Modulus Value of Real Numbers

The modulus function, also known as the absolute value function, determines the magnitude or absolute value of a number, whether positive or negative. There is never going to be a value that is in the negative for any integer or variable. The modulus function is denoted by the expression y = |x| or f(x) = |x|, where f is R (0,) and x is R. When x is a real number, the modulus of x is denoted by the notation |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. Let’s go a little more into the modulus of real numbers, shall we? 

Absolute Values of Real Numbers

Absolute value refers to the distance that separates a given number from the zero point at the beginning of a number line. It is shown by the symbol |a|, which represents the magnitude of any number that begins with the letter ‘a.’ The actual numbers, regardless of whether the integer in question has a positive or negative sign attached to it, will always be the integer’s absolute value. The modulus of an is represented by two vertical lines |a|, which is also known as the modulus of a.

How to define a modulus value of a real number?

The absolute value (or modulus) of a real number x is | x | which is its non-negative value, regardless of its sign. For instance, the absolute value of five is five, and the absolute value of five minus five is likewise five. One way to think about the absolute value of a number is as the location of that number on the real number line in respect to the value zero. 

Absolute Value symbol

The modulus symbol, ‘| |’, is used to denote absolute value, with the numbers between it. The absolute value of 9 is expressed as |9|, for example.

The absolute value of a number is the distance between it and the origin on the number line. It also displays the number’s polarity, whether positive or negative. It can never be negative since it represents distance, and distance cannot be negative. As a consequence of this, it is consistently beneficial. 

Domain and Range of Modulus Function

Any real number can be modulated using the modulus function. The domain of the modulus function is R, while the range of the modulus function is the set of non-negative real numbers (0,). (where R stands for the whole collection of real numbers). As a result, the modulus function’s domain is R and its range is (0,).

Properties of Modulus Function

Let’s look at the modulus function’s properties now:

Property 1: For all real values of x, the modulus function always returns a non-negative number. As a result, the modulus function can never be negative.

|x|＝a; a > 0 ⇒ x = ±a;

|x| = a; a = 0 ⇒ x = 0;

If |x| = a, then a can never be zero.

Property 2:

Case1: (If a > 0): Inequality for a positive number 

|f(x)| < a and a > 0 ⇒ -a < f(x) < a

|f(x) > a and a > 0 ⇒ -a < f(x) > a

Case2: (If a < 0): Inequality for a negative number

|f(x)| < a and a < 0 ⇒ there is no solution for this.

|f(x)| > a and a < 0 ⇒ this is valid for all real values of f(x).

Property 3: If x,y are real numbers, then

|-x| = |x|

|x − y| = 0 ⇔ x = y

|x + y| ≤ |x| + |y|

|x − y| ≥ ||x| − |y||

|xy| = |x| |y|

|x/y| = |x|/|y|, y not equal to zero.

Real Numbers

Except for complicated numbers, any variety we are able to think about is an actual variety. Rational numbers, which includes positive and negative integers, fractions, and irrational numbers, are all examples of real numbers. The union of the set of rational numbers (Q) with the set of irrational numbers (Q’), denoted by way of means of R, is the set of actual numbers. As a result, the set of real numbers can be written as R = Q U Q’.

Natural numbers, whole numbers, integers, rational numbers, and irrational numbers are all examples of real numbers. Real numbers include things like 3, 0, 1.5, 3/2, 5, and so on.

So, which of these numbers isn’t a genuine number? Non-real numbers are those that are neither rational nor irrational, such as √-1, 2 + 3i, and -i. The set of complex numbers, C, is included in these figures.

Conclusion

At last, we can say that rather than being a small topic, modulus has its own importance and is interrelated to many other topics. The absolute value of real numbers can be found in a variety of mathematical contexts, including complex numbers, quaternions, ordered rings, fields, and vector spaces. The absolute value is directly linked to the concepts of magnitude, distance, and conventions in everyday life.