Complex Numbers-Argument, Cube Roots of Unity

Mahavira was a great Indian mathematician and philosopher. In one of his books written in 850 A.D., “Ganitasara Sangraha”, an equation was written as X2 + 1 = 0. Though he mentioned in his book that no value of X can make the left-hand side of the equation equal zero. It was then discovered that real numbers could not make this equation to be true and thus the introduction of complex numbers happened to the world of Mathematics. So, the numbers which could make this equation true were named Complex Numbers or specifically Imaginary numbers. First in 1748, Euler told us that X2 + 1 = 0 is true only if X2 = -1 or X = +√-1, X = –√-1 and then eventually the value√-1 was named as i(iota), which is a Greek letter. Here i (iota) represents Imaginary values.

Therefore, we can say today the equation X2 + 1 = 0 and all such equations can be solved using Complex Numbers. All the real numbers and imaginary numbers together are known as Complex Numbers.

Argument of a Complex Number:

Argument of any given Complex Number ‘Z’ is denoted by “arg(Z)” 

Formula: arg(Z) = tan-1(y/x)

Where x = real value of the given complex no.

And y = imaginary value of the given complex no.

Let’s take a complex number, Z = a+bi

Now, to calculate the argument of Z, first we need to join the Z (complex no. ) shown on the graph below with the origin(O). The angle θ made with the positive real axis is the argument of complex number Z. (see graph below)

Therefore, the definition of Argument of a Complex Number can be given as the value of Angle (Θ ) made by OZ with the positive direction of real axis is known as argument of ‘Z’ or arg(Z).

Types of Argument of a Complex Number ‘Z’:

General Argument of ‘Z’: Θ is a periodic function and the value of periodic function is 2 π. Hence the value of general value of argument is written as Θ + 2k π, where k is an integer.

arg(Z) = amp(Z) + 2k π, where k is an integer.

Principal Argument of ‘Z’: When the value of Θ lies between -π and π, it is referred to as the principal value of argument ‘Z’. It is also known as Amplitude of Z and is represented mathematically as amp(Z).

-π > Principal value of argument <= π

-π > amp(Z) <= π

Let us understand these concepts in detail with the help of a practical example and graph below:

For any given complex number Z = 1+i

Base and Perpendicular will be equal to 1.

Hence tan θ = P/B = 1/1

Θ = π/4

arg(Z) = π/4

Therefore, Principal value of ‘Z’ = π/4

amp(Z) = π/4

Now, the General value of ‘Z’ will be = π/4 + 2k π, where k is an integer.

Properties of argument of a complex number:

1. Product of the argument of two complex numbers will be the sum of the arguments of individual complex numbers. 2kπ will be added to get the value in the range of the argument.

 arg (Z1*Z2) = arg(Z1) + arg(Z2) + 2kπ , k ∈ I

2. Argument of the division of two given complex numbers will be the difference of arguments of the individual complex numbers. 2kπ will be added to get the value in the range of the argument.

 arg (Z1/Z2) = arg(Z1) – arg(Z2) + 2kπ , k ∈ I

3. arg (Z1*Z2*Z3……….Zn/Y1*Y2*Y3…………Yn) = arg(Z1)  + arg(Z2) + ……….arg(Zn) –arg(Y1) – arg(Y2)- arg(Y3)-………………..arg(Yn) + 2kπ , k ∈ I
4. arg (Zn) = arg(Z) + 2kπ , k ∈ I        

Cube Roots of Unity:

Unity means one. If we write a1/2, then it means we are writing square root of a. Similarly, if we express as a1/3, then it means we are writing cube root of a.

Therefore, we can derive the cube root of unity and that can be written as 11/3.

If 11/3 = X

Taking cube root on both sides, we will get:

1 = X3

X3 – 1 = 0

Factoring for the above equation, we will get the factors as:

(X-1) (X2 + X + 1) = 0

Therefore, either X-1 = 0 OR X2 + X + 1 = 0

                             X = 1   OR    X = (-1 – √-3)/ 2 , X = (-1 + √–3 )/ 2

Hence, final values of X will be as follows:

X = 1

X = -1/2 + (√3 / 2) i

X = -1/2 – (√3 / 2) i

Note that X = 1 is the real value of X, but the other two values of X are the imaginary values (non-real values). The imaginary values are called ω and ω2.

Therefore, it is generally said that cube roots of unity have three roots as 1, ω and ω2.

ω = -1/2 + (√3 / 2) i

ω2 = -1/2 – (√3 / 2) i

Conclusion:

Arguments of a complex number and Cube roots of Unity can be said to be the complex numbers’ heart and heartbeats, respectively. In this article, we discussed in detail the concept of the argument of a complex number, types of arguments of a complex number as the general value of an argument and principal value of an argument, different properties of arguments of a complex number. We also learnt to derive all the values using graphs. Then we discussed the meaning of cube roots of unity along with their derivation. We also discussed the mathematical notation of all the terms. We hope this study material would be of great value for anybody seeking basic concept clarity on the topics covered here.