Math 6 · 7.NS.A.1A

Opposites and zero pairs

A deposit and a withdrawal of the same size, a rise and a fall of the same size. Why do some pairs of amounts cancel each other out completely, and what does that let you do?

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Negative numbersMath 6

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Points that differ only in sign: reflections across the axesMath 6 Negative fractions and decimals on the number lineMath 6 Identity and inverse elementsAlgebra 1

Where you are 6 in Integers

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Negative numbers

Opposites and zero pairs

Placing and comparing integers on a number line Adding and subtracting integers2 Absolute value Multiplying and dividing integers

Why this exists

Put 20 dollars into an account and then take 20 dollars out, and the balance is back where it started. Walk 8 steps forward and then 8 steps back, and you are standing on the same spot. A temperature that rises 5 degrees and then falls 5 degrees ends at the temperature it began with. In each case two amounts have undone each other, and together they come to nothing.

Negative numbers gave each of these amounts a number: the deposit is 20 and the withdrawal is -20, the rise is 5 and the fall is -5. Each pair is a number and its opposite. This lesson is about what happens when the two are put together: they always add to 0.

That one fact does a great deal of work. It is how sums like 8 + (-3) are worked out, by letting each negative cancel a positive and counting what is left. It lets a long sum be shortened by striking out pairs that cancel. It lets you find a missing number, since the only number that makes 0 with 14 is -14. And it is the first example of an inverse, an idea that runs right through algebra.

Definition

The opposite of a number a, written -a, is the number the same distance from 0 on the other side of 0. The opposite of 7 is -7, the opposite of -7 is 7, and the opposite of 0 is 0.

A zero pair is a number together with its opposite, such as 7 and -7. The sum of a zero pair is 0:

7 + (-7) = 0

Two numbers whose sum is 0 are also called additive inverses of each other. “Additive” because the partnership is about adding, and “inverse” because each one undoes the other.

In a situation, two amounts are opposite quantities when they have the same size and lie on opposite sides of the same zero: a gain and a loss of the same amount, a climb and a descent of the same height. Opposite quantities combine to make 0.

Counters. A handy picture of integer amounts uses two kinds of counter. A positive counter, drawn with a plus sign, stands for 1. A negative counter, drawn shaded with a minus sign, stands for -1. One of each is a zero pair, worth 1 + (-1) = 0.

Figure 1A positive counter and a negative counter make a zero pair. Together they are worth nothing.

The value of a collection of counters is what all of them add up to. Every zero pair in it contributes nothing, so the value comes from the counters left over once the pairs are taken out.

Intuition

On a number line, think of adding a positive number as a move to the right and adding a negative number as a move to the left, by as many units as its size. Then 4 + (-4) is a trip: start at 0, go 4 units right, and come 4 units back. You finish where you started.

Figure 2Four units out and four units back. The trip 4 + (-4) ends where it began, at 0.

The same is true for any number. A number and its opposite are the same distance from 0, so the move one makes is exactly undone by the move the other makes, and the trip always ends at 0.

With counters the picture is even simpler. Line the positives up in a row and put the negatives underneath, one under each positive for as long as both last. Each column of two is a zero pair and is worth nothing. Whatever sticks out at the end, all positive or all negative, is the value.

Figure 3Eight positives and three negatives. Three zero pairs cancel, and five positives are left.

Figure 3 shows 8 positive counters and 3 negative ones. Three zero pairs cancel, and 5 positives are left, so the value is 5. In symbols, 8 + (-3) = 5.

This also shows that a zero pair can be added to a collection, or taken away from it, without changing its value at all, because the pair was worth 0. So one number has many different pictures, all differing by zero pairs.

Figure 4Three collections with the same value, -2. They differ only by zero pairs.

Results

The zero pair property. For every number a,

a + (-a) = 0 \quad\text{and}\quad (-a) + a = 0

Zero pairs change nothing. Adding a zero pair to a sum, or removing one from it, leaves the sum the same. For every pair of numbers n and a,

n + a + (-a) = n

So in a sum of several numbers, a number and its opposite may be struck out together.

Only the opposite works. If a + b = 0, then b = -a. Each number has exactly one partner that adds with it to make 0, and that partner is its opposite.

Zero is its own partner. a + a = 0 only when a = 0.

Why it is true

The zero pair property. Say a is d units from 0. Its opposite, -a, is also d units from 0, on the other side. Starting from 0, the move for a carries you d units to one side, and the move for -a carries you d units back toward the other side, which lands you on 0 again. With counters, a positive whole number n is n positive counters and -n is n negative counters; every positive has a negative to pair with, there are n zero pairs, and nothing is left over.

When numbers are built up carefully in later algebra, this property is not proved at all: it is taken as one of the starting rules that say what the opposite of a number is. The pictures above are why that rule is the right one.

Zero pairs change nothing. Numbers can be added in any order (the commutative property) and grouped in any way (the associative property); both are in the order and grouping of addition. So group the pair first:

\begin{aligned} n + a + (-a) &= n + \big(a + (-a)\big) \\ &= n + 0 = n \end{aligned}

Only the opposite works. Suppose a + b = 0. Add -a to both sides. The left side becomes (-a) + a + b, and the zero pair (-a) + a is worth nothing, so the left side is just b. The right side becomes (-a) + 0 = -a. So b = -a.

On the number line it says the same thing. If you are standing at a, the move b must bring you back to 0, which is d units away on the other side. The only move that does that is the move for -a.

Zero is its own partner. If a + a = 0, then a is a partner of a, and by the previous result the only partner of a is -a. So a = -a: the number is its own opposite. A number and its opposite are on different sides of 0 unless both are 0, so a = 0.

Worked examples

Example 1

Opposite quantities in situations

Write each pair of changes as numbers and find the total change. Which pairs are zero pairs?

(a) A lift goes up 6 floors and then down 6 floors.

(b) An atom has 9 protons, each with a charge of 1, and 9 electrons, each with a charge of -1. What is its total charge?

(d) The temperature rises 5 degrees and then falls 3 degrees.

Solution

In each part, take the direction that makes the amount go up as positive.

(a) Up 6 floors is 6 and down 6 floors is -6. They are opposites, so they make a zero pair: 6 + (-6) = 0. The lift ends on the floor it started from.

(b) The protons give 9 and the electrons give -9, and 9 + (-9) = 0. The atom has no overall charge. This is why such an atom is called neutral.

(d) The rise is 5 and the fall is -3. These are not a zero pair: the sizes differ. The fall of 3 undoes 3 of the 5 degrees of rise, which leaves 2 degrees of rise, so 5 + (-3) = 2. The temperature ends 2 degrees higher than it started.

Only (a), (b) and (c) are zero pairs. Opposite directions are not enough; the sizes must match too.

Example 2

The value of a collection of counters

Find the value of each collection.

(a) 6 positive counters and 6 negative counters.

(b) 8 positive counters and 3 negative counters.

Solution

Pair positives with negatives for as long as both last. Each pair is worth 0, so only the counters left over count.

(a) All 6 positives pair with all 6 negatives. That makes 6 zero pairs and leaves nothing, so the value is 0.

(b) Only 3 zero pairs can be made, because there are only 3 negatives. That uses 3 of the positives and leaves 8 - 3 = 5 positives. The value is 5. This is Figure 3, and in symbols 8 + (-3) = 5.

(c) Now the positives run out first. The 3 positives make 3 zero pairs, leaving 5 negatives, so the value is -5, and 3 + (-8) = -5.

The collections in (b) and (c) are mirror images: every positive in one is a negative in the other. Their values, 5 and -5, are opposites.

Example 3

One number, many pictures

Explain why each collection in Figure 4 has the value -2. Then find the collection with value 3 that uses exactly 4 negative counters.

Solution

(a) Two negative counters, each worth -1, and nothing to pair with them. The value is -2.

(b) One zero pair, worth 0, and two negative counters. The zero pair changes nothing, so the value is still -2.

Each collection is the one before with zero pairs added, which is why they all have the same value.

For the second part, each of the 4 negatives must be canceled by a positive, which takes 4 positives and makes 4 zero pairs. After that, 3 more positives give the value 3. So the collection has 4 + 3 = 7 positives and 4 negatives, 11 counters in all.

Check. 7 positives and 4 negatives make 4 zero pairs and leave 3 positives, so the value is 3. With exactly 4 negatives, any other number of positives leaves a different amount over, so this is the only such collection.

Example 4

Striking out zero pairs in a sum

(a) Find -17 + 9 + 17 + (-9) + 4.

(b) Mei’s bank account holds 40 dollars. During one week she pays in 15 dollars, takes out 22 dollars, takes out 15 dollars, pays in 22 dollars and takes out 10 dollars. What is the balance at the end of the week?

Solution

(a) The numbers can be added in any order. Group the zero pairs together:

\begin{aligned} &-17 + 9 + 17 + (-9) + 4 \\ &= \big(-17 + 17\big) + \big(9 + (-9)\big) + 4 \\ &= 0 + 0 + 4 \\ &= 4 \end{aligned}

Nothing had to be worked out except noticing the pairs.

(b) As numbers, the week is 40 + 15 + (-22) + (-15) + 22 + (-10). The deposit of 15 and the withdrawal of 15 are a zero pair, and so are the withdrawal of 22 and the deposit of 22. Strike both pairs out:

\begin{aligned} &40 + 15 + (-22) + (-15) + 22 + (-10) \\ &= 40 + (-10) \end{aligned}

The withdrawal of 10 cancels 10 of the 40 dollars, so 40 + (-10) = 30. The balance at the end of the week is 30 dollars. Two of the transactions changed nothing overall, whatever order they came in.

Example 5

Finding the missing partner

(a) What number must be added to -23 to make 0?

(b) Solve x + 14 = 0.

(d) Solve -x + 8 = 0.

Solution

Each part asks for a partner that makes 0, and only the opposite works.

(a) The partner of -23 is its opposite, 23. Check: -23 + 23 = 0.

(b) x adds with 14 to make 0, so x is the opposite of 14: x = -14.

(c) y adds with -3.5 to make 0, so y is the opposite of -3.5: y = 3.5. The fact holds for decimals just as for whole numbers.

(d) Here the unknown part is -x. It adds with 8 to make 0, so -x = -8. The only number whose opposite is -8 is 8, so x = 8.

Check. With x = 8, -x + 8 = -8 + 8 = 0.

Example 6

A long sum that mostly cancels

Find the sum of all the integers from -30 to 31.

Solution

The list is -30, -29, \ldots, -1, 0, 1, \ldots, 30, 31. Adding sixty-two numbers one at a time would be slow, but almost all of them are in zero pairs.

Each of 1, 2, \ldots, 30 has its opposite in the list, so these make 30 zero pairs: -30 + 30, -29 + 29, and so on down to -1 + 1. Each pair is worth 0. The number 0 adds nothing. The only number with no partner in the list is 31, because -31 is not in it.

So the whole sum is 31.

Example 7

The opposite of a sum

Show that, for any numbers a and b,

-(a + b) = (-a) + (-b)

and check it with a = 6 and b = -2.

Solution

The opposite of a + b is, by the result above, the only number that adds with a + b to make 0. So it is enough to show that (-a) + (-b) adds with a + b to make 0. Regroup the four numbers into two zero pairs:

\begin{aligned} &(a + b) + \big((-a) + (-b)\big) \\ &= \big(a + (-a)\big) + \big(b + (-b)\big) \\ &= 0 + 0 = 0 \end{aligned}

So (-a) + (-b) is a partner of a + b, and since the partner is the opposite, -(a + b) = (-a) + (-b).

Check. With a = 6 and b = -2: a + b is 6 positives and 2 negatives, which leaves 4, so -(a + b) = -4. On the other side, (-a) + (-b) = -6 + 2 is 6 negatives and 2 positives, which leaves 4 negatives, -4. Both sides are -4.

In words: the opposite of a total is the total of the opposites. Undoing a deposit of 6 dollars and a withdrawal of 2 dollars is a withdrawal of 6 dollars and a deposit of 2 dollars.

Boundaries

The sizes must match exactly. A number and a different number of the other sign do not make 0: 5 + (-3) is 2, not 0. Only a number and its own opposite are a zero pair.

Opposite quantities need the same zero and the same units. A climb of 5 meters and a descent of 5 feet do not cancel, and a deposit of 20 dollars into one account does not cancel a withdrawal of 20 dollars from a different account. To combine, two amounts must be measured on the same scale.

A total of zero is not the same as nothing happening. A lift that goes up 6 floors and down 6 floors has a total change of 0, but it has traveled 12 floors. Zero pairs describe where something ends up, not how far it went on the way. How far, whatever the direction, is a matter of absolute value.

The partnership is about adding. 5 + (-5) = 0, but 5 \times (-5) is not 0. For multiplying, the partner that undoes 5 is \displaystyle \frac{1}{5}, which gives 1 rather than 0, and 0 has no such partner at all. That other kind of inverse belongs to identity and inverse elements.

Counters show integers only. There is no counter for -3.5 or \displaystyle \frac{3}{4}, so the counter picture stops at the integers. The zero pair property itself does not stop: \displaystyle \frac{3}{4} + \left(-\frac{3}{4}\right) = 0 and -3.5 + 3.5 = 0, as the number line shows.

Zero pairs itself. 0 is the one number that is its own partner, so “a number and its opposite” for 0 is 0 and 0 again.

Where this goes next

Zero pairs are the engine of adding and subtracting integers. To add two numbers with different signs, pair off what cancels and see what is left over, exactly as in the counter examples here, and the left-over part tells you both the size and the sign of the answer. Subtracting turns out to be adding the opposite, and a subtraction such as 2 - 5, where there are not enough positives to take away, is done by first slipping in zero pairs, which changes nothing, until there are.

The same pairing works for parts of a whole: -2.5 + 2.5 = 0, and negative fractions and decimals obey every result in this lesson.

The zero pair property is also the first of a family. Zero leaves every sum unchanged, and the opposite is what undoes a number under adding; one leaves every product unchanged, and the reciprocal undoes a number under multiplying. These are identity and inverse elements, and they are what make it possible to solve an equation: to undo +\,7 on one side, you add its opposite, -7, to both sides.

Exercises

Opposite quantities

A kite rises 12 meters and then drops 12 meters. Write the two changes as numbers and find the total change.
Ines earns 18 dollars and then spends 18 dollars. Write the two changes as numbers and find the total change.
Which of these combine to make 0? Give the total for each. (a) A climb of 40 meters and a descent of 40 meters. (b) A gain of 9 points and then a loss of 6 points. (c) 3 steps forward and then 3 more steps forward.
Describe a situation that 30 + (-30) = 0 could stand for, saying which direction is positive.

Counters

Find the value of a collection of 10 positive counters and 6 negative counters.
Find the value of a collection of 4 positive counters and 11 negative counters.
A collection has 12 positive counters and 5 negative counters. How many zero pairs can be made from it, and what is its value?
Find the collection with value 4 that uses exactly 2 negative counters.
A collection has value -3 and contains 5 positive counters. How many negative counters does it contain?

Canceling in sums

Find 19 + (-7) + (-19).
Find -31 + 12 + 31 + (-12) + (-5).
Find \displaystyle \frac{3}{4} + (-2) + \left(-\frac{3}{4}\right) + 2.
Tobias’s account holds 75 dollars. He pays in 30 dollars, takes out 12 dollars, takes out 30 dollars and pays in 12 dollars. What is the balance afterwards?

Finding the partner

Solve x + 27 = 0.
Solve -50 + y = 0.
Solve 2.8 + w = 0.
Solve -x + (-6) = 0.
Find the sum of all the integers from -20 to 22.
A number added to 13 gives 0. A second number added to the first number also gives 0. What is the second number?

Reasoning

Explain why a collection with the same number of positive and negative counters always has value 0.
A collection of counters has value 0. Must it have the same number of positive and negative counters? Explain.
Is there a number a, other than 0, with a + a = 0? Explain.
Adding one positive and one negative counter to a collection gives it two more counters. Does it change the value? Explain.
Use the fact that only the opposite of a number adds with it to make 0 to show that -(-a) = a for every number a.
A lift goes up 5 floors and then down 5 floors. One person says the total change is 0 floors and another says the lift traveled 10 floors. Explain why both are right.

Answers

12 + (-12) = 0. The rise is 12 and the drop is -12, a zero pair, so the kite ends at the height it started from.
18 + (-18) = 0. Earning is 18 and spending is -18, so Ines has the same money as before.
Only (a), since 40 + (-40) = 0. In (b) the loss of 6 cancels only 6 of the 9 points, so 9 + (-6) = 3. In (c) both changes go forward, so 3 + 3 = 6; there is no opposite.
Many answers. For example, with money held as positive, a deposit of 30 dollars followed by a withdrawal of 30 dollars; or, with up as positive, a climb of 30 meters followed by a descent of 30 meters. Any pair of the same size on opposite sides of the same zero is right.
4. The 6 negatives make 6 zero pairs with 6 of the positives, leaving 4 positives: 10 + (-6) = 4.
-7. The 4 positives make 4 zero pairs, leaving 7 negatives: 4 + (-11) = -7.
5 zero pairs, and the value is 7. Each of the 5 negatives pairs with a positive, and 12 - 5 = 7 positives are left: 12 + (-5) = 7.
6 positive and 2 negative counters, 8 in all. Each negative needs a positive to cancel it, and then 4 more positives give the value: 6 + (-2) = 4.
8 negative counters. Five of them pair with the 5 positives, and 3 more are needed to leave the value -3: 5 + (-8) = -3.
-7. 19 and -19 are a zero pair, so only -7 is left.
-5. Strike out the zero pairs -31 + 31 and 12 + (-12); only -5 is left.
0. \displaystyle \frac{3}{4} and \displaystyle -\frac{3}{4} are a zero pair, and so are -2 and 2.
75 dollars. The deposit and withdrawal of 30 cancel, and so do the withdrawal and deposit of 12: 75 + 30 + (-12) + (-30) + 12 = 75.
x = -27. The only number that makes 0 with 27 is its opposite.
y = 50. The partner of -50 is its opposite.
w = -2.8. The partner of 2.8 is its opposite.
x = -6. The part -x makes 0 with -6, so -x = 6, and the only number whose opposite is 6 is -6. Check: -(-6) + (-6) = 6 + (-6) = 0.
43. The integers from -20 to 20 make 20 zero pairs and a 0, so they add to 0. Only 21 and 22 have no partner, and 21 + 22 = 43.
13. The first number makes 0 with 13, so it is -13. The second makes 0 with -13, so it is the opposite of -13, which is 13.
Each positive can be paired with its own negative, and with equal numbers every counter is in a pair. Each pair is worth 1 + (-1) = 0, and there is nothing left over, so the value is 0.
Yes. Pair the counters off. If there were more positives, some positives would be left over and the value would be positive; if there were more negatives, some negatives would be left over and the value would be negative. A value of 0 leaves nothing over, so the numbers are equal.
No: a + a = 0 only when a = 0. If a + a = 0, then a is a partner of a, so a = -a. A number other than 0 and its opposite are on different sides of 0, so they are never equal.
No. The two new counters are a zero pair, worth 1 + (-1) = 0, and adding 0 to the value changes nothing. The number of counters is not the value.
By the zero pair property applied to the number -a, the sum (-a) + \big(-(-a)\big) is 0, so -(-a) is a partner of -a. But (-a) + a = 0 as well, so a is also a partner of -a. A number has only one partner, so -(-a) = a.
The changes are 5 and -5, a zero pair, so the total change is 5 + (-5) = 0: the lift ends on the floor where it started. The distance traveled ignores direction and counts both journeys, 5 + 5 = 10 floors. The two people are answering different questions.