Math 7 · 7.NS.A.1 · 7.NS.A.1C

Subtracting integers

How do you take away a number you do not have, such as subtracting -3 from 4?

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Adding and subtracting integers2

Adding integers

Subtracting integers

Why this exists

Among the whole numbers (0, 1, 2, 3 and so on), 3 - 5 has no answer: no whole number added to 5 gives 3. The Integers were made so that it does. But they bring a new puzzle. Read as “take away”, 4 - 2 is easy: four things, take two. Now try 4 - (-3). It asks you to take three negatives away from a collection that has no negatives in it. And -8 - 3 asks you to take three positives from a collection of negatives only.

This lesson shows that every subtraction of integers can be turned into an addition, and the addition is one you already know how to do from adding integers. Then it puts subtraction to two uses: finding how far apart two numbers sit on the number line, and finding changes in height and money when some of the values are below zero. Changes in temperature have a lesson of their own, Temperatures below zero and temperature change.

Definition

In plain words: subtracting a number gives the same result as adding its opposite.

a - b = a + (-b)

This holds for all integers a and b. The opposite of b, written -b, is the number the same distance from zero on the other side (Opposites and zero pairs). The opposite of a negative number is positive, -(-b) = b, so subtracting a negative number is the same as adding a positive one:

a - (-b) = a + b

The rule rests on what subtraction means. a - b is the number that, added to b, gives a (Subtraction as an unknown addend). That meaning was true for whole numbers and stays true, word for word, for integers. “Take away” is one picture of it, and it still works with integers once zero pairs are allowed, as the next section shows.

Two words go with it. The difference of a and b is a - b, in that order; it can be positive, negative or zero. The distance between two numbers on a number line is how many units apart they are, counted with no direction; it is never negative.

Intuition

Two pictures give the same answer, and each explains the rule in its own way.

Counters. An open counter stands for +1 and a filled counter for -1. One of each makes a zero pair, worth 0, so putting zero pairs into a collection never changes its value. To find 4 - (-3), start with four open counters. There are no negatives to take away, so put in three zero pairs: the collection is still worth 4, and now it has three negatives. Take them away. Seven open counters are left.

Figure 14 - (-3) with counters. Three zero pairs go in, three negatives come out, and seven positives are left.

Look at what happened. Removing the three negatives left their three partners behind, so the collection gained three positives. Taking away -3 did exactly what adding 3 does:

4 - (-3) = 4 + 3 = 7

The number line. Read a - b as the question “what do I add to b to reach a?” To get from -3 to 4 you move 7 to the right, so you add 7, and 4 - (-3) = 7.

Figure 24 - (-3) on a number line. To get from -3 to 4 you add 7.

The arrow always starts at the number being subtracted and ends at the number it is subtracted from. If the arrow points right, the difference is positive; if it points left, the difference is negative. Swapping the two numbers reverses the arrow, so -3 - 4 = -7: same length, opposite direction.

Results

Subtraction is adding the opposite. For all integers a and b,

a - b = a + (-b)

Subtracting a negative is adding a positive.

a - (-b) = a + b

Swapping the order gives the opposite.

b - a = -(a - b)

Distance. The distance between a and b on a number line is the absolute value of their difference, in either order:

|a - b| = |b - a|

Change. When a quantity goes from a starting value to an ending value, the change is the ending value minus the starting value. A positive change is a rise, a negative change is a fall.

Why it is true

Subtraction means: a - b is the number that, added to b, gives a. So the job is to show that a + (-b) is that number, and that no other number is.

Add b to a + (-b) and see what comes out:

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

The first line regroups, which the associative property of addition allows. The second uses the fact that a number and its opposite add to zero. The third uses the fact that adding zero changes nothing. So a + (-b), added to b, gives a.

No other number does. Suppose c + b = a. Add -b to both sides: the left becomes c + b + (-b) = c + 0 = c and the right becomes a + (-b). So c = a + (-b), and there is only one answer. That proves a - b = a + (-b).

The other results follow from this one.

A negative. The opposite of -b is b, so a - (-b) = a + b.
Swapping. The sum (a - b) + (b - a) is a + (-b) + b + (-a). Reorder and regroup (the commutative and associative properties of addition) to get \big(a + (-a)\big) + \big(b + (-b)\big), which is 0 + 0 = 0. Two numbers that add to zero are opposites, so b - a = -(a - b).
Distance. If a is to the right of b, you reach a from b by adding a positive amount, and a - b is that amount: it is the distance. If a is to the left of b, then a - b is negative, the walk from b to a goes left, and the distance is the opposite, b - a. Either way the distance is |a - b|, the absolute value of the difference. If a = b, both are 0.

Worked examples

The lesson is taught in two parts. Examples 1 to 5 belong to the first: turning a subtraction into an addition, checking the answer, and working with several signs at once. Examples 6 to 10 belong to the second: the distance between two numbers on a number line, and changes in money and height.

Example 1

Four subtractions with 8 and 3

Find each difference: (a) 8 - 3, (b) -8 - (-3), (c) -8 - 3, (d) 8 - (-3).

Solution

(a) Eight positive counters, take away three positives: five are left, so 8 - 3 = 5.

(b) Eight negative counters, take away three negatives: five negatives are left, so -8 - (-3) = -5. Parts (a) and (b) mirror each other, and in both the counters to be taken away are already there.

(c) Eight negatives, and three positives to take away, but there are none. Put in three zero pairs, which leaves the value at -8. Take away the three positives. Their three negative partners stay behind, so there are now eleven negatives: -8 - 3 = -11. Adding the opposite gives the same thing directly:

-8 - 3 = -8 + (-3) = -11

(d) Eight positives, and three negatives to take away. Put in three zero pairs, then take the three negatives away. Eleven positives are left:

8 - (-3) = 8 + 3 = 11

In (c) and (d) the counters model needed zero pairs, and the rule did the same work in one line.

Example 2

Rewriting each subtraction as an addition

Find (a) 14 - 23, (b) -16 - 9, (c) 12 - (-17), (d) -21 - (-6).

Solution

In each part, keep the first number, change the subtraction to addition, and replace the second number by its opposite. Then add, using the rules for adding integers.

(a) 14 - 23 = 14 + (-23). The signs differ, so the two partly cancel: 23 - 14 = 9 is left over, and it carries the sign of -23, the number farther from zero. So 14 - 23 = -9.

(b) -16 - 9 = -16 + (-9). Both are negative, so their distances from zero add: 16 + 9 = 25, and the sum is negative. So -16 - 9 = -25.

(d) -21 - (-6) = -21 + 6. The signs differ: 21 - 6 = 15 is left over, with the sign of -21. So -21 - (-6) = -15.

Example 3

Checking a difference by adding back

Find -9 - (-15), and check the answer.

Solution

Subtracting -15 is adding 15:

-9 - (-15) = -9 + 15 = 6

Check. By the meaning of subtraction, the difference added to the number subtracted must give back the first number. Here 6 + (-15) = -9, which is the first number, so the answer is right. On a number line, -9 sits 6 units to the right of -15, which says the same thing.

Example 4

A string of additions and subtractions

Simplify -7 - 12 + 5 - (-9) - 3.

Solution

Subtraction does not let you regroup or reorder freely (see Boundaries), so there are two safe ways in: work strictly from left to right, or first rewrite every subtraction as an addition, after which the commutative and associative properties of addition let you add in any order. Take the second way:

-7 + (-12) + 5 + 9 + (-3)

Reorder so the negatives sit together (commutative property) and group them (associative property). The negatives give -7 + (-12) + (-3) = -22 and the positives give 5 + 9 = 14. Then

-22 + 14 = -8

Check. Left to right: -7 - 12 = -19, then -19 + 5 = -14, then -14 - (-9) = -5, then -5 - 3 = -8. Both ways give -8.

Example 5

Evaluating a difference of two letters

Evaluate x - y and y - x when x = -4 and y = -13.

Solution

Put each value in with its own brackets, so the two minus signs stay apart:

x - y = -4 - (-13) = -4 + 13 = 9

y - x = -13 - (-4) = -13 + 4 = -9

The two answers are opposites, as the result b - a = -(a - b) says they must be. On the number line, -4 is 9 units to the right of -13, so going from y to x is +9 and going back is -9.

Example 6

The distance between a negative and a positive number

Find the distance between -5 and 4 on a number line.

Solution

Figure 3The distance between -5 and 4 is 5 + 4 = 9, and 4 - (-5) = 9.

Counting: from -5 to 0 is 5 units and from 0 to 4 is 4 more, so the distance is 5 + 4 = 9. When the two numbers lie on opposite sides of zero, their distances from zero add.

The difference gives the same number without counting:

4 - (-5) = 4 + 5 = 9

Subtracting the other way round gives -5 - 4 = -5 + (-4) = -9, the opposite. The distance is the absolute value of either one: |9| = |-9| = 9.

Example 7

The distance between two negative numbers

Find the distance between -17 and -5.

Solution

Both numbers lie to the left of zero, 17 and 5 units away, so they are 17 - 5 = 12 units apart. The difference agrees:

-5 - (-17) = -5 + 17 = 12

In the other order, -17 - (-5) = -17 + 5 = -12, and |-12| = 12. Here the distances from zero are subtracted, not added, because the two numbers are on the same side of zero. The formula |a - b| handles both cases without the reader having to decide which one applies.

Example 8

A bank balance that rises and falls

On Monday Kofi’s bank account was overdrawn, with a balance of -30 dollars: he owed the bank 30 dollars. By Friday the balance was 45 dollars. (a) What was the change from Monday to Friday? (b) Over the weekend he paid out 60 dollars. What was the balance then? (c) What was the change from Monday to the end of the weekend?

Solution

(a) A change is the ending value minus the starting value:

45 - (-30) = 45 + 30 = 75

The change is positive, so the balance rose 75 dollars.

(b) Paying out 60 dollars means subtracting 60:

45 - 60 = 45 + (-60) = -15

The balance was -15 dollars, overdrawn again.

-15 - (-30) = -15 + 30 = 15

The change is positive: the balance ended 15 dollars higher than on Monday, even though it was still below zero.

Check. A rise of 75 followed by a fall of 60 is a total change of 75 + (-60) = 15, which matches (c).

Example 9

Above and below sea level

A diver is 35 meters below sea level, and a gull flies 12 meters above the water directly over her. (a) Write both heights as integers, taking sea level as 0. (b) How far apart are they? (c) The diver rises 20 meters. What is her height now, and how far is she from the gull?

Solution

(a) Heights above sea level are positive and heights below are negative, so the diver is at -35 and the gull at 12.

(b) The distance between them is the difference of their heights:

12 - (-35) = 12 + 35 = 47

They are 47 meters apart.

12 - (-15) = 12 + 15 = 27

Check. The diver closed the gap by 20 meters, and 47 - 20 = 27. The answers agree.

Example 10

Working back to a starting value

A caver climbs down a rope 7 meters and is then at -18 meters, that is, 18 meters below the cave mouth. At what height was she before the climb?

Solution

Call her starting height s, measured from the cave mouth. Climbing down 7 meters adds -7, so

s + (-7) = -18

By the meaning of subtraction, the number that added to -7 gives -18 is -18 - (-7):

s = -18 - (-7) = -18 + 7 = -11

She was at -11 meters, 11 meters below the cave mouth.

Check. -11 - 7 = -11 + (-7) = -18. Subtraction undid the addition, which is exactly the relationship that defines it.

Boundaries

Order matters. Subtraction is not commutative: a - b and b - a are opposites, and they are equal only when a = b (both are then 0). It is not associative either: (10 - 6) - 3 = 1 but 10 - (6 - 3) = 7. That is why a string of subtractions is worked left to right, or rewritten as additions first. Once every step is an addition, reordering and regrouping are allowed.

The rewrite touches only the number being subtracted. In a - b = a + (-b) the first number a keeps its sign. So -5 - 3 becomes -5 + (-3), and the -5 is left alone.

One sign, two jobs. Between two numbers, a minus sign means “subtract”. In front of a single number, it means “the opposite of”. The rule a - b = a + (-b) trades the first job for the second.

Difference and distance are different things. A difference has an order and a sign. A distance has neither and is never negative. “How far apart” asks for a distance, |a - b|; “how much did it change” asks for a signed difference, ending minus starting.

A negative value needs an agreed zero and direction. Sea level, 0 degrees and a zero bank balance are choices, and so is which way counts as positive. Two values can be subtracted only when they are measured from the same zero in the same units: a change in temperature compares two readings on the same scale.

Which numbers. Among the whole numbers, subtraction can fail, as 3 - 5 shows. Among the integers it never does: the difference of two integers is always an integer, because it is a sum of two integers. The rule a - b = a + (-b) holds just as well for fractions, decimals and every real number; this lesson keeps to integers.

Where this goes next

Because every difference of integers is again an integer, the integers are closed under subtraction while the whole numbers are not. That idea, a set that an operation never leads out of, is closure, and subtraction is the first operation where the difference between two sets of numbers shows up.

The same rule carries fractions and decimals below zero straight through, with nothing new to learn:

\tfrac{1}{2} - \left(-\tfrac{3}{4}\right) = \tfrac{1}{2} + \tfrac{3}{4}

In algebra it is used constantly. Subtracting an expression in brackets means adding the opposite of every term inside, and x - (-y) = x + y is used without comment once letters stand for numbers of either sign.

The distance |a - b| grows into the distance between two points in the coordinate plane: two points on the same horizontal line are |x_2 - x_1| apart, and the general distance formula is built from two such differences. Temperatures, heights, scores and bank balances keep appearing as numbers placed on a line, and change as ending minus starting is the first step toward rates of change.

Exercises

Part 1: subtracting one integer from another

Find each difference.

9 - 15
-11 - 7
6 - (-13)
-20 - (-8)
-4 - (-19)
0 - (-25)
38 - 52
-27 - 46

Part 1: several steps and letters

Simplify -3 - 8 - (-5).
Simplify 15 - (-6) - 30 + (-2).
Evaluate a - b when a = -7 and b = 12.
Evaluate x - y - z when x = 5, y = -8 and z = 14.
Find n when n - (-6) = -2.

Part 2: distance on a number line

Find the distance between -7 and 6.
Find the distance between -15 and -4.
A number is 10 units from -3 on a number line. What can the number be?

Part 2: problems in words

In a board game, Ama had 4 points at the start of a round and -15 points at the end of it. What was the change in her score during the round?
A miner works at -120 meters, that is, 120 meters below the surface. She climbs 45 meters, then goes down 70 meters. What is her height now?
A walk starts beside a lake whose shore is 86 meters below sea level and ends on a ridge 1{,}240 meters above sea level. What is the change in height from start to finish?
A bank account is overdrawn, with a balance of -35 dollars. A deposit of 120 dollars goes in, then a payment of 64 dollars comes out. What is the balance now?
In a quiz game where wrong answers lose points, Ben ends with 25 points and Priya with -40 points. How many more points does Ben have than Priya?
After a deposit of 13 dollars, a bank balance is -2 dollars. What was it before the deposit?

Reasoning

Show that a - b and b - a are opposites for all integers a and b.
Using counters, explain why subtracting -4 from a number gives the same result as adding 4.
Is the distance between a and b always equal to a - b? Say when it is and when it is not.
True or false: subtracting a negative integer from a number always gives a result greater than the number. Explain.
Show that (a - b) - c = a - (b + c) for all integers a, b and c, and test it with a = 9, b = 4 and c = 2.
Explain why 3 - 5 has no answer among the whole numbers but does have one among the integers.

Answers

-6. 9 - 15 = 9 + (-15); the signs differ, 15 - 9 = 6 is left over, and -15 is farther from zero.
-18. -11 - 7 = -11 + (-7) = -18.
19. 6 - (-13) = 6 + 13 = 19.
-12. -20 - (-8) = -20 + 8 = -12.
15. -4 - (-19) = -4 + 19 = 15.
25. 0 - (-25) = 0 + 25 = 25.
-14. 38 - 52 = 38 + (-52); 52 - 38 = 14 is left over, and -52 is farther from zero.
-73. -27 - 46 = -27 + (-46) = -73.
-6. -3 + (-8) + 5 = -11 + 5 = -6.
-11. Rewritten as additions it is 15 + 6 + (-30) + (-2), and that is 21 + (-32) = -11.
-19. a - b = -7 - 12 = -7 + (-12) = -19.
-1. x - y - z = 5 - (-8) - 14, which is 5 + 8 + (-14) = 13 + (-14) = -1.
n = -8. Subtracting -6 is adding 6, so n + 6 = -2 and n = -2 - 6 = -8. Check: -8 - (-6) = -8 + 6 = -2.
13. 6 - (-7) = 6 + 7 = 13. The numbers are on opposite sides of zero, 7 and 6 units away.
11. -4 - (-15) = -4 + 15 = 11. Both are left of zero, 15 and 4 units away, and 15 - 4 = 11.
7 or -13. Ten to the right is -3 + 10 = 7; ten to the left is -3 - 10 = -13. Check: 7 - (-3) = 10 and -3 - (-13) = 10.
-19 points: her score fell 19 points. Ending minus starting is -15 - 4 = -15 + (-4) = -19.
-145 meters, that is, 145 meters below the surface. -120 + 45 - 70 = -75 + (-70) = -145.
A rise of 1{,}326 meters. The heights are -86 and 1{,}240, and ending minus starting is 1{,}240 - (-86), which is 1{,}240 + 86 = 1{,}326.
21 dollars. -35 + 120 - 64 = 85 - 64 = 21.
65 points. 25 - (-40) = 25 + 40 = 65.
-15 dollars, an overdraft of 15 dollars. The starting balance s satisfies s + 13 = -2, so s = -2 - 13 = -15. Check: -15 + 13 = -2.
Add them: the sum (a - b) + (b - a) is a + (-b) + b + (-a). Reordering and regrouping (the commutative and associative properties of addition) give \big(a + (-a)\big) + \big(b + (-b)\big) = 0 + 0 = 0. Two numbers that add to zero are opposites.
To take four negatives from any collection, first put in four zero pairs; that adds 0, so the value is unchanged. Then remove the four negatives. Their four positive partners stay behind, so the collection ends up with four more positives than it started with, which is adding 4.
Only when a \ge b, that is, when a is at or to the right of b. Then a - b is zero or positive and equals the distance. When a < b, a - b is negative and the distance is its opposite, b - a. For example, with a = 2 and b = 7, a - b = -5 but the distance is 5. In every case the distance is |a - b|.
True. If n is a positive integer, a - (-n) = a + n, and adding a positive number moves to the right on the number line, so the result is greater than a. For example, -10 - (-3) = -7, and -7 > -10.
(a - b) - c = a + (-b) + (-c). Also a - (b + c) = a + \big(-(b + c)\big), and the opposite of b + c is (-b) + (-c), since (b + c) + (-b) + (-c) = 0. So both sides equal a + (-b) + (-c). Test: (9 - 4) - 2 = 5 - 2 = 3 and 9 - (4 + 2) = 9 - 6 = 3.
The whole numbers are 0, 1, 2, 3 and so on. 3 - 5 asks for a number that, added to 5, gives 3, but adding a whole number to 5 always gives 5 or more. Among the integers, -2 + 5 = 3, so 3 - 5 = -2.