Question

What is an absolute value? (6th grade math)

Answer 1

The absolute value (or modulus) | x | of a real number x is the non-negative value of x without regard to its sign. For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. The absolute value of a number may be thought of as its distance from zero along real number line.

Answer 2

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Answer:

In mathematics, the absolute value or modulus of a real number x, denoted |x|, is the non-negative value of x without regard to its sign. Namely, |x| = x if x is positive, and |x| = −x if x is negative, and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3.

Derivative

Hence, a simple way of expressing this in an expression is |x|/x which turns out to be 1 when x is positive and -1 when x is negative. Also, it's value cannot be determined at 0. Therefore, the derivative of absolute value of x is abs(x)/x.

Properties

Absolute value has the following fundamental properties:

Non-negativity |a| ≥ 0.

Positive-definiteness |a| = 0a = 0.

Multiplicativity |ab| = |a| |b|

Subadditivity |a + b| ≤ |a| + |b|

Idempotence ||a|| = |a|

Symmetry |−a| = |a|

Identity of indiscernible |a − b| = 0 ⇔ a = b.

Triangle inequality |a − b| ≤ |a − c| + |c − b|