Absolute Value Calculator

Two boxes, one idea. An absolute value calculator answers a single question: how far is this number from zero? The absolute value calculator on this page takes that question in two forms - a bare number, or the arithmetic hiding between the bars.

• Value (x) - one number, such as -7. This box handles the everyday $|x|$ case, and -7 sits in it as the placeholder. The absolute value calculator reads the box the moment Calculate is pressed.
• Expression - the inside of the bars, such as 2 - 5*3. Type the arithmetic only; the bars are implied, because an absolute value calculator adds them itself and closes them again. The example buttons swap whole samples in and out.

Two result rows come back, one per box, and both rows are distances - an absolute value calculator never reports a sign, only a size. The first row is $|x|$, the distance from your number to 0. For -7 the absolute value calculator returns 7. Both rows update together, because one absolute value calculator backs both boxes. The second row evaluates the expression first and then measures its result from zero. Enter 2 - 5*3 and the box computes -13, so the second row reads 13.

This page is more than a static absolute value calculator - it is also an AI tutor you can ask questions:

Both boxes feed the same definition, so start anywhere, and the absolute value calculator keeps the two rows side by side. If you have never met the bars before, the next section builds the picture from scratch. If you already know what $|x|$ means, skip ahead - the absolute value calculator will keep working either way.

No bars on the keyboard? The pipe character | usually hides above the enter key - shift plus backslash on US layouts. Typing abs(x) is the long-standing stand-in, and the AI box on this page accepts it happily. In the Expression box you can skip the character entirely: the absolute value calculator there already knows where the bars belong, so 8 - 12 simply means |8 - 12|. Three notations, one idea - |x|, abs(x), and the two boxes of the absolute value calculator above. Pick whichever your homework uses; the distance answer does not change. The absolute value calculator on this page simply accepts all three dialects - and measures the same distance under each.

The absolute value of a number is its distance from zero on the number line. Distance ignores direction, so a trip 5 units left and a trip 5 units right count the same. An absolute value calculator turns that one-fact picture into answers, and the picture in turn explains every answer the absolute value calculator gives. The picture shows both trips - and why they land on one answer. Reading it takes three seconds; the absolute value calculator underneath takes even less. Point it at 5 or at -5 and the reply is the same 5, because both arrows reach equally far. Distance is the whole story, and an absolute value calculator is that story told in one number.

Ray's New Higher Algebra (1866) calls the same idea the numerical value of an expression - what it is worth once the signs are settled. Today the definition splits on the sign of $x$:

The $x$ in the table is the same $x$ as the Value box - one letter, one input, exactly the pairing an absolute value calculator needs to accept AI-filled forms. For $x = -7$ the second case applies, so $|-7| = -(-7) = 7$. An absolute value calculator runs exactly this case split on every click. It never drops the minus that flips a negative input - the flip is where hand calculations usually slip. Whenever you doubt a result, ask what the absolute value calculator would print for the same $x$ and compare cases. Feed -3 into the Value box and it lands in the same second case as $x = -7$ did. For $x = -9$ it walks the second case too: $|-9| = -(-9) = 9$.

One consequence deserves its own line: $|x| \ge 0$ for every real $x$. Zero is the floor, and it is also the one guard every absolute value calculator applies before printing. Zero input, zero out - any absolute value calculator will show it. If a computation hands you a negative "absolute value", something upstream went wrong - Mistake 1 below shows the usual culprit. An absolute value calculator is a handy referee for exactly those disputes.

Subtract, then take the bars - that is the whole distance formula. The gap between $a$ and $b$ on the number line is $|a - b|$, and the order cannot matter: $|3 - (-2)|$ and $|(-2) - 3|$ both equal 5. One expression field turns an absolute value calculator into a distance meter - the Expression box above is wired for exactly this. The absolute value calculator is the fast path; the number line is the proof. Type either order and the absolute value calculator erases the sign for you, same 5 both times. It is also the quiet answer to why the bars exist at all: without a sign-eraser, $3 - (-2)$ and $(-2) - 3$ could not share one distance.

Read the picture left to right: 3 sits 2 units right of 0, and -2 sits 2 units left. Between them stretches a 5-unit gap. Any "how far apart" question is an absolute value in disguise, and that gap - five units of pure distance - is exactly what an absolute value calculator reports.

The Expression box above computes such gaps directly. Type 3 - (-2) and the absolute value calculator answers 5 - no sign bookkeeping required. Reverse the entry to (-2) - 3 and the absolute value calculator still answers 5. Temperature spans, elevation gaps, and profit-versus-loss spreads all fold into the same $|a - b|$ shape. A morning of -6 degrees and an afternoon of 9 degrees differ by $|9 - (-6)| = 15$ degrees. A gain of 120 dollars against a loss of 45 spans $|120 - (-45)| = 165$. Both readings fall out of the absolute value calculator in one entry each, which is why a general-purpose absolute value calc earns its keep beyond plain $|x|$ runs.

A dedicated absolute value calculator could stop at $|x|$; this one follows the bars wherever they appear.

Two quick facts follow from the distance reading, and both come in handy:

• Same absolute value, two sources. $|5| = |-5|$ because both numbers stand 5 units from 0 - an absolute value calculator shows both on one number line. The bars erase the direction, not the size.
• Order flips, distance holds. Swapping $a$ and $b$ only changes the sign inside the bars, and the bars erase that sign - one more thing the absolute value calculator does on every run.

With the definition and the distance formula on the table, the three examples below run the absolute value calculator through its paces - three signs, an expression, and a real gap. Watch what each one feeds in and what comes back.

Evaluate $|12|$, $|0|$, and $|-9|$ - the three sign cases an absolute value calculator meets every day. Each lands in a different branch of the definition, so an absolute value calculator treats all three differently - worth one look each.

1. $|12|$: 12 is non-negative, so the first case passes it through unchanged: $|12| = 12$.
2. $|0|$: 0 sits at the center, zero units from itself: $|0| = 0$. The absolute value calculator is at its most literal here - it reports the distance it sees, even when that distance is nothing.
3. $|-9|$: -9 is negative, so the sign flips: $|-9| = -(-9) = 9$.

Check the pattern: 12, 0, and 9 are all non-negative, as every absolute value must be. The drill itself is old - sign-flip exercises like $|-8| = 8$ fill the practice sections of a century of algebra texts. An absolute value calculator compresses all three cases to clicks; the case list above is what those clicks mean. Run 12, then 0, then -9 through the Value box of the absolute value calculator: three clicks, three one-step answers. Each result matches the rule case it came from. An absolute value calculator shows no work for these - none is needed - but it never fumbles the flip on $|-9|$, which is the one place a rushed pencil does. Three cases, three answers, no surprises.

Evaluate $|2 - 5 \cdot 3|$, the sample preloaded in the Expression box.

1. Inside first: $5 \cdot 3 = 15$.
2. Finish the subtraction: $2 - 15 = -13$.
3. Only now take the absolute value: $|-13| = 13$.

A wrong turn starts with the bars. Splitting them over the subtraction gives $|2| - |5 \cdot 3| = 2 - 15 = -13$, a negative number - impossible for an absolute value, so the split must be wrong. An absolute value calculator catches this slip by construction: it never opens the bars until the inside is a single number. Inside first, bars last, every time - an absolute value calculator enforces that order for you. This two-step routine is precisely what the absolute value calculator automates. The Expression box reduced 2 - 5*3 to -13 internally; the absolute value calculator then measured -13 from zero, and the second row reported 13.

Change the sample to 8 - 12 and the same machinery returns 4, ready for checking by hand in two lines - subtract to -4, flip to 4. That is the division of labor on this page: the absolute value calculator answers fast, and the hand routine above lets you audit every answer it gives.

A diver rests on a ledge 18 m below the surface, which reads $-18$ m on the depth scale. A gull rides the swell 7 m above it, at $+7$ m on the same scale. How far apart are they?

1. Gap formula: $|7 - (-18)|$.
2. Inside: $7 - (-18) = 7 + 18 = 25$.
3. Bars change nothing: $|25| = 25$.

They are 25 m apart, and one Expression entry did all the work. The same line answers any vertical-gap question - the Expression box of the absolute value calculator takes 7 - (-18) unchanged and returns 25. Note that the absolute value calculator never asks which one is higher - distance has no hierarchy. Whatever the context - sea level, ground level, or zero on a price chart - an absolute value calculator turns paired readings into one clean gap. A balcony 12 m up and a basement 3 m down? The absolute value calculator takes 12 - (-3) and hands back 15 m, no diagram required.

The reverse question runs the other direction. Given the 25 m gap and the gull's 7 m, where can the diver be? Two depths fit: $7 - 25 = -18$ and $7 + 25 = 32$ - each exactly 25 m from 7, as the absolute value calculator's distance reading promised. Turning a distance back into candidates is solving an equation, not evaluating one. An absolute value calculator evaluates; it does not hunt candidates. That job belongs to the absolute value equations calculator on this site, and its guide on how to solve absolute value equations walks the two branches step by step.

The bars obey three short rules, and each one says something about distance:

Try the product rule on the absolute value calculator: $|-3 \cdot 4|$ is 12, and so is $|-3| \cdot |4|$. One absolute value calculator run covers the left side, two runs the right - entries -34, then -3, then 4 - the absolute value calculator settles it either way. The Expression box takes -34 in one entry, while the same absolute value calculator confirms the split version through two Value runs. The square-root link matters later in algebra - it is the reason $\sqrt{(-6)^2}$ equals 6 and not -6. Enter (-6)^2 and the Expression box evaluates it to 36 internally; the bars, wherever you place them, return 6.

Run -7 the same way: squared it is 49, rooted it is 7, and the absolute value calculator confirms |(-7)^2| = 7 without any prompting. The absolute value calculator agrees here, because the bars and the root both erase signs.

Not every tempting pattern is a rule, though. Sums resist the bars: $|-3| + |4| = 7$ while $|-3 + 4| = 1$, a gap no absolute value calculator papers over, because the two expressions genuinely differ and honest arithmetic says so. That failed sum rule is exactly Mistake 3 below - the properties table earns its keep by marking where the analogy stops. A quick check on the absolute value calculator settles each row in seconds.

Three slips cause most of the lost points around the bars:

1. Letting the outside minus sneak inside. $-|-3| = -(3) = -3$. The minus in front is not part of the bars; it survives the whole operation, and the result really is negative. Asked about $|-3|$ alone, an absolute value calculator returns 3, never -3.
2. Distributing the bars over subtraction. $|9 - 12| = 3$, but $|9| - |12| = -3$; an absolute value calculator takes the single entry 9 - 12 and never makes the split. The bars seal one box around the entire expression - finish the inside, then measure the result.
3. Trusting $|a + b| = |a| + |b|$. With $a = -3$ and $b = 4$, the left side is $|1| = 1$ while the right side is $7$ - feed both into the absolute value calculator and the disagreement is immediate. Multiplication does factor, $|a \cdot b| = |a| \cdot |b|$; addition does not.

Each slip vanishes under one habit: reduce whatever sits inside the bars to a single number before judging any signs. That habit is exactly the two-step routine the absolute value calculator above executes on every Calculate click. Watch it run Mistake 2 as a demo: enter 9 - 12 and the box reads 3, not -3, because the bars waited for the whole subtraction. That patience is what separates an absolute value calculator from a hasty rewrite of the same line. Keep the habit and the absolute value calculator becomes a checker; lose it and no absolute value calculator can rescue the answer, because the entry itself is wrong.