The modulus function, which also goes by the name absolute value function, is responsible for determining the magnitude of a number, commonly known as its absolute value, regardless of whether that value is positive or negative. It never returns a value that is negative, whether it be an integer or a variable. The modulus function is denoted by the expression y = |x| or f(x) = |x|, where f is R (0,) and x is R. 

Where x is a real number, |x| is the modulus of x. If x is not negative, then the value of f(x) will be the same as x. If the value of x is negative, the function f(x) will return the magnitude of x; that is, f(x) will equal -x. This is a brief summary of the formula for the modulus function. 

Rules of modulus

y = |x| or f(x) = |x| denotes the modulus function, where f is R (0,) and x is R. Where x is a real number, |x| is the modulus of x. If x is not negative, f(x) has the same value as x. If x is negative, f(x) will be the magnitude of x, i.e., f(x) = -x.

Modulus Function Significance

The result of the modulus function is always a positive value of any variable or a number. It’s also called the absolute value function since it returns a non-negative result for any independent variable, positive or negative. A modulus function, in other words, determines the magnitude of a number.

Property of Modulus Function

Let’s have a look at the modulus function’s qualities now that we have its formula and graph in front of us. These are the characteristics: 

Property 1: 

For all real values of x, the modulus function always returns a non-negative integer. As a result, the modulus function can never be negative.

|x| = a; a > 0 x = a ; |x| = a; a = 0 x = 0 ; 

If |x| equals a, then a can never be 0.

Property 2:

Case 1: Inequality for a positive number

|f(x)| > a and a > 0 a f(x) (or) f(x) > a

 Case 2: Inequality for a negative number (if a 0).

There is no solution for this with |f(x)| a and a 0.

|f(x)| > a and a 0 This holds true for all real f values (x).

Property 3:

If x and y are both real numbers, then |-x| = |x| |x y| = 0

 |x + y| ≤ |x| + |y|

 |x − y| ≥ ||x| − |y||

 |xy| = |x| |y|

 |x/y| = |x|/|y|, where y is not zero.

Modulus of Complex Number

The modulus of a complex number is the distance between that number and the origin in the argand plane. If z = x + iy is a complex number with real x and y and I = -1, then the value that is not negative and has the form (x2 + y2) is referred to as the modulus of the complex number. Absolute value is another name for the modulus of a complex number, which is another name for the complex number. 

In this post, we will learn how to comprehend the notion of modulus of complex numbers algebraically and visually, as well as its formula and several solved cases.

Modulus of Complex Number Formula

The formula |z| = (x² + y²) gives the modulus of a complex number z = x + iy, represented by |z|, where x is the real component and y is the imaginary part of the complex number z. The conjugate of z may also be used to derive the modulus of a complex number z. 

Modulus of Complex Number Using Graph

The distance between the complex number’s coordinates and the origin on a complex plane is called the modulus of the complex number when it is shown on a graph. The distance that a complex number would have to travel on the argand plane for it to be represented as a point is the modulus of that number (a, b). This distance is denoted by the equation r = (a² + b²), and it is calculated by moving in a direction perpendicular to the origin (0, 0). (a, b). 

Furthermore, the Pythagoras theorem may be used to determine the modulus of a complex integer, where the hypotenuse represents the modulus, the real component is the base, and the imaginary part is the height of the right-angled triangle. The magnitude (or length) of the vector encoding a + bi is the same as the modulus of a complex number a + bi.

Properties of Modulus of Complex Number

Some of the features of a complex number are listed below. When z and w are both complex numbers, we get:

1. The modulus of complex numbers z and -z is the same, i.e., |z| = |-z|.
2. |z| = 0 if z = 0 If and only if the complex number in question is zero, the modulus of that number will be equal to zero. 
3. |z.w| = |z|.|w| The modulus of complex numbers multiplied together yields the same result as multiplying complex numbers by their own modulus. 
4. |z/w| = |z|/|w| There is a one-to-one correspondence between the modulus of the quotient of two complex numbers and the modulus of the complex numbers themselves. 
5. |z|=|z| The modulus of a complex number is equal to the modulus of the complex number’s conjugate; in other words, the modulus of the complex number is equal to itself. 
6. Since the modulus of the nth power of a complex number is equal to the modulus of the complex number itself, this means that |zn| equals |z|n. 

Important Notes on Modulus of Complex Number

The distance between the origin (0, 0) and the complex number represented as a point on the argand plane is known as the modulus of the complex number (a, b). 

If and only if the complex number itself is zero, the modulus of the complex number will be 0. 

The magnitude of the vector that encodes the complex number a + bi, also known as its length, is the same as the modulus of the complex number a + bi. 

The modulus of a complex number may be calculated by taking the square root of the sum of the squares that represent the real and imaginary components of the complex number. 

Conclusion

The magnitude of a number may be determined by a modulus function regardless of the sign of the number. It is also known as the absolute value function in certain circles. The modulus of a real number x may be calculated using the modulus function, which in mathematical notation is written as |x|. It gives back the value of x that is not negative. The distance between the origin and zero is also known as the modulus or absolute value of a number. In this article we conclude the Rules and Significance of Modulus. Function with its rules of modulus, modulus function significance and property of modulus function.