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Absolute value

In mathematics, the absolute value or modulus of a real number x , is the (non-negative) magnitude measured without regard to its sign.

Version
v1 · 2026-09-28 · History
Domain-specific #
7826
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Elementary Algebra, Real Analysis → Mathematics

Core Idea

Absolute value is treated here as the recurring crossdomainmodelsstructuresrepresentations identity summarized by this source-grounded definition: In mathematics, the absolute value or modulus of a real number x , is the (non-negative) magnitude measured without regard to its sign. In mathematics, the absolute value or modulus of a real number x , is the (non-negative) magnitude measured without regard to its sign. Namely, |x|=x if x is a positive number, and |x|=-x if x is negative (in which case -x is positive), and For example, the absolute value of 3 and the absolute value of −3.

How would you explain it like I'm…

How Far from Zero

The absolute value of a number is how far it is from zero, no matter which way. 3 is three steps from zero, and −3 is also three steps from zero the other way. So both have an absolute value of 3.

Size Without the Sign

The absolute value of a number is its size without its sign, and it is never negative. We write it with bars, like |−3|. If the number is positive, the absolute value is the same number. If it's negative, you drop the minus sign, so |−3| = 3 and |3| = 3. You can think of it as the distance from the number to zero on a number line.

Magnitude Ignoring Sign

The absolute value (or modulus) of a real number x, written |x|, is its non-negative size, ignoring its sign. By definition, |x| = x if x is positive or zero, and |x| = −x if x is negative, which makes the result positive. So |3| = |−3| = 3, and |x| can be read as the distance from x to zero on the number line. The same idea is extended to complex numbers, vectors and other systems, where it becomes related to magnitude, distance and 'norm'. Be careful: vertical bars around a vector sometimes mean a norm, which is a related but different thing.

 

For a real number x, the absolute value or modulus |x| is its non-negative magnitude without regard to sign: |x| = x for x ≥ 0 and |x| = −x for x < 0, so |3| = |−3| = 3. Geometrically, |x| is the distance from x to 0 on the real line, and |x − y| gives the distance between two reals. Generalizations appear across mathematics: absolute values are defined for complex numbers, quaternions, ordered rings, fields and vector spaces, and the concept is closely tied to magnitude, distance and norm. A related but distinct usage puts single bars around a vector to denote its Euclidean or sup norm; double bars with subscripts are the more common and less ambiguous notation. The identity of this entry is the real-number case: the sign-free non-negative magnitude.

Scope of Application

  • Terminology and notation. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English.

  • Definition and propertiesReal numbers. The notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference.

  • Definition and propertiesReal numbers. This is equivalent to the definition above, and may be used as an alternative definition of the absolute value of real numbers.

  • Definition and propertiesReal numbers. The absolute value has the following four fundamental properties ( a , b are real numbers), that are used for generalization of this notion to other domains.

  • Definition and propertiesReal numbers. The absolute value, as "distance from zero", is used to define the absolute difference between arbitrary real numbers, the standard metric on the real numbers.

Clarity

A clear use of Absolute value names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, the absolute value or modulus of a real number x , is the (non-negative) magnitude measured without regard to its sign.

Manages Complexity

Absolute value compresses multiple crossdomainmodelsstructuresrepresentations details into a stable diagnostic relation. The source shows both the central mechanism—these are either immediate consequences of the definition or implied by the four fundamental properties above.—and the practical consequence—the formulas can be derived by considering each case s>t and t>s separately. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit.

Abstract Reasoning

  1. Type the carrier. Identify the crossdomainmodelsstructuresrepresentations entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: In mathematics, the absolute value or modulus of a real number x , is the (non-negative) magnitude measured without regard to its sign.
  3. Check operation and conditions. The absolute value of a complex number is defined by the Euclidean distance of its corresponding point in the complex plane from the origin.
  4. Demand recognition evidence. This can be computed using the Pythagorean theorem: for any complex number.

Knowledge Transfer

Within the home domain. Knowledge about Absolute value transfers literally when a new case preserves the same carrier type, relation, and recognition test. The term absolute value has been used in this sense from at least 1806 in French and 1857 in English. The notion of an abstract distance function in mathematics can be seen to be a generalisation of the absolute value of the difference. Beyond the home domain. No canonical parent is asserted for Absolute value.

Neighborhood in Abstraction Space

Absolute value sits in a crowded region of the domain-specific corpus (27th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Named Analytic Theorems & Operators (39 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08