Square Root of a Complex Number

What are Complex Numbers?

Complex number is part of the number system containing real numbers along with imaginary numbers. A complex number is denoted by i which is equivalent to the root of -1. Every complex number can be expressed with an addition of two variables along with i  where the two variables are real numbers. As real numbers cannot satisfy the equation for square root of complex numbers, an imaginary number given Descartes is used. A set of complex numbers is often denoted by C and historical nomenclature will regards imaginary and real numbers as part of mathematical science.

The formula for the square root of a complex number can be used to solve polynomial equations along with every fundamental theorem of algebra. The fundamental theorem also asserts that every polynomial function with no constants and coefficients will have a solution consisting of a complex number. 

Complex Number Identities

In square root of complex numbers, the additive and multiplicative identities are naturally defined and they can be combined with the associative, commutative, and distributive laws. The multiplicative inverse is present in every non-zero complex number and finding the square root of a complex number will present us with a real vector space of dimension two with 1 and i as standard numbers. 

The geometric interpretation of a complex number along with the square root of complex numbers is possible along the cartesian plane or a complex plane. The real numbers are represented in a real line which is the horizontal axis of the complex plane. As finding the square root of a complex number is an absolute value, it forms the value of an Euclidean norm. Therefore, the complex numbers are an algebraically closed field along with an Euclidean vector space of second dimension. 

Square Root of Complex Numbers 

The square root of complex numbers are simply the squares of the original complex numbers that can be easily defined. The square root of complex numbers can be determined with a simplified formula just like natural numbers. The square root of the complex number ‘c + id’ in terms of algebraic expression, is given by √(c + id) = ±(y + iz), where y and z are real numbers.

It is quite easy to find the square root of complex numbers. We can assume the square root of complex number c + id to be √(c + id) = y + iz, so that we will have (y + iz)2 = c + id. The easiest way would be to compare the real and imaginary parts of the equation (c + id) = y + iz by simply squaring both sides. We can then find the values of y and z through the formula for the square root of a complex number.

Visualisation of Finding the Square Root of a Complex Number

A complex number is represented by an ordered pair of real numbers that can be charted in a two-dimensional space. When there are two arbitrary values, the cartesian coordinates in a horizontal axis is used to display the real numbers in the square root of complex numbers. The imaginary part is always represented on the vertical or the imaginary axis with the increasing values in the upward direction. A charted number on a plane can be viewed as a coordinated point or as a position vector. The coordinated point is represented by z and can be expressed in Cartesian, rectangular, or algebraic form.

The alternative to a cartesian coordinate is the polar coordinate system that expresses the absolute value of z. In the complex plane, the polar coordinate system uses the distance from the origin to the point z along with the angle subtended between the positive side of the horizontal axis and the line segment counterclockwise. By applying Pythagoras’ theorem, we can find the absolute value of a complex number by computing the distance to the origin that represents the complex number in the plane. 

Applications of Formula of the Square Root of a Complex Number

Square root of complex numbers have varied applications in the field of mathematics and various scientific areas like signal processing, fluid dynamics, and cartography. In mathematics, finding the square root of a complex number can help with geometrical shapes, fractal geometry, triangles, algebraic number theory, and improper integrals. Square root of complex numbers also find their way in electromagnetism, quantum mechanics, and theory of relativity. You are also likely to find the square root of complex numbers in analytic number theory, dynamic equations, and linear algebra. 

Conclusion

Although the formula for the square root of a complex number is quite a task to remember, remembering the similarities between the rectangular and polar coordinates can help you a lot.