Absolute Value: Free Response

Each number has a position on the line, and the bars ask only how far that position is from zero.

The number $-13$ sits thirteen units to the left of zero, $4$ sits four units to the right, and $11$ sits eleven units to the right:

$\lvert -13 \rvert = 13, \qquad \lvert 4 \rvert = 4, \qquad \lvert 11 \rvert = 11.$

One of the three inputs was negative and all three outputs came out positive. That is not luck. The bars report a count of unit steps, and the side of zero a number sits on is not part of that count.

An integer with absolute value $6$ is an integer six units from zero, so start at zero and walk six units. There are two directions to walk, so there are two places to land:

$\lvert 6 \rvert = 6, \qquad \lvert -6 \rvert = 6.$

No third integer qualifies, because every other integer sits some other number of units from zero. The two that do qualify are opposites, and that is exactly why they tie: opposites are mirror images across zero, so they are the same distance from it.

Now walk zero units from zero. You never leave, so there is only one place to land:

$\lvert 0 \rvert = 0.$

The second list is shorter than the first because zero is the one number whose opposite is itself.

No integer has an absolute value of $-4$, and the reason sits in the definition rather than in any calculation.

The bars report a distance: the number of unit steps between the number and zero on the line. A count of steps starts at none and grows from there, so the smallest value it can take is $0$, and that value belongs to zero alone:

$\lvert x \rvert \ge 0 \quad \text{for every integer } x.$

A value of $-4$ is less than $0$, so nothing produces it.

Trying numbers could not have settled this. Testing twenty integers rules out twenty integers, and the claim is about every integer. The argument above needs no list, because it works from what the bars measure.

The two expressions differ only in where the bars are drawn, and that is enough to change what gets measured.

In the first, one pair of bars encloses the whole sum, so the addition is finished before any distance is read. Adding a positive number to a negative one moves right along the line, and starting at $-7$ and moving two units right lands at $-5$:

$\lvert (-7) + 2 \rvert = \lvert -5 \rvert = 5.$

In the second, each number carries its own pair of bars, so two distances are read off first and only then added:

$\lvert -7 \rvert + \lvert 2 \rvert = 7 + 2 = 9.$

Same two numbers, same addition sign, two different results.

In the first expression the bars enclose a subtraction, so carry it out before measuring. Taking eleven away from $4$ moves eleven units left and crosses zero:

$\lvert 4 - 11 \rvert = \lvert -7 \rvert = 7.$

The second expression has a symbol written outside the bars, which means the bars never enclosed it. They act on $-8$ alone and return the distance $8$, and the minus sign standing in front then takes the opposite of that:

$-\lvert -8 \rvert = -(8) = -8.$

The second result is negative and nothing has gone wrong, because it is not an absolute value. It is the opposite of one. The rule that an absolute value is never negative is a promise about what comes out from between the bars, and this minus sign was never between them.

Read each expression as an instruction about what to do first.

In $\lvert (-7) + 2 \rvert$ the bars enclose the whole sum, so the sum is the thing whose distance is measured. The numbers combine into one number before anything is measured, and the positive $2$ pulls that number back toward zero:

$\lvert (-7) + 2 \rvert = \lvert -5 \rvert = 5.$

In $\lvert -7 \rvert + \lvert 2 \rvert$ each number is enclosed separately, so nothing combines before measuring. Two distances are read off and then added, and neither distance remembers which side of zero its number came from:

$\lvert -7 \rvert + \lvert 2 \rvert = 7 + 2 = 9.$

So the two forms may not be exchanged. Stripping the signs before adding throws away the cancellation that adding a positive number to a negative one produces, which is why the second arrangement comes out the larger of the two here. Rewriting $\lvert a + b \rvert$ as $\lvert a \rvert + \lvert b \rvert$ is the same kind of error as replacing a quantity grouped in parentheses by the loose pieces inside it.

How far apart two numbers sit on the line is the absolute value of their difference, so subtract the two readings and then measure what comes out.

Subtracting $5$ from $-14$ moves five units further left:

$-14 - 5 = -19.$

The bars then discard the sign and keep the size of the gap:

$\lvert -14 - 5 \rvert = \lvert -19 \rvert = 19.$

The temperature moved by $19$ degrees Celsius. Counting along the line agrees: fourteen degrees to climb from $-14$ up to zero, then five more to reach $5$.

Take one station at a time, each as the absolute value of a difference.

Station Fern went from $-9$ to $4$:

$\lvert -9 - 4 \rvert = \lvert -13 \rvert = 13.$

Station Ridge went from $7$ to $-5$. Subtracting a negative number moves right rather than left, which is where a dropped sign would cost the answer:

$\lvert 7 - (-5) \rvert = \lvert 7 + 5 \rvert = 12.$

So Fern's temperature moved $13$ degrees Celsius and Ridge's moved $12$ degrees Celsius, and Fern moved further.

Notice what the comparison is between. It is between the two gaps, not between the readings: Ridge finished at the colder temperature of the two stations and still covered the shorter distance.

Subtract one pair of readings both ways round and set the results beside each other. For a dawn reading of $-14$ degrees and an afternoon reading of $5$ degrees,

$-14 - 5 = -19, \qquad 5 - (-14) = 19.$

The two results are opposites, and that is what always happens: reversing a subtraction reverses the direction you travelled, and reversing a direction leaves the number of steps alone. A number and its opposite are mirror images across zero, so they are the same distance from it, and one pair of bars turns both differences into one answer:

$\lvert -14 - 5 \rvert = \lvert 5 - (-14) \rvert = 19.$

What the bars discard is the sign, and the sign is not part of the size. It records the direction. Subtract afternoon minus dawn, and a positive result means the temperature rose while a negative one means it fell. That is real information, but it answers a different question: a station that rose nineteen degrees and a station that fell nineteen degrees moved by the same amount, and only the signed difference tells them apart.

The definition sorts an integer by which side of zero it lies on, so check the sign of each number before doing anything else.

The number $-19$ is negative, so the case for a negative input applies and the absolute value is the opposite of the number. The opposite of a negative integer is positive:

$\lvert -19 \rvert = -(-19) = 19.$

The number $23$ is positive, so the other case applies and the number already counts its own distance from zero:

$\lvert 23 \rvert = 23.$

Both results came out positive, and neither could have come out negative, because a distance is never less than zero. The two cases got there by doing different things to their inputs.

The student has read $-x$ as a description, a negative number, when it is an instruction: take the opposite of $x$. What sign the result carries depends on what $x$ stands for. Take the opposite of a negative integer and the result is positive:

$-(-19) = 19.$

So the minus sign in the case does not make the answer negative. It undoes the sign that $x$ already carries, which is exactly why the case is stated only for negative inputs, and never for the positive ones.

Now test the replacement. If $\lvert x \rvert = x$ held for every integer, then at $x = -19$ it would claim

$\lvert -19 \rvert = -19,$

and no point sits $-19$ units from zero. A single negative integer is enough to disprove a rule that claims to cover every integer, so the replacement is not a repair. The one-case rule the student wants is correct only for inputs that are zero or positive, and those are the inputs that never needed a second case.

Stand on each side of zero and count the units back to it.

To the right of zero, a number already is its own count of units: the point $23$ is reached by twenty-three steps from zero, so nothing has to be done to it. To the left, the number and the count disagree in sign. The point $-19$ is nineteen steps from zero, and nineteen is not $-19$ but its opposite. So one side needs the opposite taken and the other does not, and so neither $x$ nor $-x$ alone can serve both sides:

$\lvert x \rvert = x \quad \text{when } x \ge 0, \qquad \lvert x \rvert = -x \quad \text{when } x < 0.$

That is what forces the split. Now check the boundary. The first case is stated for $x \ge 0$, so zero falls under it, and it returns

$\lvert 0 \rvert = 0,$

which is right, since zero is no distance from itself. What a definition may not do is leave an integer out of both cases. That integer would still sit some distance from zero. The definition, though, would have no case to apply to it, so it would return no absolute value at all for that integer. A definition that reports nothing for some of its inputs is incomplete.

The two orderings answer different questions, so carry them out one at a time.

As signed numbers, position along the line decides, and further left means smaller. The number $-16$ sits furthest left, then $-3$, then $9$, then $12$:

$-16 < -3 < 9 < 12.$

For the second ordering, replace each integer by its distance from zero and forget which side it sat on:

$\lvert -3 \rvert = 3, \qquad \lvert 9 \rvert = 9, \qquad \lvert 12 \rvert = 12, \qquad \lvert -16 \rvert = 16.$

So by absolute value the order runs $-3$, $9$, $12$, $-16$. The ends of the two lists are the giveaway: $-16$ is the least of the four integers and also the furthest of the four from zero.

Which integer is greater is a question about position. The number $11$ lies to the right of zero and $-14$ lies to the left of it, and everything to the right of zero is greater than everything to the left:

$-14 < 11.$

Which has the greater absolute value is a question about distance, so measure both from zero:

$\lvert -14 \rvert = 14, \qquad \lvert 11 \rvert = 11.$

Since $14$ is more than $11$, the number $-14$ has the greater absolute value while being the smaller integer.

There is no contradiction here, because the two comparisons measure different things. Greater asks how far to the right a number has got, while absolute value asks how far from zero it has got, counting both directions the same. That is why a number far to the left can lose the first comparison and win the second.

Take the pair $-20$ and $6$. As signed numbers,

$-20 < 6,$

so the shortcut would have the absolute value of $-20$ come out the smaller of the two. Measure them:

$\lvert -20 \rvert = 20, \qquad \lvert 6 \rvert = 6.$

Here $20$ is more than $6$, so the smaller integer has the larger absolute value and the shortcut fails on this pair. One pair is enough, because the shortcut was offered for every pair.

What it runs together is position and distance. Ordering integers reads position along the line, where further left is smaller. Absolute value reads distance from zero, where the two directions count the same. Below zero those two run opposite to each other: the further left an integer moves, the smaller it becomes as a number and the larger its distance from zero grows. The shortcut does hold when both integers are zero or positive, which is very likely where it came from, and that is precisely the region where position and distance are measuring the same thing.