Roots of Complex Numbers

The imaginary root of a quadratic or polynomial function is called a complex root. These complex roots, which are expressed as α = a + ib, and β = c + id, are a type of complex number. The imaginary roots of quadratic equations with a discriminant value less than zero  (D<0)  are represented as complex integers. Complex roots have a real and imaginary part, and the formula i² = -1 can be used to calculate complex roots.

Roots of Complex Numbers

Complex roots are the imaginary roots of quadratic equations expressed as complex integers.

Because the square root of a negative number is impossible, we convert it to a complex number. Equations with discriminant values smaller than zero are quadratic equations.

b²-4ac<0 , i get the complex roots by transforming it using. i²D stands for -D in this case.

The complex integers a + ib are used to express complex roots. A real component and an artificial party make up the complex root. Z = a + ib is a common representation of the complex root. Here, ‘a’ denotes the real component of the complex number, which is Re(Z), and ‘b‘ denotes the imaginary part, which is Im (Z). The imaginary number ib is used here.

The letter i is referred to as iota in the imaginary component of a complex number. To determine the square root of any negative number, the iota – i is quite useful. Here, i² = -1, and the negative number -N is written as i²N, which has now become a positive integer.

Complex Roots’ Properties

Complex Roots’ Magnitude

The complex root α= a + ib is represented on the argand plane as a point (a, +b), and the distance between this point and the origin (0, 0) is called the complex number’s modulus. 

The distance is calculated as a basic linear distance.

 r= √a²+b²

The modulus of the complex root is represented by the hypotenuse of a right triangle, the base is the real component, and the height is the imaginary part, which can be easily understood using Pythagoras theorem.

Complex Roots’ Argument

The complex root can be represented as a point in the argand plane, and the line connecting this point to the origin forms an angle with the positive x-axis in the argand plane, which is known as the complex number’s argument. 

The inverse of the trigonometric tan of the imaginary component divided by the real part yields the argument of the complex root. which is equal to Arg z(θ)=b / a

Complex Roots Polar Representation

The modulus and argument of the complex number in the argand plane can be used to represent the complex root in polar form.

r(cosθ + isinθ ) is the polar form of the complex root α= a + ib. The complex root’s modulus is r, and its argument isθ.

Complex Roots Reciprocal

With the use of the reciprocal of a complex root, it is feasible to divide one complex root by another complex root. The product of one complex root with the reciprocal of another complex root equals the division of one complex root by another complex root.

A complex root’s reciprocal α=a+ib is a-1=1 / a+ib=a-ib / a²+b²=a / a²+b²

Determining the roots of complex numbers

When looking for the roots of complex numbers, keep the following procedures in mind.

• Make sure to transform the complex number to polar form if it is still in rectangle form.
• Find or raise to the power of 1/n‘s nth root
• If we need to discover the nth root, we will utilize the formula k= {0,1,2..n- 1}.
• Begin by dividing θ by n to find the argument of the first root.

Conclusion

We conclude in this article, that we can easily get the roots of complex numbers by taking the modulus root and dividing the complex numbers’ argument by the supplied root. When the complex numbers are in polar form, we may readily discover the roots of distinct complex numbers and equations with complex roots. Any polynomial with real-number coefficients can be factored entirely across the field of complex numbers, according to the Fundamental Theorem of Algebra. When the discriminant is negative in the case of quadratic polynomials, the roots are complicated.