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Absolute Value

In this article, we will discuss about the absolute value and it’s properties.

In 1806, Jean-Robert Argand coined the term “module”, which define the unit of measure in French and refers to the complex absolute value. It was borrowed into English as the Latin equivalent modulus in 1866. Since at least 1806 in French and 1857 in English, the word absolute value has been used in this sense. Karl Weierstrass introduced the notation |x| in 1841, featuring a vertical bar on each side.

|x|, which is pronounced as ‘Mod x’ or ‘Modulus of x,’ represents the absolute value of a variable x. ‘Modulus’ is a Latin term that literally means’ measure.’ The term “absolute value” is sometimes known as “numerical value” or “magnitude.” The absolute value solely represents the numeric value and excludes the numeric value’s sign.

On the number line, the absolute value of the number or any integer is the number’s actual distance from origin i.e., zero. As a result, the absolute value is always greater the zero and never be negative.

Because the absolute value converts the sign of the numbers into positive, yet zero has no sign, there is no absolute value for it.

If the number is greater than zero, a positive number will be returned. And if the given number is negative, then the modulus value of that number will be positive as well. It’s written as |x|, where x is a positive integer.

Representation of Absolute Value

The modulus symbol, ‘| |’, is used to denote absolute value, with the numbers between it.

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 for which result, it always been a good thing.

The absolute value or modulus of any real number x is indicated by |x|, with a vertical bar on each side of the quantity, and is defined as 

                              |x|= x if x≥0

                                       -x if x<0

The absolute value of a real number is its distance from zero along the real number line, and the absolute value of a difference of two real numbers is the distance between them, according to analytic geometry. In mathematics, the concept of an abstract distance function can be considered as a generalisation of the absolute value of the difference.

Properties of Absolute Value

  1. |X|=0 if and only if X=0
  2. |X|>0 for all X >0
  3. |-X|=X
  4. |X – Y|= |Y – X|
  5. |X|² = X²
  6. |X| =√ X²
  7. |X*Y|= |Y*X|
  8. |X/Y|= |X|/|Y| if Y not equals to 0
  9. |X|- |Y| ≤|X – Y|
  10. ||X|- |Y|| ≤ |X – Y|

Absolute Value of Real Number

The absolute value or modulus of any real number x is indicated by |x|, with a vertical bar on each side of the quantity, and is defined as 

                              |x|= x if x≥0

                                       -x if x<0

On the number line when we see the absolute value of 2 The distance of 2 from 0 is represented by |2|. As a result, both +2 and -2 are the same distance from the origin. However, because the distance is never calculated in the negative, it would be accepted as 2.

Absolute Value of Complex Number

The absolute value |z| of a complex number z = x + iy is defined as the distance between z and 0 in the complex plane C. Because the absolute value |x| of a real number x can be read as the distance from x to 0 on the real number line, this will broaden the definition of absolute value for real numbers.

Real numbers with imaginary numbers make up complex numbers. As a result, unlike integers, finding the absolute value for them is challenging. Assume that the supplied complex number is x+iy.

Let A = x + iy then, |A|= √(real(A))² + (img(A))² = √x²+y².

Points To Remember

The following bullet points assist in the representation of absolute values.

  • |x| or abs are used to represent the absolute value of x. (x).
  • Any number’s absolute value always yields a non-negative outcome.
  • |x| is pronounced as ‘mod x’ OR ‘modulus of x.’


A number’s absolute value is its distance from 0. We all know that distance is usually a positive number. Because the absolute value is a measure of distance, it is never negative. 

In addition to the value, a sign is sometimes assigned to a numeric value to indicate the direction. A positive or negative value is occasionally assigned to a numeric value to explain an increase or decrease in quantity, values above or below the mean value, profit, or loss in a transaction. The sign of the numeric value is ignored in absolute value, and only the numeric value is examined.


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