1.1.22 · D5Arithmetic & Number Systems
Question bank — Absolute value - modulus — definition, number line interpretation
This page attacks the ideas behind absolute value, not the arithmetic. Each item is a trap that a confident-but-hasty student walks into. Read the question, cover the answer, and force yourself to justify — a bare "yes/no" earns nothing.
True or false — justify
for every real number .
False. This only holds when ; for negative the bars flip the sign, e.g. .
for all .
False. Reflection across never changes distance, so , which is ; but is . They agree only at .
is always strictly positive.
False. It is non-negative: . The one exception is , where . "Strictly positive" wrongly excludes zero.
for every real .
False. The square root symbol returns the non-negative root, so , not . For , , not .
and can differ.
False. Distance is symmetric: and differ only by a sign, which the bars erase, so always.
If then .
False. Same distance from allows two positions: but . The correct conclusion is or .
can be true.
True — but only when and point the same way (both or both , including when one is ). Then no cancellation occurs and the triangle inequality becomes equality.
has no solutions.
True. Distance can never be negative, so no real satisfies it. Contrast with , which does have the single solution .
The equation has two solutions.
False. It has zero solutions: no number sits at distance from , because distances aren't negative.
has infinitely many solutions.
False. Distance means the point is ; exactly one solution, . Only makes .
Spot the error
": drop the bars and the minus, so . Therefore dropping bars always just deletes minus signs, so ."
The trap is treating a symbol as always negative. is negative when ; then . You can only "delete a minus" when you know the inside is negative.
", and equality is rare, so I'll assume it's always strict."
Equality is not rare — it holds for every same-sign pair (and whenever either term is ). Writing where is meant loses all these cases.
"Since , by analogy ."
Absolute value distributes over multiplication but not addition. For : yet . Addition can cancel; multiplication of magnitudes cannot.
"To solve I write , so . Done."
Half the answer. A point at distance from can lie on either side, so also , giving . Both directions must be checked.
" means or ... so I square both sides to be safe: ."
Squaring is legal (both sides are non-negative here) but you must then solve both roots of . Squaring doesn't remove the two-case structure; it hides it.
" where : that's , so is always positive because the outer minus makes it positive."
The result is right but the reasoning is wrong. regardless of the visible sign; the bars, not the minus, guarantee non-negativity. Try : too.
"Distance from to is ; I'll skip the bars since is bigger."
Only valid if you know . If then , which can't be a distance. The bars in protect you when you don't know the order.
Why questions
Why does the definition use (not "" or " with a minus") for the negative case?
When , the number is genuinely positive (negating a negative). So is the distance, correctly stated without knowing a numeral in advance.
Why is written with a piecewise definition instead of one formula?
Because the rule genuinely changes at : keep the value on the right side, flip it on the left. Two behaviours need two branches — this makes a piecewise function.
Why does deserve the name "distance"?
It is , is only when , and is symmetric — exactly the properties a distance must have. This is why is the standard metric on in Distance and metric spaces.
Why does the triangle inequality use "" rather than ""?
Because walking then can involve backtracking, making the straight-line displacement shorter than the total path . Only same-direction travel avoids waste. See Triangle inequality.
Why does (with ) always give two answers but gives one?
Two points sit at any positive distance from (one each side), but distance pins you at — a single point. See Solving absolute value equations.
Why does describe an interval rather than two separate pieces?
"Within distance of " is the single stretch from to , i.e. . This links absolute value to Inequalities and interval notation.
Why can we say but not ?
The radical is defined to output the non-negative root, and is exactly the non-negative number whose square is . Choosing would give a negative answer when , contradicting the radical's rule.
Edge cases
What is , and why isn't it a special exception to the definition?
. Since , the first branch applies and returns unchanged — it fits the rule, no exception needed.
What is when is a very large negative number, say ?
grows without bound toward ; the farther left you go, the larger the distance from . Absolute value has no upper limit.
Does together with force ?
Yes. Restricting to non-negatives removes the mirror-image possibility, so equal distances mean the same point. The ambiguity only exists when signs are unknown.
What happens to exactly at ?
It equals — the single point where the "V-shaped" distance function touches its minimum. This is the lowest value can ever take.
Is still meaningful if one of is zero?
Yes, and it becomes equality: . Zero is a degenerate "same direction" case — nothing to cancel.
For complex numbers, does the number-line picture of "flip the sign" still work?
No — a complex number has no left/right on a single line. Its modulus is its straight-line distance from the origin in the plane, generalising the same distance idea. See Complex numbers — modulus.
Can and both hold at once?
Only at , where . Elsewhere exactly one branch applies, so both can't be true together.
Recall One-line summary to carry away
Every trap on this page is defused by remembering: = distance from , so it is , symmetric, single-valued only at , and splits into two directions everywhere else.
Connections
- Number line and ordering — the left/right structure that "distance" quietly relies on
- Piecewise functions — why needs a two-branch definition
- Solving absolute value equations — the two-case machinery behind
- Inequalities — how becomes an interval
- Triangle inequality — the that these traps keep testing
- Distance and metric spaces — the properties that make a genuine distance
- Complex numbers — modulus — where the "flip the sign" picture must be upgraded to plane distance