How this instrument works
Absolute value answers one question: how far is this number from zero, ignoring which side it sits on? Formally it is a piecewise rule — |x| = x when x is zero or positive, and |x| = −x when x is negative — but the two branches are really one idea wearing two faces. Negative input, negate it; positive input, leave it alone. Either way the number line's leftward and rightward halves fold onto the same non-negative ray.
There is a second, algebraic route to the same answer: |x| = √(x²). Squaring a number erases its sign before the square root ever sees it, so this version needs no case split at all. That identity is more than a curiosity — it is how absolute value gets extended to complex numbers and vectors, where 'is x negative' stops making sense but 'how big is x' still does.
One consequence trips up more students than any other: the minus sign inside |x| = −x is not an instruction to make the output negative. If x is already negative, negating it makes it positive — a double negative doing exactly what double negatives do. The only input where the two branches meet is x = 0, the single point the number line folds around, and the only place where distance from zero is itself zero.
- Enter any real number — positive, negative, or zero — into the Number field.
- Read Absolute value: it reports the size of that number with the sign removed.
- Try a negative entry, then its positive twin, and confirm both return the same figure.
- Enter 0 to see the one input where the result and the input coincide exactly.
Worked example — how far is −7 from zero?
Set Number to −7 and read Absolute value: it returns 7. The rule sees only how far −7 sits from the origin on the number line — seven units — and discards which direction it sits in, so the negative branch of the definition fires and reports |−7| = −(−7) = 7.
Enter 7 instead and Absolute value still reports 7, since 7 is already zero or greater and the first branch, |x| = x, applies with nothing to flip. The two runs land on an identical output despite starting seven units apart on opposite sides of zero — that collapse is the entire purpose of the function.
Questions
What does absolute value actually mean?
It is the distance a number sits from zero on the number line, with direction thrown away. |−7| and |7| both equal 7 because both points are seven units from the origin — one to the left, one to the right — and the sign carries no weight in the final answer.
Why does |x| equal −x when x is already negative?
Because negating a negative number makes it positive: if x = −7, then −x = −(−7) = 7, the correct non-negative result. That minus sign is not a request to make the output negative — it is the double-negative arithmetic that turns a negative input the right way up.
Is |x| the same thing as the square root of x squared?
Yes — squaring erases the sign of x before the root is taken, and the principal square root returns a non-negative value, so √(x²) and |x| agree for every real x. This identity is how absolute value gets defined for complex numbers and vectors, where a two-branch piecewise rule no longer applies.
What is the triangle inequality and how does absolute value fit in?
It states |a + b| ≤ |a| + |b| for any real a and b, one of the load-bearing facts of analysis: combining two quantities can never produce more size than adding their sizes separately. Try a = −7 and b = 7: |a + b| = |0| = 0, while |a| + |b| = 14, so the gap between the two sides can be large.
Can absolute value ever come out negative?
No. By definition |x| is zero only when x itself is zero, and strictly positive for every other real input. If a chain of arithmetic ever produces a negative absolute value somewhere upstream, the error is in that arithmetic, not in what absolute value means.
Does |a + b| always equal |a| + |b|?
Only when a and b share the same sign, or one of them is zero. With mixed signs the left side is strictly smaller, as −7 and 7 show: |−7 + 7| = 0 while |−7| + |7| = 14. Treating absolute value as though it distributes over a sum is one of the most common mistakes made with this operator.