Absolute Value

The absolute value of a number is the non-negative value of that number without regard to its sign. It represents the distance of the number from zero on the number line.

What is the Absolute Value of a Number?

The absolute value of a number is its distance from zero on the number line, irrespective of its direction. This means that both positive and negative numbers have the same absolute value. For instance, the absolute value of -5 is 5, and the absolute value of 5 is also 5. Denoted by vertical bars, such as ∣−7∣ = 7 and ∣3∣ = 3, absolute value is always a non-negative number. This concept is crucial in various mathematical contexts, including distance measurement and solving equations.

Absolute value of a Number Examples

• ∣−3∣=3
• ∣−20∣=20
• ∣4∣=4
• ∣0∣=0
• ∣8∣=8
• ∣10−3∣=∣7∣=7
• ∣5+6∣=∣11∣=11
• ∣−4×6∣=∣−24∣=24

Absolute Value Function

The absolute value function is a function that returns the absolute value of a number, which is the number’s distance from zero on the number line, regardless of its sign. It is denoted by f(x) = ∣x∣. The function can be defined piecewise as:

• |x| = +x for x > 0 
• |x| = -x for x < 0

This means:

• If x is positive or zero, the absolute value of x is x.
• If x= is negative, the absolute value of x is −x.

Examples:

• f(3) = ∣3∣ = 3
• f(−4) = ∣−4∣ = 4
• f(0) = ∣0∣ = 0

The graph of the absolute value function f(x)=∣x∣ forms a “V” shape, with the vertex at the origin (0,0).

Absolute Value Properties

Property	Mathematical Expression
Non-negativity	| x | ≥ 0
Positive-definiteness	| x | = 0 ↔ a = 0
Multiplicativeness	| x × y| = |x| × |y|
Subadditivity	| x + y| ≤ | x | + | y |
Symmetry	|-x| = |x|
Identity of indiscernible (Positive definiteness)	| x – y | = 0 ↔ a = b
Triangle inequality (Subadditivity)	| x – y | ≤ | x – z | + | z – x |
Preservation of division (Multiplicativeness)	| x / y| = | x | / | y |
Equivalent to subadditivity	| x – y | ≥ | | x | – | y | |

Examples:

1. Non-Negativity: ∣3∣ = 3 and ∣−3∣ = 3
2. Symmetry: ∣−5∣ = 5 and ∣5∣ = 5
3. Triangle Inequality: ∣3+4∣ = 7 and ∣3∣+∣4∣ = 7
4. Multiplicativity: ∣2⋅−5∣ = 10 and ∣2∣⋅∣−5∣ = 10

Absolute Value of Complex Number

The absolute value (or modulus) of a complex number is a measure of its distance from the origin in the complex plane. Complex numbers are often represented in the form z = a + bi, where a and b are real numbers, and i is the imaginary unit, defined by i² = −1. the absolute value is denoted by ∣z∣ and is defined as: ∣z∣=a²+b² This formula is derived from the Pythagorean theorem and represents the Euclidean distance from the point (a,b) in the complex plane to the origin (0,0).

Geometric Interpretation

In the complex plane, a complex number z = a + bi can be represented as a point (a,b). The absolute value ∣z∣ is the length of the line segment connecting the origin to this point. This distance is always non-negative and provides a way to quantify the “size” of a complex number irrespective of its direction.

Example Calculations

1. Real Numbers: If z is purely real, say z=a, then its absolute value is simply: ∣a∣ = √a² = ∣a∣
   For example, if z=3, then ∣z∣=3. If z=−4, then ∣z∣=4.
2. Imaginary Numbers: If z is purely imaginary, say z=bi, then its absolute value is: ∣bi∣ = √(0)² + (b)² = ∣b∣
   For example, if z=5, then ∣z∣=5. If z=−7, then ∣z∣=7.
3. Complex Numbers: For a general complex number z = a + bi:
   ∣z∣=√a²+b²
   For example, if z = 3 + 4i:
   ∣z∣ = √32 + 42 = √9 + 16 = √25 = 5

Absolute Value in Number Line

The absolute value of a real number on a number line represents its distance from zero, regardless of direction. It is always a non-negative value. The absolute value is denoted by vertical bars, such as ∣x∣.

∣x∣ = x if x≥0
∣x∣ = −x if x<0

Geometric Interpretation

On the number line, the absolute value of a number represents its distance from zero. Here are a few examples to illustrate this concept:

1. Positive Numbers: For a positive number x, the absolute value is the number itself. For example, if x = 5, then ∣5∣ = 5. This means that the distance from 5 to 0 on the number line is 5 units.
2. Negative Numbers: For a negative number xxx, the absolute value is the positive counterpart of the number. For example, if x=−7, then ∣−7∣=7. This means that the distance from -7 to 0 on the number line is 7 units.
3. Zero: The absolute value of zero is zero itself. For example, if x=0, then ∣0∣=0. This means that the distance from 0 to 0 on the number line is 0 units.

Examples

1. |3|: The absolute value of 3 is 3, as it is already positive. The distance from 3 to 0 on the number line is 3 units. Number line: …−3  −2  −1  0  1  2  3…
2. |-4|: The absolute value of -4 is 4, as we take the positive counterpart. The distance from -4 to 0 on the number line is 4 units. Number line: …−4  −3  −2  −1  0  1  2…
3. |0|: The absolute value of 0 is 0. The distance from 0 to 0 on the number line is 0 units. Number line: …−2  −1  0  1  2…