Magnitude And Argument

We write complex numbers in the form \(z = a + bi\), where a is the real part and b is the imaginary part. To understand this better, we look at two important properties of complex numbers: magnitude and argument.
 

Magnitude, also known as the modulus of a complex number, tells us the distance of the number from its origin. It represents how far the number is from the origin in the complex plane.  

In simple terms, it tells us how big a number is, regardless of the direction.  The argument tells us the direction of the complex number in the complex plane. The argument is the angle θ, measured in radians. It is the angle between the complex number’s vector and the positive real axis.

The distance between the point (x, y) that represents it in the complex plane and the origin (0,0) is called the magnitude, or modulus, of a complex number


We often denote the magnitude of a complex number by | z |  and is given by formula:

\(|z| = \sqrt{x^2 + y^2}\)

Here:

x → real part,

y → imaginary part of the complex number.

On a complex plane, magnitude represents the length of the vector from the origin to the point (x, y). Note that the distance from the origin to the point is always a non-negative real number. For example, given the complex number:  \(z = 3 + 4i\). 

The formula we use to find the magnitude is:

\(|z| = \sqrt{x^2 + y^2}\)
 

Here, x = 3 and b = 4.
 

\(|z| = \sqrt{3^2 + 4^2}\) \(= \sqrt{9 + 16} = \sqrt{25} = 5\)
 

So the magnitude of 3 + 4i is 5.

Some key points to know about magnitude and arguments are:
 

• Magnitude tells us the length or absolute value of a complex number. This is useful for operations like distance calculations.
   
• Argument is essential in visualizing the complex number in the polar coordinate system and for performing multiplication/division in polar form.
   
• The value of the argument depends on the quadrant in which the complex number lies. Special attention is needed when the number lies on the real or imaginary axis (e.g., when x = 0 or y = 0).
   
• Polar and Euler forms are highly efficient for operations on complex numbers. This is especially useful when we want to perform operations such as multiplying, dividing, or raising complex numbers to powers and simplifying exponential and trigonometric operations.

The argument of a complex number is the angle between the positive real axis and the line connecting the origin to the point in the complex plane. The angle is measured in radians, it is calculated as 
 

Use \( θ = tan⁻¹(y/x)\) and adjust for quadrant:
 

If \(x < 0\), add π; if \(x = 0\), \( θ = π/2\) or \(-π/2\) depending on y.
 

If:

• x > 0 indicates that the argument is in the first or fourth quadrant.
• x < 0 indicates that the argument is in the second or third quadrant.
   

There are some cases that occur when x = 0, in which case the argument is 𝛑/2 or 3𝛑/2 depending on the sign of y. 


To understand the argument of a complex number, let us take an example with the complex number \(z = 1 + 1i\).


To find its argument θ, we use the formula: 
 

\(θ = tan-1 (y/x)\)

Where, x = 1 and y = 1

\(θ = tan-1 (1/1) =  tan-1 (1) \)

Since tan 45° = 1, we get:

Θ = 45° = 0.785 radians

So, the argument of 1 + 1i is 0.785 radians.
 

A complex number, represented as \( z = a + bi\), can be expressed in terms of its magnitude and argument using polar form. The magnitude |z| is the length of the complex number in the complex plane, while the argument θ is its direction.
 

Considering these, we can express a complex number as:
 

\( z = |z| (\cos \theta + i \sin \theta) \)

This is known as the polar form of a complex number. 
 

Alternatively, we can use Euler’s formula  \( e^{i\theta} = \cos \theta + i \sin \theta \), and we can then write the complex number as:  \( z = |z| e^{i\theta} \)

This relationship helps in simplifying multiplication, division, and powers of complex numbers. 
 

Mastering magnitude and argument helps in understanding the geometric meaning of complex numbers. Visualizing them on the complex plane makes solving and interpreting problems easier.

• Understand that magnitude represents the distance of a complex number from the origin, while the argument represents its angle with positive x-axis.
   
• Use the formula to find the magnitude quickly.
   
• Find the argument using \( \tan \theta = \frac{y}{x} \), and adjust the angle based on the quadrant of the complex number.
   
• Practice converting complex numbers between rectangular and polar forms to strengthen understand.
   
• Visualize points on the complex plane to interpret magnitude as length and argument as direction.

Here are a few real-world applications where the magnitude and argument of complex numbers are utilized:

Electrical Engineering: Engineers use the magnitude to know how strong a current or voltage is, while the argument tells them the phase or timing of the signal.

GPS Navigation: When you’re using GPS, the magnitude helps calculate how far you are from a point, and the argument shows which direction to go.

Robotics and Motion: For robots moving around, the magnitude represents their speed, and the argument shows the exact angle or direction they’re heading.

Aerospace: Pilots and flight systems use magnitude to measure an aircraft’s speed, and argument to determine its heading or flight path.

Signal Processing: When working with signals, the magnitude shows how strong the signal is, and the argument (or phase) helps in timing and synchronization for smooth communication.