Negative Numbers and the Number Line: Free Response

Zero is the reference, so start there and count ticks. A dot to the left of zero names a negative number and a dot to the right names a positive one, so the count gives the size and the side gives the sign.

Counting out from zero, P sits on the seventh tick to the left, Q on the fourth tick to the left, R on the second tick to the right, and S on the seventh tick to the right:

$P = -7, \qquad Q = -4, \qquad R = 2, \qquad S = 7.$

Notice that the two readings are independent. The count alone would make P and S the same, and the side alone would make P and Q the same; it takes both readings together to name a single integer.

On a number line the order is the picture itself: of any two points, the one farther to the right is the greater and the one farther to the left is the smaller. So reading the dots off from the left-hand end already sorts them from least to greatest.

From the left the dots come in the order P, Q, R, S, which gives

$-7 < -4 < 2 < 7.$

No digits were compared to get this. Comparing the bare digits of $-7$ and $-4$ would suggest the wrong order, since $7$ is more than $4$, while the picture puts the dot for $-7$ farther left and so lower.

Two numbers are opposites when they sit the same distance from zero and on opposite sides of it, so the test has two conditions and both have to hold.

Count from zero out to each dot: P is seven ticks to the left, Q is four ticks to the left, R is two ticks to the right and S is seven ticks to the right. Now check the pairings. P and Q are on the same side, so they fail the second condition. Q and R are on opposite sides but at four ticks against two, so they fail the first. P and S pass both:

$P = -7 \quad \text{and} \quad S = 7.$

A quick way to see it in the picture is to fold the page along the line through zero. The dots for P and S come down on top of each other, and no other pair does.

The opposite of a number is where a reflection of the line about zero sends it: the distance from zero is kept, and the two sides of zero trade places. Each of these four numbers sits some units out on one side, so its opposite sits that same count out on the other.

The number $14$ is fourteen units to the right of zero, so its opposite is fourteen units to the left. The number $-11$ is eleven units to the left, so its opposite is eleven units to the right, and $7$ and $-23$ flip the same way:

$\text{opposite of } 14 = -14, \qquad \text{opposite of } -11 = 11$

$\text{opposite of } 7 = -7, \qquad \text{opposite of } -23 = 23$

The distance from zero is the same in every one of these pairs. Only the side changes, which is exactly what the minus sign in front of a number records.

Read the expression from the inside out. First, $-16$ names the point sixteen units to the left of zero. The outer minus sign is then an instruction to take the opposite of that point, which flips it across zero to the matching point sixteen units to the right:

$-(-16) = 16.$

So two flips undo each other, and that is general rather than special to $16$. Reflecting the line about zero and then reflecting it again puts every point back where it started, whatever its distance from zero and whichever side of zero it began on, so taking the opposite twice always returns the number you began with.

The claim describes what the flip does to a number sitting away from zero, and for those numbers it is right: a point some ticks to the right of zero has its opposite the same number of ticks to the left, and the two are different points.

But the flip is made about zero, and zero is the point the fold runs through. Reflecting it moves it nowhere at all, so it comes down on itself:

$-(0) = 0.$

The claim therefore fails, and it fails at exactly one integer. Zero is the only one at no distance at all from zero; every other integer is some positive count of ticks out on one side, and its opposite is that same count out on the other, so the two are genuinely different points. This is also why $-0$ is not a new number: writing a minus sign in front of zero asks for a flip that changes nothing.

Sea level is the reference, so the sign records which side of it a site falls on and the digits record how far. Above sea level is the positive side of the line and below sea level is the negative side.

The lookout is above, so its value stays positive. The mine floor and the salt flat are both below, so each takes a minus sign, and the jetty sits at the reference itself, which is neither above nor below:

$320, \qquad -175, \qquad -60, \qquad 0.$

The jetty is worth a second look. Its zero is not a missing measurement; it is a genuine reading, and it says the site sits exactly at the level the other three are measured from.

Lowest to highest is left to right along the number line, so the two sites below sea level come first, then the reference itself, then the site above it.

Between the two negative values, compare positions and not digits. The value $-175$ lies one hundred seventy-five units to the left of zero while $-60$ lies only sixty units to the left, so $-175$ is farther left and therefore the smaller:

$-175 < -60 < 0 < 320.$

The chain matches the physical picture, which is the point of recording sites this way. The deepest site is the least number in the list and the highest site is the greatest.

Two different quantities are being run together. The numbers $175$ and $60$ record how far each site lies from sea level, so comparing those two settles which site is farther from sea level, not which one is higher.

Height is a position, and position is what the signed values carry. Both sites lie below sea level, so both sit to the left of zero on the line, and the site farther from sea level is the site farther to the left:

$-175 < -60.$

So the mine floor is the lower of the two, not the higher. The reasoning failed at the moment the signs were dropped, because on the negative side of the line a greater distance from zero carries a point farther left, which makes it the smaller number. The member's own numbers point the right way once they are read as depths: being farther below sea level is exactly what makes the mine floor the lower site.

Take the pairs one at a time and read positions.

Both $-14$ and $-9$ lie to the left of zero, fourteen ticks out and nine ticks out, and fourteen ticks out is farther left, so that one is the smaller. For $0$ and $-4$, every negative number lies to the left of zero, so zero is the greater of those two. For $-58$ and $6$, the whole negative side of the line lies to the left of the whole positive side, so no counting is needed at all:

$-14 < -9, \qquad 0 > -4, \qquad -58 < 6.$

Only the first pair needed its ticks counted. The other two were settled by which side of zero each number falls on, with no counting at all.

Greatest to least runs right to left along the line, so read the positions starting from the right-hand end and work back.

The two positive values sit to the right of zero, at twelve ticks and seven ticks, so $12$ comes first and $7$ second. Zero comes next, since it lies to the right of every negative number. Then the three negatives, ordered by how far to the left of zero each one sits: three ticks, eight ticks, sixteen ticks.

$12 > 7 > 0 > -3 > -8 > -16.$

The negatives come out in the reverse of what their numerals suggest. The largest numeral anywhere in the list belongs to $-16$, and $-16$ is the smallest number in it.

Start from what the numeral of an integer measures. It gives the number of one-unit ticks between zero and the point, and the sign gives the side of zero that count is made on. So a larger numeral always means a point farther from zero, whichever side it is on.

On the right of zero, moving farther from zero means moving to the right, and farther right is greater. So among positive numbers a larger count is a greater number, which is the behaviour you already expect:

$4 < 11.$

On the left of zero, moving farther from zero means moving to the left, and farther left is smaller. So among negative numbers a larger count sits farther left, which makes it the smaller number:

$-11 < -4.$

The two statements are one fact seen from opposite sides. Nothing about the two counts changed between them; what changed is the direction the counting runs in, and order along the line is fixed by direction, not by how large the numeral looks. The same reading also explains why no counting at all is needed to compare a negative number with a positive one: they sit on different sides of zero, so the negative one is farther left however large its numeral is.

Both $4$ and $9$ sit to the right of zero, at four ticks and nine ticks, so $9$ is farther right and therefore the greater:

$4 < 9.$

Their opposites sit at those same counts on the left-hand side, four ticks left and nine ticks left. Now the larger count is the one farther to the left, so it is the smaller:

$-4 > -9.$

The pair started with the four-tick number smaller and finished with it greater. Nothing was recomputed along the way; the two points simply swapped which of them was on the left.

One pair where the test gives the wrong answer is enough to defeat it. Look for a pair where distance from zero and position on the line disagree, so put the number that is farther from zero on the negative side.

Take $-13$ and $5$. Their distances from zero are thirteen steps and five steps, so the test compares thirteen against five and calls $-13$ the greater. The line says otherwise: $-13$ sits thirteen steps to the left of zero while $5$ sits five steps to the right, so $-13$ is farther left and therefore the smaller:

$-13 < 5.$

The test fails here for a reason that repeats. Distance from zero is a length and carries no direction, while order asks which point lies to the left. Every negative number is less than every positive number, so any far negative paired with a near positive breaks the test the same way.

Both $6$ and $11$ sit to the right of zero, at six steps and eleven steps. The eleven-step point is farther right, so it is the greater:

$6 < 11.$

Now take the opposites. The point $-6$ sits six steps to the left of zero, and $-11$ sits eleven steps to the left. Each has kept its count of steps, but the counting now runs the other way. So the eleven-step point is the one farther left, and farther left means smaller:

$-6 > -11.$

The comparison came back turned around, and the reason is entirely about position. Nothing was recomputed from the digits. The bigger count puts $11$ farther out on the right, where farther out is greater, and it puts $-11$ farther out on the left, where farther out is smaller.