- The real axis is horizontal (x-axis).
- The imaginary axis is vertical (y-axis).
- Any point on the real axis has the form a + 0i.
- Any point on the imaginary axis has the form 0 + bi.
6. How do you add complex numbers on the complex plane?
To add complex numbers on the complex plane, add their real parts and imaginary parts separately: (a + bi) + (c + di) = (a + c) + (b + d)i.
- Add horizontal components (real parts).
- Add vertical components (imaginary parts).
7. How is multiplication of complex numbers represented on the complex plane?
Multiplication of complex numbers in the complex plane multiplies their moduli and adds their arguments: |z₁z₂| = |z₁||z₂| and arg(z₁z₂) = arg(z₁) + arg(z₂).
- In rectangular form, use expansion: (a + bi)(c + di).
- In polar form, multiply magnitudes and add angles.
8. What is the polar form of a complex number?
The polar form of a complex number is written as z = r(cos θ + i sin θ), where r is the modulus and θ is the argument.
- r = |z| = √(a² + b²)
- θ = arg(z)
9. What is the geometric meaning of the conjugate of a complex number?
The conjugate of a complex number reflects the point across the real axis in the complex plane.
- If z = a + bi, then its conjugate is z̄ = a − bi.
- The real part stays the same.
- The imaginary part changes sign.
10. Why is the complex plane important in mathematics?
The complex plane is important because it provides a geometric way to visualize and operate on complex numbers.
- It simplifies addition, subtraction, multiplication, and division.
- It explains rotations and scaling using polar form.
- It is essential in solving polynomial equations and studying complex functions.
