Absolute Value: Definition, Equation and Properties with Examples

Q: What's the difference between a linear function and an absolute value function?

A linear function graphs as a straight line, while an absolute value function graphs as a V-shape. The absolute value function has a "corner" where it changes direction, unlike the continuous slope of a linear function.

Q: What's the relationship between absolute value and distance on a number line?

The absolute value of the difference between two numbers represents the distance between them on a number line. For example, |a - b| gives the distance between points a and b.

Q: What's the difference between -|x| and |-x|?

-|x| means the negative of the absolute value of x, while |-x| is the absolute value of the negative of x. They are not the same: -|x| will always be non-positive, while |-x| will always be non-negative.

Absolute Value: Definition, Equation and Properties with Examples

Edited By Team Careers360 | Updated on Jul 02, 2025 05:15 PM IST

The separation of a number from a number line's origin is referred to as a number's absolute value. It is denoted by the symbol |a|, which specifies the size of any integer 'a'. Any integer's absolute value, regardless of its sign or whether it is positive or negative, will be a non-negative real number. To write the modulus of a, two vertical lines |a| are used to symbolize it.

This Story also Contains

What Is A Number's Absolute Value?
Symbol For Absolute Value
Absolute Value in Equations or Inequalities.
Examples For Absolute Value Of A Number
Absolute Value Function
Properties

Absolute Value: Definition, Equation and Properties with Examples

What Is A Number's Absolute Value?

The absolute value of an integer or number is how far it is from zero on a number line. Because of this, the absolute value is always positive or zero and never negative. The following definition applies to the absolute values:

|a| = ( a if a ≥ 0 )

( -a if a < 0 )

Only positive numbers will be generated if the number is positive. Additionally, the number's modulus will also be negative if it is negative. It is represented by the symbol |n|, where n is an integer.

Symbol For Absolute Value

The absolute value sign is represented by the modulus symbol "| |" and the numbers inside the symbol. For example |9| is used to represent the absolute value of nine.

Any number's absolute value is determined by how far it is from the line's origin. Additionally, it indicates whether a number is positive or negative and its polarity. It can never be negative since it represents distance, which cannot be negative. It is therefore always positive.

‘| |’ – Absolute value symbols, ‘| | pipes’ are used to represent the absolute values.

The number "a'' is the one whose absolute value needs to be determined, for example: "|a|".

Absolute Value in Equations or Inequalities.

The symbol |a| is used to signify a number's absolute value. On a number line, this value or number signifies the separation (absolute difference) between a and 0. An absolute value equation is one that contains an absolute value expression. Absolute value inequalities come in two different varieties. One with |a| < b or less, and the other with |a|> b or more.

Examples For Absolute Value Of A Number

Let's look at a few illustrations of the absolute value of numbers.

|-1| = 1

|-14| = 14

|1| = 1

|0| = 0

|7| = 7

Absolute Value Function

If f(x) = |x|, the absolute value function is defined as follows:

f(x) = +x if x > 0,

And, f(x) = -x if x < 0

Properties

If x and y are real numbers, and the absolute values meet the criteria listed below:

Property	Expression
Non-negativity	\| x \|≥ 0.
Positive-definiteness	\|x\| = 0 => x = 0.
Multiplicatively	\| x × y\| = \|x\| × \|y\|.
Subadditivity	\|x + y\| ≤ \|x\| + y\|.
Symmetry	\|-x\| =\|x\|
Identity of indiscernible (equivalent to positive definiteness)	\|x – y\| = 0 ↔ x = y
Triangle inequality	\|x – y\| ≤ \| x – z \| + \|z – x\|
Preservation of division (equivalent to multiplicatively)	\|x / y\| = \| x \| / \| y\|
Equivalent to subadditivity	\|x – y\| ≥\| \| x \|–\| y \| \|

Frequently Asked Questions (FAQs)

1. What is a number's absolute value?

The distance a number is from zero on a number line is represented by its absolute value.

2. What does 4 mean in absolute terms?

|4| = 4 is the absolute value of four.

3. What is -12 in absolute terms?

In absolute terms, -12 equals 12. Given that there are 12 spaces between -12 and 0 on a number line.

4. What is the absolute value symbol?

Absolute value is denoted by the symbol |x|, where x is an integer.

5. What is -7 in absolute terms?

The absolute value of -7 is 7. Its sign is |-7|, which equals 7.

6. Can absolute value of an integer be Negative?

Any integer's absolute value is always positive and never negative.

7. How do you graph y = |x|?

The graph of y = |x| is V-shaped, with the vertex at the origin (0,0). It consists of two line segments: y = x for x ≥ 0, and y = -x for x < 0, forming a "V" shape.

8. What's the difference between a linear function and an absolute value function?

A linear function graphs as a straight line, while an absolute value function graphs as a V-shape. The absolute value function has a "corner" where it changes direction, unlike the continuous slope of a linear function.

9. What's the relationship between absolute value and distance on a number line?

The absolute value of the difference between two numbers represents the distance between them on a number line. For example, |a - b| gives the distance between points a and b.

10. What's the difference between -|x| and |-x|?

-|x| means the negative of the absolute value of x, while |-x| is the absolute value of the negative of x. They are not the same: -|x| will always be non-positive, while |-x| will always be non-negative.

11. What's the difference between |x + y| and |x| + |y|?

|x + y| is the absolute value of the sum of x and y, while |x| + |y| is the sum of their individual absolute values. They're not always equal. For example, if x = 3 and y = -5, |x + y| = |-2| = 2, but |x| + |y| = 3 + 5 = 8.

12. How do you solve an equation like |x - 3| = 4?

To solve |x - 3| = 4, consider two cases: (x - 3) = 4 and (x - 3) = -4. Solve each equation: x = 7 or x = -1. The solution set is {-1, 7}.

13. How does absolute value affect inequalities?

When solving inequalities with absolute values, you often need to consider two cases. For example, |x| < 3 means -3 < x < 3, as x could be positive or negative while having an absolute value less than 3.

14. What does the equation |x| = a mean geometrically?

The equation |x| = a represents two points on a number line: one at +a and one at -a. It means x is either a units to the right of zero or a units to the left of zero.

15. How do you solve a system of equations involving absolute values?

Solve systems with absolute values by considering different cases based on the possible signs inside the absolute value symbols. Solve each case separately and combine the results, checking solutions in the original equations.

16. What's the triangle inequality in terms of absolute values?

The triangle inequality states that for any real numbers a and b, |a + b| ≤ |a| + |b|. This means the absolute value of a sum is always less than or equal to the sum of the absolute values.

17. How do you find the absolute value of a fraction?

The absolute value of a fraction is the absolute value of its numerator divided by the absolute value of its denominator. For example, |-3/4| = |3|/|4| = 3/4.

18. What's the difference between |x²| and (|x|)²?

|x²| is always equal to x², because x² is always non-negative. (|x|)² is also equal to x². In this case, they're equivalent, but it's important to understand why.

19. How does absolute value relate to complex numbers?

For a complex number z = a + bi, its absolute value |z| represents its distance from the origin in the complex plane. It's calculated as √(a² + b²).

20. What's the relationship between absolute value and the square root function?

For any real number x, |x| = √(x²). This relationship shows that the absolute value can be expressed using squares and square roots, which are always non-negative for real inputs.

21. How does absolute value relate to the concept of parity?

Absolute value preserves the parity (evenness or oddness) of integers. If n is even, |n| is even. If n is odd, |n| is odd. This is because absolute value only changes the sign, not the magnitude.

22. How do you write the absolute value of a number mathematically?

The absolute value of a number x is written as |x|. For example, the absolute value of -5 is written as |-5|, which equals 5.

23. How does absolute value relate to the concept of absolute error in measurements?

Absolute error is often expressed using absolute value. If x is the true value and x̂ is the measured value, the absolute error is |x - x̂|, representing the magnitude of the difference regardless of whether it's an overestimate or underestimate.

24. What's the relationship between absolute value and the concept of a limit point in topology?

In the real number line, a point p is a limit point of a set S if for every ε > 0, there exists an x in S such that 0 < |x - p| < ε. This definition uses absolute value to measure the "closeness" of points.

25. What's the difference between |x - y| and ||x| - |y||?

|x - y| represents the distance between x and y on a number line. ||x| - |y|| represents the difference in the magnitudes of x and y. They're not always equal. For example, if x = -3 and y = 2, |x - y| = |-5| = 5, but ||x| - |y|| = |3 - 2| = 1.

26. What's the relationship between absolute value and the concept of a metric in topology?

The absolute value difference |x - y| serves as the standard metric (distance function) on the real number line. It satisfies the properties of non-negativity, symmetry, and the triangle inequality, which are required for a metric.

27. What is the absolute value of a number?

The absolute value of a number is its distance from zero on a number line, regardless of whether it's positive or negative. It's always non-negative and represents the magnitude of a number without considering its sign.

28. Why is the absolute value of a negative number positive?

The absolute value represents distance from zero, which is always positive. A negative number's distance from zero is the same as its positive counterpart's distance, so they have the same absolute value.

29. Can the absolute value of a number ever be negative?

No, the absolute value of a number is always non-negative. It represents the magnitude or distance from zero, which can never be negative.

30. How does absolute value relate to the concept of magnitude?

Absolute value directly represents magnitude. It tells us the size or amount of a quantity without considering its direction or sign. For example, |-5| and |5| both have a magnitude of 5.

31. What's the connection between absolute value and the concept of a norm in linear algebra?

Absolute value is the simplest example of a norm, which measures the "size" of a mathematical object. In one-dimensional space, the absolute value norm |x| gives the distance of x from 0.

32. What's the relationship between absolute value and even/odd functions?

The absolute value function is neither even nor odd. An even function f(-x) = f(x) is symmetric about the y-axis, while an odd function f(-x) = -f(x) is symmetric about the origin. The absolute value function doesn't satisfy either condition.

33. How does absolute value affect the properties of limits?

The limit of an absolute value function may not exist at points where the function inside the absolute value changes sign. For example, lim(x→0) |x|/x doesn't exist because the function approaches 1 from the right and -1 from the left.

34. How do you differentiate an absolute value function?

The derivative of |x| is 1 for x > 0, -1 for x < 0, and undefined at x = 0. This is because the absolute value function has a "corner" at x = 0, where it's not differentiable.

35. What's the relationship between absolute value and the concept of a modulus in number theory?

In number theory, the modulus operation (as in a mod n) is related to absolute value. The result of a mod n is always non-negative, similar to how |a| is always non-negative.

36. How do you integrate an absolute value function?

Integrating an absolute value function often requires splitting the integral at the points where the function changes sign. For example, to integrate |x| from -1 to 1, you'd split it into two integrals: from -1 to 0 (where |x| = -x) and from 0 to 1 (where |x| = x).

37. How does the absolute value function behave under composition with other functions?

The composition of absolute value with other functions can lead to interesting behaviors. For example, |sin(x)| creates a function that "bounces" off the x-axis instead of going below it, effectively "folding" the sine wave.

38. What's the connection between absolute value and the concept of supremum and infimum?

For any set of real numbers S, the supremum of the set of absolute values of elements in S (sup{|x| : x ∈ S}) gives the maximum distance of any element in S from zero.

39. How do you solve an equation like ||x| - 2| = 3?

To solve ||x| - 2| = 3, consider two cases: |x| - 2 = 3 and |x| - 2 = -3. Then solve each case considering the possible signs of x. The solution set is {-5, -1, 1, 5}.

40. What's the relationship between absolute value and the concept of a vector's magnitude?

In one-dimensional space, the absolute value of a number is equivalent to the magnitude of a vector. In higher dimensions, the concept generalizes to the Euclidean norm, which is analogous to absolute value for vectors.

41. How does absolute value relate to the concept of convexity in functions?

The absolute value function is convex, meaning that a line segment between any two points on its graph lies on or above the graph. This property generalizes to functions involving absolute values, influencing their optimization behavior.

42. What's the difference between |f(x)| and f(|x|) for a function f?

|f(x)| takes the absolute value of the function's output, while f(|x|) applies the function to the absolute value of x. They're generally not equivalent. For example, if f(x) = x², then |f(-2)| = |-4| = 4, but f(|-2|) = f(2) = 4.

43. How do you find the inverse of an absolute value function?

An absolute value function is not one-to-one, so it doesn't have a unique inverse. However, you can define a restricted inverse by considering only the non-negative or non-positive part of the domain.

44. What's the relationship between absolute value and the concept of a p-norm?

The absolute value is equivalent to the 1-norm (also called taxicab norm) in one-dimensional space. In higher dimensions, different p-norms generalize the concept of absolute value to vectors.

45. How does absolute value affect the continuity of a function?

The absolute value function is continuous everywhere, including at x = 0 where it has a "corner". When composing absolute value with other functions, it preserves continuity but can introduce corners where the inner function changes sign.

46. What's the connection between absolute value and the concept of a Lipschitz condition?

The Lipschitz condition, which bounds the rate of change of a function, is often expressed using absolute value. A function f is Lipschitz continuous if there exists a constant K such that |f(x) - f(y)| ≤ K|x - y| for all x and y in the domain.

47. How do you solve an inequality like |x² - 4| < 3?

To solve |x² - 4| < 3, consider two cases: -3 < x² - 4 < 3. Solve each inequality: 1 < x² < 7. Taking the square root (and considering both positive and negative roots), the solution is -√7 < x < -1 or 1 < x < √7.

48. How does absolute value relate to the concept of a norm-preserving linear transformation?

A linear transformation T is norm-preserving if |T(x)| = |x| for all x in the domain. In one-dimensional space, this means T(x) = ±x, corresponding to reflection or identity transformations.

49. What's the connection between absolute value and the concept of a metric space?

The absolute value difference |x - y| defines the standard metric on the real number line, making it a metric space. This concept generalizes to other spaces where a suitable "distance" function can be defined.

50. How do you solve an equation like |x + 1| + |x - 1| = 2?

To solve |x + 1| + |x - 1| = 2, consider different cases based on the possible signs inside the absolute value symbols. In this case, the solution is -1 ≤ x ≤ 1, as this is where the sum of the distances from x to -1 and 1 equals 2.

51. What's the relationship between absolute value and the concept of a uniform norm?

The uniform norm (or supremum norm) of a function f on an interval [a,b] is defined as ||f||∞ = sup{|f(x)| : a ≤ x ≤ b}. This uses absolute value to measure the maximum magnitude of the function over the interval.

52. How does absolute value affect the behavior of sequences and series?

The absolute value is crucial in defining convergence of sequences and series. For example, a series ∑an converges absolutely if ∑|an| converges, which is a stronger condition than regular convergence.

53. What's the connection between absolute value and the concept of a contraction mapping?

A function f is a contraction mapping if there exists a constant 0 ≤ k < 1 such that |f(x) - f(y)| ≤ k|x - y| for all x and y in the domain. This definition uses absolute value to measure how much the function "shrinks" distances.

54. How do you find the domain of a function involving absolute values?

The domain of a function with absolute values is typically all real numbers, unless there's a restriction from other parts of the function. For example, the domain of f(x) = √(|x| - 2) is x ≤ -2 or x ≥ 2, because |x| - 2 must be non-negative.

55. What's the relationship between absolute value and the concept of a bounded function?

A function f is bounded if there exists a constant M such that |f(x)| ≤ M for all x in the domain. This definition uses absolute value to measure the maximum "size" of the function's output.

56. How does absolute value relate to the concept of a distance function in abstract algebra?

In abstract algebra, a distance function d(a,b) generalizes the properties of |a - b|. It must be non-negative, symmetric (d(a,b) = d(b,a)), and satisfy the triangle inequality (d(a,c) ≤ d(a,b) + d(b,c)), just like absolute value difference does on the real line.