Complex Number - Introduction & Uses


Complex Number

A complex number is composed of two parts: a real number and an imaginary number. Complex numbers serve as the foundation for more advanced mathematics such as algebra. They have numerous real-world applications, particularly in electronics and electromagnetism.

The standard complex number format is a + bi, with the real number coming first and the imaginary number coming last. Because either part can be zero, any real or imaginary number can be considered a complex number. Complex does not imply complicated; it simply means that the two types of numbers combine to form a complex, such as a housing complex — a collection of buildings joined together.

Real numbers, such as fractions, integers, or any other countable number, are tangible values that can be plotted on a horizontal number line. You can use imaginary numbers to find the square root of a negative number.

Complex Numbers & Mathematical Operations

For Complex number addition, subtraction, and multiplication can be naturally defined by combining the rule  i2 = −1with the associative, commutative, and distributive laws. Every complex number that is not zero has a multiplicative inverse. As a result, complex numbers become a field with real numbers as a subfield. The complex numbers also form a two-dimensional real vector space with 1, ‘i’ as a standard basis.

This standard basis transforms the complex numbers into a Cartesian plane known as the complex plane. This allows for a geometric interpretation of complex numbers and their operations, as well as the expression of geometric properties and constructions in terms of complex numbers. 

For example, the real numbers form the real line, which corresponds to the complex plane's horizontal axis. The unit circle is formed by complex numbers of absolute value one. A complex number's addition is a translation in the complex plane, and a complex number's multiplication is a similarity centered at the origin. The reflection symmetry with respect to the real axis is the complex conjugation. A Euclidean norm is the complex absolute value.

What is the Polar Form and the Argand Plane?

Like the XY plane, the Argand plane, also known as the complex plane, is a system of rectangular coordinates in which the complex number a+ib is represented by the point whose coordinates are a and b. The real and complex components are calculated in terms of r and, where r is the vector's length and is the angle formed with the real axis.

Use of Complex Numbers

In electronics and electromagnetism, complex numbers are used. A complex number combines two real quantities, making them easier to work with. In electronics, for example, the voltage (V) and current (I) define the state of a circuit element (I). 

Circuit elements can also have capacitance (c) and inductance (L), which describe how well the circuit resists changes in V and I. Rather than describing the state of the circuit element using V and I, it can be expressed as z = V + Ii. The laws of electricity can then be expressed using complex number addition and multiplication.

As previously stated, this can also be applied to electromagnetism. Instead of being described as electric field strength and magnetic field strength, you can create a complex number with real and imaginary numbers for the electric and magnetic components.

Cuemath Website

Solving a problem involving Complex numbers will not feel like a problem if you have conceptual clarity on the subject, and this is where Cuemath comes into play. Go to the Cuemath website. Cuemath is the best online math services platform for laying strong mathematical foundations. If you want to learn about these concepts in-depth and in an interactive way, go to the Cuemath website.

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