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Last updated on September 19, 2025
The angle between the line representing a complex number and the positive x-axis in the Argand plane is called its argument. A complex number can be written as Z = a + ib. The argument is calculated as the inverse tangent of the ratio of the imaginary part to the real part of the complex number. The formula for the argument is: θ = tan⁻¹(b/a) This basic form is applicable for numbers lying in the first quadrant, where a is the real part and b is the imaginary part. For complex numbers in other quadrants, adjustments are needed to determine the correct argument. In this topic, we will explore the argument of a complex number and its significance in detail.
A complex number has two parts: a real part and an imaginary part. The argument is the angle between the positive x-axis and the line representing the direction or angle of the complex number relative to the positive real axis. In the Argand plane, we can plot any complex number with its real part on the x-axis and its imaginary part on the y-axis. The complex number Z = a + ib can be plotted as a point A (a, b), and the angle can be found using the inverse tangent of the imaginary part divided by the real part.
Any complex number can be written as:
Z = a + ib, where a is the real part and b is the imaginary part.
When the complex number lies in the first quadrant, we can use the formula for finding the argument:
θ = tan - 1 (b/a)
The inverse tangent function is represented as tan -1.
For other quadrants, different formulas and some adjustments are needed to find the correct argument.
In the Argand plane, the argument of complex numbers is determined by the quadrant where the point (a, b) is located. Therefore, the formula for the argument varies depending on the quadrant.
The complex number is located in the first quadrant, and the argument is calculated using:
θ = tan - 1(b/a)
When the complex number is located in the second quadrant, the direction can be adjusted by adding π or 180°.
The formula for argument is:
θ = π - tan - 1(b/a)
π or 180° is added to adjust the direction when the complex number is in the third quadrant. The formula is:
θ = π + tan - 1(b/a)
The argument of a complex number is negative when the number is located in the fourth quadrant. The principal value of the argument is:
θ = -tan -1(b/a)
There are some special cases where the formula varies, mainly when the real part (a) or the imaginary part (b) is zero.
An angle has both a principal value and a general value, giving us the principal and general arguments. The argument of the complex number is determined using the inverse tangent function, which follows the general solution of the trigonometric tangent function.
The values of the principal argument of a complex number is denoted as Ard(z) and range from -π < θ π for the first and fourth quadrants. In the first and second quadrants, angles are measured counterclockwise from the positive x-axis (0 < θ < π). In the third and fourth quadrants, angles are measured clockwise (-π < θ < 0).
2nπ + θ is the general argument of complex numbers, where θ is the principal argument, and n is any integer. The argument of the complex number has both a principal and a general argument, determined using the tangent function.
Two fundamental characteristics, the modulus and the argument completely describe a complex number in the Argand plane. The modulus of a complex number explains how far it is from the origin, while the argument is the angle it makes with the line representing the number and the positive x-axis. Now, let us look at each characteristic in detail.
The point A (a, b) with the origin point O (0, 0) represents the complex number.
Understanding the real-life applications of a complex number's argument helps students use it effectively. The real-world applications of the argument of a complex number are listed below:
Students often make some mistakes when they calculate the argument of complex numbers. Here are some common mistakes and their solutions to avoid them on finding the argument.
Find the argument of the complex number Z = 2 + 4i.
1.107 radians.
Here, the given complex number is Z = 2 + 4i
In the form Z = a + ib
So, we have to find the real and imaginary parts:
Real part (a) = 2
Imaginary part (b) = 4
Now, we can use the formula for argument:
θ = tan - 1 (b/a)
Next, substitute the values:
θ = tan - 1 (4/2)
θ = tan⁻¹(2) ≈ 1.107 radians
Now, find the value:
θ = 1.107 radians.
Since Z is in the first quadrant, no adjustment is needed.
Therefore, the argument of the complex number Z = 2 + 4i is 1.107 radians.
Find the argument of the complex number Z = 6 + 5i.
tan - 1 (5/6)
Here, the given complex number is Z = 6 + 5i
The real part (a) = 6
The imaginary part (b) = 5
Since a > 0 and b > 0, the complex number is located in the first quadrant of the Argand plane.
The argument θ is given by:
θ = tan - 1 (b/a)
Now, we can substitute the values:
θ = tan - 1 (5/6)
Now, to determine the value of θ, use a calculator:
θ = tan - 1 (5/6) ≈ 40.6°
Therefore, the argument of the complex number is approximately 40.6°.
Find the argument of the complex number Z = 3 + 4i.
θ = 53.13° or 0.93 radians.
The given complex number is Z = 3 + 4i
The real part = 3
The imaginary part = 4
Now, we can use the argument formula:
θ = tan - 1 (b/a)
Next, substitute the values:
θ = tan - 1 (4/3)
Thus, for Z = 3 + 4i, the argument is tan - 1 (4/3).
We can use a calculator to determine the value.
tan - 1 (4/3) ≈ 53.13°
Or in radians:
θ ≈ 0.93 radians
Since a > 0 and b > 0, the number is in the first quadrant, and the argument is positive.
Therefore, the complex number 3 + 4i forms an angle of 53.13° with the positive x-axis in the Argand plane.
Find the argument of Z = √3 + i
arg (z) = π/6
We need to find the real and imaginary parts of the given expression:
Real part a = √3
Imaginary part = 1
Next, we need to determine the quadrant:
If a > 0 and b > 0, the number is in the first quadrant.
Now, calculate the argument:
θ = tan - 1 (b/a)
Next, substitute the values:
θ = tan - 1 (1/√3) = π/6
Since tan (π/6) = 1/√3
Since z is in the first quadrant, no adjustment is needed.
Thus, the argument of a complex number Z = √3 + i is π/6.
Find the argument of Z = -3 + 0i.
arg (z) = π
First, we must find the real and imaginary parts of the expression.
Real part a = -3
Imaginary part b = 0
Since b = 0, the number lies on the real axis.
If a < 0 and b = 0, the complex number is on the negative real axis, and arg (z) = π.
Now, calculate the argument:
θ = tan - 1 (b/a)
Next, substitute the values:
θ = tan - 1 (0/-3) = tan - 1(0) = 0.
Here, we adjust for the negative real axis: θ = π.
Thus, the argument of the complex number Z = -3 + 0i is π.
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
: She loves to read number jokes and games.