What is meant by roots?

The statement given by the Fundamental Theorem of Algebra is that each polynomial comprising a degree one or more has a minimum single root in the complex number system (note that the complex numbers may be real in nature if the imaginary part is zero). In certain cases, such as the Linear Factorization Theorem implies that any polynomial with a degree ‘n’ has precisely ‘n’ linear factors and every factor is depicted in the format of (x – c), where ‘c’ is the root. In the Roots of Higher Degree concept, all these ‘n’ complex roots (inclusive of certain real roots) are evaluated with multiplicity. In case of 1^st degree polynomials, the factors are real roots of the initial polynomial, whereas, in 2^nd degree polynomials, the factors are irreducible by utilizing real numbers but with the help of complex numbers, solving those quadratic equations is possible and determining the ‘n’ roots of the degree ‘n’ polynomial is feasible. A quadratic equation of the standard form ax² + bx + c = 0, where, b² -4ac < 0 has two complex roots: 

The statement given by the Fundamental Theorem of Algebra is that each polynomial comprising a degree one or more has a minimum single root in the complex number system (note that the complex numbers may be real in nature if the imaginary part is zero). In certain cases, such as the Linear Factorization Theorem implies that any polynomial with a degree ‘n’ has precisely ‘n’ linear factors and every factor is depicted in the format of (x – c), where ‘c’ is the root. In the Roots of Higher Degree concept, all these ‘n’ complex roots (inclusive of certain real roots) are evaluated with multiplicity. In case of 1^st degree polynomials, the factors are real roots of the initial polynomial, whereas, in 2^nd degree polynomials, the factors are irreducible by utilizing real numbers but with the help of complex numbers, solving those quadratic equations is possible and determining the ‘n’ roots of the degree ‘n’ polynomial is feasible. A quadratic equation of the standard form ax² + bx + c = 0, where, b² -4ac < 0 has two complex roots: 

Difference between real and complex roots

The main dissimilarity between real and complex roots is that the real roots are obtained as real numbers, in contrast with the complex roots, which are indicated as imaginary numbers. The standard format for depicting any complex number is in the form of a + bi.

For instance, a real root of √4 is 2 and a complex root will be is -2+i.

Important terms used in the roots of complex numbers

Argument: The angle produced by the positive real axis and the segment joining the origin to the plotting of a complex number on the graph in the complex plane.

Complex Conjugate: The complex conjugate of a complex number a + bi is always expressed as a - bi.

Complex Number: The set of all the numbers in the format a + bi, where ‘a’ and ‘b’ are real numbers and they all come under the category of complex numbers.

Complex Plane: A plane depicted with two perpendicular axes which contain the real axis and the imaginary axis over which a complex number a + bi is marked having the coordinates (a, b). It is common for the real axis to intersect with the x-axis and for the imaginary axis to intersect with the y-axis of the rectangular coordinate system.

Imaginary Axis: The axis in the complex plane that usually intersects with the y-axis of the rectangular coordinate system and also on which the imaginary parameter ‘bi’ of the complex number a + bi is displayed.

Imaginary Number: A number that is written in the form of ‘bi’, where ‘b’ is a real number.

Imaginary Part: The ‘bi’ term is the imaginary one in each complex number a + bi.

Imaginary Unit: The imaginary unit is expressed as √ (-1).

Modulus: The length between the coordinates of the complex number plotted and the origin of the graph is termed as modulus. It is computed using the equation √ (a² + b²).

Polar Form of a Complex Number: It is one of the ways to depict any complex number equation. It is written as z = r (cos θ + i sin θ), where r = | z | and θ = argument of z.

Real Axis: The axis in the complex plane that usually intersects with the x-axis of the rectangular coordinate system and also on which the real parameter ‘a’ of the complex number a + bi is displayed.

Real Part: The ‘a’ term is the real one in each complex number a + bi.

The Square Root of a complex number

One of the easiest methods to evaluate the square root of a complex number a + ib is to compare the real and imaginary parts of the equation √ (a + ib) = x + iy by squaring both sides and then determining the values of x and y.

Square Root of Complex Number in Rectangular Form

The square root of a complex number with rectangular coordinates is using the equation:

$\sqrt{a+ib}=(\sqrt{\frac{\left|z\right|+1}{2}}+i\frac{b}{\left|b\right|}\sqrt{\frac{\left|z\right|-a}{2})}$, where $z = a + ib$ and $b\ne 0$

Here,

• If b < 0, then b/|b| = -1, and x and y have opposite signs
• If b > 0, then b/|b| = 1, and x and y have same signs

Square Root of Complex Number in Polar Form

The formula is,

$z^{\frac{1}{2}} = r^{\frac{1}{2}} \left\{\right[\cos \left(\frac{\theta + 2k\pi }{2}\right)] + i\sin [\frac{\theta + 2k\pi }{2}\left]\right\},$where k = 0, 1

How to determine roots?

How to determine roots?

1. Transforming complex numbers into the rectangular form or the polar form as well as vice-versa.
2. Application of the De Moivre’s theorem along with its working for computing the roots of complex numbers

Finding the roots of complex numbers graphically

In order to evaluate the roots of complex numbers plot these roots graphically on a complex plane and the chronological order that needs to be followed is:

1. Convert the rectangular format of the complex number into the polar format, in case the given number is not in the form of a polar complex number.
2. Divide 360° or 2π by ‘n’.
3. Make the 1^st root on the graph by connecting the origin with the segment ‘r’ units in length.
4. Mark that complex root by using the equation z = r (cos θ + isin θ), where r = | z |, θ = argument of z and k = 0. Here z = (a + bi)ⁿ were ‘n’ are the number of roots.
5. Illustrate the following root by ensuring that it is 2π/n or 360°/n apart from the following roots.

Finding the roots of complex numbers geometrically.

In order to evaluate the roots of a complex number geometrically on a complex plane, the chronological order that needs to be followed is:

1. Convert the rectangular format of the complex number into the polar format, in case the given number is not in the form of a polar complex number.
2. Find the nth root of ‘r’ or r^1/n.
3. Make the 1^st root on the graph by connecting the origin with the segment ‘r’ units in length.
4. For finding the nth root, use k = (0, 1, 2 … n – 1)
5. Initiate by determining the argument of the 1^st root by dividing θ by ‘n’.
6. Repeat the above procedure keeping in mind θ + 2 πk or θ + 360°k until the ‘n’ roots are achieved.

Formulae used in Roots of Complex Numbers

1. A quadratic equation of the standard form ax² + bx + c = 0, where, b² -4ac < 0 has two complex roots: $x = \frac{-b+\sqrt{b^2-4ac}}{2a} and \frac{-b-\sqrt{b^2-4ac}}{2a}$.
2. Square root of the equation in rectangular form: $\sqrt{a+ib}=(\sqrt{\frac{\left|z\right|+1}{2}}+i\frac{b}{\left|b\right|}\sqrt{\frac{\left|z\right|-a}{2})}$ $where z=a+ib and b\ne 0$.
3. Square root of the equation in polar form: $z^{\frac{1}{2}}=r^{\frac{1}{2}}\left\{\right[\cos \left(\frac{\theta +2k\pi }{2}\right)]+i\sin [\frac{\theta +2k\pi }{2}\left]\right\}, where k=0,1$.
4. The equation to determine roots graphically: z = r (cos θ + isin θ), where r = | z |, θ = argument of z and k = 0. Also, z = (a + bi)n were ‘n’ are the number of roots.
5. Another method to find square roots is to compare the real and imaginary parts of the equation $\sqrt{a+ib}=x+iy$ , squaring both sides and then determining the values of x and y.

Context and Applications

This concept is extensively used by architects, engineers, accountants, and many other professionals in their daily lives. It is also taught widely in schools and in the following courses:

1. Bachelor of Technology (Engineering)
2. Masters in Science (Algebra, Geometry, and Cryptography)
3. Masters in Science (Mathematics and Applications)

Related Concepts

• Euler’s Identity
• Unit complex number
• Eisenstein Integer
• Algebraic surface
• Trigonometric equations

Practice Problems

Q1. What is known as the interval between the plot and the origin of a complex number?

1. Modulus
2. Coordinate
3. Slope
4. Unit

Correct Option: 1.

Explanation: Modulus is known as the interval between the plot and the origin of a complex number.

Q2. How many roots does a quadratic equation have?

1. 1
2. 2
3. 3
4. 4

Correct Option: 2.

Explanation: A quadratic equation has 2 roots.

Q3. What is the value of √4 and state the nature of the root?

1. 2, Imaginary root
2. 2, Contradictory root
3. 2, Real root
4. None of the above

Correct Option: 3.

Explanation: The value of √4 is 2 and the root is a real root in nature.

Q4. Which theorem is used for determining the roots of complex equations?

1. Pythagoras theorem
2. Remainder theorem
3. Greens theorem
4. De Moivre’s theorem 

Correct Option: 4.

Explanation: De Moivre’s theorem is used for determining the roots of complex equations.

Q5. What is/are the way(s) in which square roots of complex numbers can be found?

1. Rectangular form
2. Polar form
3. All of the above
4. None of the above

Correct Option: 4.

Explanation: The way(s) in which square roots of complex numbers can be found is in rectangular and polar forms.