Argument of Complex Numbers

Argument of Complex Number

In Mathematics, complex planes play an extremely important role. We also call it a z-plane which consists of lines that are mutually perpendicular known as axes. The real numbers are represented by the horizontal line and are therefore known as real axis whereas the imaginary numbers are represented by the vertical line and are therefore known as an imaginary axis. We basically use complex planes to represent a geometric interpretation of complex numbers. It is just like the Cartesian plane which has both the real as well as imaginary parts of a complex number along with the X and Y axes. Complex numbers are branched into two basic concepts i.e., the magnitude and argument. But for now we will only focus on the argument of complex numbers and learn its definition, formulas and properties. 

What Is A Complex Number?

A complex number is written as a + ib, where “a” is a real number and “b” is an imaginary number. The complex number consists of a symbol “i” which satisfies the condition \[i^{2}\] = −1. Complex numbers are referred to as the extension of one-dimensional number lines. In a complex plane, a complex number denoted by a + bi is usually represented in the form of the point (a, b). We have to note that a complex number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a complex number with absolutely no imaginary part is known as a real number.

What Is Argument of Complex Number? 

The argument of a complex number is an angle that is inclined from the real axis towards the direction of the complex number which is represented on the complex plane. We can denote it by “θ” or “φ” and can be measured in standard units “radians”.


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In the diagram above, the complex number is denoted by the point P. The length OP is the magnitude or modulus of the number, and the angle at which OP is inclined from the positive real axis is known as the argument of the point P.

How To Find Argument Of a Complex Number?

There are few steps that need to be followed if we want to find the argument of a complex number. These steps are given below: 

Step 1) First we have to find both real as well as imaginary parts from the complex number that is given to us and denote them x and y respectively. 

Step 2) Then we have to use the formula θ = \[tan^{-1}\] (y/x) to substitute the values.

Step 3) If by solving the formula we get a standard value then we have to find the value of  θ or else we have to write it in the form of \[tan^{-1}\] itself.

Step 4) The final value along with the unit “radian” is the required value of the complex argument for the given complex number.

With this method you will now know how to find out argument of a complex number.

Argument of Complex Number Examples

Example 1) Find the argument of -1+i and 4-6i

Solution 1) We would first want to find the two complex numbers in the complex plane. This will make it easy for us to determine the quadrants where angles lie and get a rough idea of the size of each angle. For, z= --+i 

We can see that the argument of z is a second quadrant angle and the tangent is the ratio of the imaginary part to the real part, in such a case −1. The tangent of the reference angle will thus be 1. Write the value of the second quadrant angle so that its reference angle can have a tangent equal to 1. If the reference angle contains a tangent which is equal to 1 then the value of reference angle will be π/4 and so the second quadrant angle is π − π/4 or 3π/4. 

For z = 4 − 6i: 

This time the argument of z is a fourth quadrant angle. The reference angle has a tangent 6/4 or 3/2. None of the well known angles consist of tangents with value 3/2. Therefore, the reference angle is the inverse tangent of 3/2, i.e. \[tan^{-1}\] (3/2). Hence the argument being fourth quadrant itself is 2π − \[tan^{-1}\] (3/2).

In order to get a complete idea of the size of this argument, we can use a calculator to compute 2π − \[tan^{-1}\] (3/2) and see that it is approximately 5.3 (radians). In degrees this is about 303o.


Solved Example

Question: Find the argument of a complex number 2 + 2\[\sqrt{3}\]i.

Solution: Let z = 2 + 2\[\sqrt{3}\]i.

The real part, x = 2 and the Imaginary part, y = 2\[\sqrt{3}\]

We already know the formula to find the argument of a complex number. That is

arg (z) = \[tan^{-1}\](y/x)

arg (z) = \[tan^{-1}\](2\[\sqrt{3}\]/2)

arg (z) = \[tan^{-1}\](\[\sqrt{3}\])

arg (z) = \[tan^{-1}\](tan π/3)

arg (z) = π/3

Therefore, the argument of the complex number is π/3 radian.

FAQ (Frequently Asked Questions)

1. What is the difference between general argument and principal argument of a complex number?

The value of the principal argument is such that -π < θ =< π.

However, because θ is a periodic function having period of 2π, we can also represent the argument as (2nπ + θ), where n is the integer. This is referred to as the general argument.

Now, consider that we have a complex number whose argument is 5π/2. This is a general argument which can also be represented as 2π + π/2. Here π/2 is the principal argument.


2. What are the properties of complex numbers?

The properties of complex number are listed below:

  1. If a and b are the two real numbers and a + ib = 0 then a = 0, b = 0

  2. When the real numbers are a, b and c; and a + ib = c + id then a = c and b = d.

  3. A set of three complex numbers z1, z2, and z3 satisfy the commutative, associative and distributive laws.

  4. The sum of two conjugate complex numbers is always real.

  5. The product of two conjugate complex numbers is always real.

  6. If both the sum and the product of two complex numbers are real then the complex numbers are conjugate to each other.

  7. For two complex numbers z3 and z3 : |z1 + z2|≤ |z1| + |z2|