Integers and the Number Line: Chapter Test

A floor above the ground and a floor below it lie on opposite sides of one reference point, so the sign records the side and the count records how far.

$12, \qquad -5, \qquad -12, \qquad 0$

Lowest to highest is left to right along the number line, so the two floors below ground come first, and of those two the one farther below ground is the one farther to the left.

$-12 < -5 < 0 < 12$

Two numbers are opposites when they lie the same distance from zero on opposite sides of it. Among the four values only the roof deck and the vault do.

$-(12) = -12, \qquad -(-12) = 12$

The other two floors have opposites that are not on the list. The opposite of $-5$ is five floors ABOVE ground, and zero is the one number that is its own opposite, because it sits at the fold rather than to one side of it.

$-(-5) = 5, \qquad -0 = 0$

Nothing about whether a numeral carries a sign settles an order. What settles it is position: of two floors, the one farther to the left on the line is the lower.

$-12 < -5 < 0 < 12$

Zero is the dividing point of the line, not its bottom end. Every negative number lies to its left and every positive number to its right, so the ground floor stands between the two floors below ground and the roof deck above them, third of the four counting up. The lowest is the vault at $-12$, which is the farthest left of the four points.

What misleads is that $0$ is written without a plus sign and without a minus sign. That absence marks it as belonging to neither side, which is a different thing from sitting below both.

Each change is a step from wherever the balance stands. A deposit steps to the right and a withdrawal steps to the left.

$-23 + 40 = 17$

The signs differ there, so the steps cancel in pairs: twenty-three of the forty dollars bring the balance up to zero and seventeen carry past it. The withdrawal is then a step of thirty-one to the left from $17$.

$17 + (-31) = -14$

Reaching zero from a point means stepping exactly the distance back, in the direction of zero. The balance stands fourteen dollars to the left of zero, so the step is fourteen dollars to the right.

$-14 + 14 = 0$

The number that does this is always the opposite of the one you start from, and the two together are called a zero pair.

Adding a positive number is a step to the RIGHT, and the size of the number is the length of the step. Neither of those depends on where the step begins.

$-23 + 40 = 17, \qquad 4 + 40 = 44$

Both balances moved forty dollars to the right. They simply started in different places.

What a starting balance below zero does change is where the step lands, and the zero pair shows exactly how. From $-23$ the first twenty-three dollars are spent bringing the balance up to zero, and only what is left over carries past it.

$-23 + 40 = (-23 + 23) + 17 = 0 + 17 = 17$

So the deposit does the same amount of work in both cases. An account holder starting below zero has less to show for it afterwards, but that is a fact about where the account started, not about what the deposit did.

Six bins each recorded at the same amount is six groups of that amount, which is one multiplication rather than a sum of six terms.

$6 \times (-17) = -102$

The sizes multiply as usual, $6 \times 17 = 102$, and neither factor is zero, so the single negative factor makes the product negative. The reading in the stockroom agrees: six shortfalls can only add up to a larger shortfall.

The number of bins is the missing factor: it is the number which, multiplied by the amount in one bin, gives the total.

$(-105) \div (-15) = 7 \quad \text{because} \quad 7 \times (-15) = -105$

Neither number is zero and the two signs match, so the quotient is positive, and the sizes divide as usual: $105 \div 15 = 7$. Multiplying back returns the recorded total, so the count stands.

The three numbers are not three of the same kind of thing, and separating them is most of the answer.

$(-105) \div (-15) = 7$

The $-105$ is a signed shortfall, the whole week's. The $-15$ is a signed shortfall too, one bin's. The $7$ is neither: it is a count of how many bins, and a count of things has no sign to carry.

The sign rule agrees with that reading. Neither number going into the division is zero, and the two signs match, so the quotient is positive. It could not have been otherwise without the arithmetic contradicting the situation, since a negative number of bins would mean nothing in a stockroom.

Multiplying back shows the same three roles in the other order: seven bins, each fifteen units short, come to one hundred and five units short in all.

Each pair of bars asks the same question: how far is the enclosed number from zero?

$\lvert -57 \rvert = 57, \qquad \lvert 0 \rvert = 0$

The point $-57$ sits fifty-seven units to the left of zero, and zero is no distance at all from itself. In the third expression the bars enclose the $24$ and nothing else, so they report $24$, and the minus sign written in front of them is applied to that result afterwards.

$-\lvert 24 \rvert = -(24) = -24$

In the first line one pair of bars encloses the whole subtraction, so the subtraction is settled first and the single number it produces is measured.

$\lvert 7 - 22 \rvert = \lvert -15 \rvert = 15$

In the second line each number carries its own pair, so each is measured first and the subtraction then happens between two distances.

$\lvert 7 \rvert - \lvert 22 \rvert = 7 - 22 = -15$

A distance counts unit steps along the line and can never be below none, so only the $15$ could be one. The $-15$ is a perfectly good number; it is simply not a distance.

Deleting minus signs describes what you often see happen, not what the bars do. What they do has two stages, and their order is the whole of it: everything inside is settled down to a single number, and only then is that number's distance from zero reported.

Part B shows both stages mattering. One arrangement settles $7 - 22$ first and measures the result; the other measures $7$ and $22$ first and subtracts afterwards.

$\lvert 7 - 22 \rvert = 15, \qquad \lvert 7 \rvert - \lvert 22 \rvert = -15$

The same two numbers and the same subtraction give different values. No rule about deleting signs could produce that difference, because the signs on the page are identical in the two lines.

A minus sign written outside the bars is reached by neither stage, since it is not inside them.

$-\lvert 24 \rvert = -24$

There the bars measure $24$ and hand back $24$, and the minus sign in front is applied to that. An absolute value is never negative; the NEGATIVE OF an absolute value can be, and this is one.

Four rounds each changing the score by the same amount is four groups of that amount, which is one product.

$4 \times (-6) = -24$

Neither factor is zero and exactly one of them is negative, so the change is negative, as a run of losses must be. Applying it to the starting score is then an addition of two negatives, whose distances pile up.

$-9 + (-24) = -33$

The run of wins is again one product, this time of two positive factors.

$5 \times 11 = 55$

Adding it to the standing score puts two different signs together, so the steps cancel in pairs and the greater distance decides the side, which here is the rightward $55$.

$-33 + 55 = 22$

The closing loss is a single step of seven to the left.

$22 - 7 = 15$

The two runs are the same calculation with one factor changed.

$4 \times (-6) = -24, \qquad 5 \times 11 = 55$

The size of each total change is the number of rounds multiplied by the size of one round's change: twenty-four points in the first run, fifty-five in the second. That part is identical in kind.

The sign is where they part, and it parts for one reason only. The count of rounds is a positive number in both runs, and neither factor is zero, so the count of rounds contributes no sign flip at all. Everything therefore rests on the single round: a loss makes one factor negative, so exactly one flip happens and the product is negative, while two positive factors leave the product positive.

That is why one multiplication serves both runs. Multiplication was never told what a round means; it multiplies the sizes and counts the negative factors, and it is the sign written on one round's result that reports whether the run is driving the score down or up.

The gap between two numbers is the absolute value of their difference, so subtract first and measure afterwards. Subtracting a negative stop number adds its opposite.

$\lvert 9 - (-14) \rvert = \lvert 9 + 14 \rvert = \lvert 23 \rvert = 23$

The second pair sits on the same side of the depot, and the same rule handles it without any change.

$\lvert -30 - (-14) \rvert = \lvert -30 + 14 \rvert = \lvert -16 \rvert = 16$

Nearness is settled by the two gaps, not by the two stop numbers.

$16 < 23$

So stop $-30$ is the nearer to a passenger standing at stop $-14$. Order among the stop numbers themselves is settled by position on the line and runs the opposite way.

$-30 < 9$

The two comparisons are asking different questions of the same pair of numbers, so there is nothing strange in their disagreeing.

Take the pair already in hand, stops $-30$ and $9$. The rule of thumb compares the stop numbers.

$-30 < 9$

So it names stop $9$ as the farther from the depot. Now measure both distances, which is what the question actually asks.

$\lvert -30 \rvert = 30, \qquad \lvert 9 \rvert = 9$

Stop $-30$ is thirty stops out and stop $9$ is nine, so on this pair the rule of thumb has the answer exactly backwards. One counterexample is enough to sink it.

The two questions it has collapsed into one are these. Position along the line asks which side of the depot a stop is on and how far along it stands in that direction; distance from the depot throws the direction away and counts only the stops. Whenever both stops lie on the positive side the two answers agree, which is why the rule of thumb looks sound at first. Among the negative stops they run in opposite directions: moving farther out from the depot there makes the stop number smaller, not larger. So the rule of thumb can start going wrong as soon as one of the two stops is negative, though it need not: on stops $-5$ and $9$ it happens to be right, since $9$ is both the larger number and the farther from the depot.

A pair of neighbours is out of order when the second of the two lies to the left of the first on the number line. Walk the output and test each of the five neighbouring pairs.

$-6 < 0, \qquad -6 < 8, \qquad -51 < 8, \qquad -51 < 61, \qquad -74 < 61$

The first, third and fifth of those say that the SECOND entry of its pair is the smaller, so those three places are out of order: $0$ before $-6$, $8$ before $-51$, and $61$ before $-74$. The second and fourth say the first entry is the smaller, so those two steps already stand as asked.

Every place that is wrong has a negative reading arriving late, which is the first sign of what the program's rule is doing to them.

Genuine order is by position, so $-67$ stands sixty-seven units to the left of zero: farther left than $-51$, and not as far out as $-74$.

$-74 < -67 < -51$

The program compares instead the digits $67$ against the digits already in its list, which are $0$, $6$, $8$, $51$, $61$ and $74$. That puts it after $61$ and before $74$, and writing the signs back on leaves it between the entries $61$ and $-74$.

The two placements disagree because the digits $67$ put the reading near the END of a digit-sorted list, while the reading itself sits near the BEGINNING of the line.

Separate the program's output by side of zero and the pattern is plain. The readings that are not negative come out as

$0, \qquad 8, \qquad 61$

which is the relative order that was asked for. The negatives come out as

$-6, \qquad -51, \qquad -74$

which is exactly the reverse of what was asked for, since $-74 < -51 < -6$.

So the rule fails in two ways at once, and a repair has to answer both. The two sides are interleaved, when every negative belongs ahead of $0$ and ahead of every positive; and the negatives run in the wrong direction among themselves. Repairing it means splitting the column at zero first, then running the negatives from the largest numeral down while everything from $0$ rightward keeps the program's own increasing order.

Why that works comes down to one fact about what a numeral measures. The numeral of a reading counts units out from zero, and the sign records which way that counting ran. Counting out on the right moves toward the greater numbers; counting out on the left moves toward the smaller ones. So a larger numeral makes a positive reading greater and a negative reading smaller, and a rule reading the numeral with the sign set aside can only ever get one of the two sides right.

The column the program was asked for is

$-74 < -51 < -6 < 0 < 8 < 61.$

Work each product left to right. The first two factors are the same in both, and neither is zero, so their two sign flips undo each other.

$(-3) \times (-8) = 24$

The third factor is where the two products part.

$24 \times 5 = 120, \qquad 24 \times 0 = 0$

In both products exactly two factors carry a minus sign, since zero carries none: it is neither positive nor negative.

The rule counts negative factors, and both products give it the same count.

$(-3) \times (-8) \times 5 = 120, \qquad (-3) \times (-8) \times 0 = 0$

So the rule as stated calls both of them positive. The first really is. The second is $0$, and zero is neither positive nor negative, so the rule has made a claim about it that is simply not true.

The repair is not a special case bolted onto the answer, but a restriction on what the rule is about, and the second product shows exactly where the restriction belongs. Every flip in the rule's account is contributed by a factor with a sign, and zero has none. So the rule should read: with no factor equal to zero, a product is positive when an even number of its factors are negative, and negative when an odd number are.

Both products of part A satisfy that statement, the first by falling under it and the second by falling outside it.

Every step in the counting rule is a sign flip, and a sign flip needs a sign to flip. Zero has none: it sits at the dividing point of the line and is neither positive nor negative, so it enters the count with nothing to contribute.

What it does instead is stronger than any flip. It replaces the product.

$(-3) \times (-8) \times 0 = 0$

Whichever numbers stand beside it, and however many of them are negative, the product is $0$. So a zero factor does not adjust the sign of an answer; it settles the answer, and the counting rule is left with nothing to decide.

Division by zero fails at an earlier stage still. A quotient is the missing factor: $(-3) \div 0$ is the number that gives $-3$ when multiplied by $0$. But every product with a factor of zero is zero, never $-3$, so there is no such number.

$(-3) \div 0 \;\text{ is undefined}$

The contrast is worth holding on to. The product with a zero factor has a value, $0$, which simply is not what the sign rule predicted. The quotient has no value at all, so asking after its sign is asking about something that does not exist.

Subtracting a number is adding its opposite, so both subtractions are rewritten first. The opposite of $-12$ is $12$, and the opposite of $25$ is $-25$.

$-18 + 7 - (-12) - 25 = -18 + 7 + 12 + (-25)$

Now combine left to right, one step at a time.

$-18 + 7 = -11, \qquad -11 + 12 = 1, \qquad 1 + (-25) = -24$

Once every step is an addition, the steps may be gathered by direction rather than taken in order.

$7 + 12 = 19, \qquad 18 + 25 = 43$

Nineteen units of travel go to the right and forty-three to the left. Those cancel in pairs, and the direction with travel left over is the one the walk finishes on.

$43 - 19 = 24$

The leftover travel is leftward, so the walk ends twenty-four units to the left of zero, at $-24$. The two routes agree.

Follow the chain one expression at a time. The first expression is the problem as set, so it cannot be wrong. The second is where it breaks.

$-18 + 7 - (-12) - 25 \quad \text{became} \quad -18 + 7 - 12 - 25$