Why does subtracting a negative number make the answer bigger?

Because subtracting means moving in the opposite direction from the number you subtract. Subtracting 5 moves you 5 units left, so subtracting -5 moves you 5 units right. The rule a - b = a + (-b) captures this: the opposite of a negative is positive, so 3 - (-5) = 3 + 5 = 8. Real check: if the temperature is 3 degrees and you find it is 5 degrees warmer than yesterday's -5, that fits.

Is |a - b| the same as |a| - |b|?

No, and assuming they are equal is one of the most common errors in this lesson. Try a = -7 and b = 4: |-7 - 4| = 11, but |-7| - |4| = 3. Absolute value bars act like parentheses, so finish all the arithmetic inside them before taking the absolute value.

When should my answer be negative and when should it be positive?

Look at what the question asks. "What was the change?" or "Find the difference" wants a signed answer, so a decrease stays negative. "How far apart?", "How much did it drop?", or "What is the distance?" wants a positive number, because the words already say the direction. Writing "negative 27 degrees colder" double-counts the direction.

Do I have to rewrite subtraction as addition every single time?

Not forever, but do it in writing while you are learning. The rewriting step is what prevents sign errors, especially with double negatives and mixed numbers. Later, when you work with the order of operations and with expressions containing variables, being fluent at turning every subtraction into addition of the opposite makes long expressions much easier to reorder and simplify.

M7MATH-2.2

Subtracting Rational Numbers

Learn to rewrite any subtraction as adding the opposite, a − b = a + (−b), work with integers, fractions, and decimals, and find distance using absolute value.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Subtracting Rational Numbers, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Subtracting Rational Numbers free

What this lesson covers

Subtraction used to mean "take away," and that worked fine when you started with the bigger number. But what is

3 - 8

? Or

-4 - (-9)

? Once negative numbers enter the picture, "take away" stops being a reliable guide, and you need a rule that always works.

That rule is short: subtracting is the same as adding the opposite. In symbols,

a - b = a + (-b)

. Every subtraction problem in this lesson — integers, fractions, decimals, mixed signs — becomes an addition problem you already know how to do. Along the way you'll see why the distance between two numbers on the number line is

|a - b|

, which is the reason weather reports can say the temperature "dropped 27 degrees" even when it crossed zero.

The Rule: Subtracting Is Adding the Opposite

The opposite (or additive inverse) of a number is the number the same distance from zero on the other side. The opposite of

8

-8

; the opposite of

-\frac{2}{3}

\frac{2}{3}

; the opposite of

0

0

.

The rule

a - b = a + (-b)

says: keep the first number exactly as it is, change the subtraction sign to addition, and replace the second number with its opposite. Two things change, never one.

Original	Rewritten as addition	Result
$3 - 8$	$3 + (-8)$	$-5$
$-4 - 9$	$-4 + (-9)$	$-13$
$-4 - (-9)$	$-4 + 9$	$5$
$6.5 - (-2.1)$	$6.5 + 2.1$	$8.6$

Notice row 3 and row 4: subtracting a negative turns into adding a positive, so the answer gets larger. This surprises a lot of students, but the number line makes it sensible. Subtracting means moving in the opposite direction from the number you subtract. Subtracting

-9

means moving 9 units in the opposite direction from "9 units left," which is 9 units right.

The most common error is changing only the operation and forgetting to flip the sign of the second number — writing

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

. Slow down and rewrite the whole expression on paper before computing. Once it says

-4 + 9

, you are just adding rational numbers, which you already practiced.

One more caution: subtraction is not commutative.

3 - 8 = -5

but

8 - 3 = 5

. Addition lets you reorder; subtraction does not, so rewrite first, then reorder if you want.

Integers, Fractions, and Decimals — Same Rule, Different Bookkeeping

The rewriting step is identical for every kind of rational number. What changes is the arithmetic that follows.

With integers, after rewriting, apply your addition rules: same signs, add the absolute values and keep the sign; different signs, subtract the smaller absolute value from the larger and keep the sign of the number farther from zero. So

-15 - 6 = -15 + (-6) = -21

, and

12 - 20 = 12 + (-20) = -8

.

With fractions, rewrite first, then find a common denominator. For

\frac{1}{4} - \frac{5}{6}

, rewrite as

\frac{1}{4} + \left(-\frac{5}{6}\right)

, convert to twelfths:

\frac{3}{12} + \left(-\frac{10}{12}\right) = -\frac{7}{12}

. A negative fraction can carry its sign in three equivalent places:

-\frac{5}{6} = \frac{-5}{6} = \frac{5}{-6}

. Putting the sign on the numerator is usually easiest to track.

Mixed numbers deserve extra care. For

-2\frac{1}{2} - 1\frac{3}{4}

, remember that

-2\frac{1}{2}

means

-\left(2 + \frac{1}{2}\right) = -\frac{5}{2}

, not

-2 + \frac{1}{2}

. Rewriting:

-\frac{5}{2} + \left(-\frac{7}{4}\right) = -\frac{10}{4} - \frac{7}{4}

... written as addition,

-\frac{10}{4} + \left(-\frac{7}{4}\right) = -\frac{17}{4} = -4\frac{1}{4}

.

With decimals, line up the decimal points after rewriting. For

-3.4 - 1.75

, you get

-3.4 + (-1.75)

; both are negative, so add

3.4 + 1.75 = 5.15

and keep the negative:

-5.15

. Writing

3.40

instead of

3.4

prevents place-value slips.

Distance as the Absolute Value of a Difference

On a number line, the distance between two points is always positive (or zero), but a difference can be negative. Absolute value bridges the gap:

\text{distance between } a \text{ and } b = |a - b| = |b - a|

Both orders give the same distance because

a - b

and

b - a

are opposites, and opposites have the same absolute value. That is worth remembering: if you subtract in the "wrong" order, the absolute value fixes it.

Find the distance between

-7

and

4

. Compute

|-7 - 4| = |-7 + (-4)| = |-11| = 11

. Check the other order:

|4 - (-7)| = |4 + 7| = |11| = 11

. Same answer, and you can count 11 units on a number line to confirm.

A frequent mistake is computing

|-7| - |4| = 7 - 4 = 3

. Absolute value is not distributive over subtraction:

|a - b| \neq |a| - |b|

. Do all the arithmetic inside the bars first, then take the absolute value. The bars work like parentheses that way.

When both numbers are on the same side of zero, the distance is the difference of their distances from zero: from

-9

-2

|-9 - (-2)| = |-9 + 2| = 7

. When they straddle zero, the distance is the sum of their distances from zero: from

-9

2

|-9 - 2| = 11 = 9 + 2

. Sketching a quick number line and checking which case you are in is a fast way to catch a sign error before you finish.

Reading Real Situations as Subtraction

Word problems rarely say "subtract." They describe a change, a drop, a gap, or a difference, and you decide what to compute.

Phrase in the problem	What to write	Why
"How much did it change?"	final $-$ initial	A negative result means a decrease
"How far apart are they?"	$\lvert a - b \rvert$	Distance is never negative
"How much colder?"	$\lvert a - b \rvert$ , then say "colder"	The word already carries the direction
"She owes 12 dollars and pays 5"	$-12 + 5$	A payment is added toward the debt

Change is always final minus initial, in that order. If the temperature went from 6 degrees to negative 21 degrees, the change is

-21 - 6 = -21 + (-6) = -27

degrees: a drop of 27 degrees. The negative sign is the information — it tells you the direction — so do not drop it when the question asks for the change.

But if the question asks how much the temperature dropped, or how many degrees separate the two readings, the answer is 27 degrees, positive, because the words "dropped" and "separate" already supply the direction. Answering "negative 27 degrees warmer" is a garbled answer even if the arithmetic was right.

Elevation problems work the same way. A hiker at 340 feet above sea level and a diver at 85 feet below sea level are

|340 - (-85)| = |340 + 85| = 425

feet apart vertically. Notice that the phrase "below sea level" is what makes the second number negative — translate every direction word into a sign before you compute.

Key terms

Rational number.: Any number that can be written as a fraction $\frac{a}{b}$ with integers $a$ and $b$ , $b \neq 0$ . This includes integers, fractions, mixed numbers, and terminating or repeating decimals, positive or negative.
Opposite (additive inverse).: The number the same distance from zero on the other side of the number line. The opposite of $b$ is $-b$ , and $b + (-b) = 0$ .
Absolute value.: The distance of a number from zero, written $|x|$ . It is never negative: $|-11| = 11$ and $|11| = 11$ .
Difference.: The result of a subtraction. The difference $a - b$ can be positive, negative, or zero, unlike a distance.
Distance between two numbers.: How far apart two points are on the number line, computed as $|a - b|$ , which equals $|b - a|$ .
Change.: Final value minus initial value. A negative change means a decrease; a positive change means an increase.
Common denominator.: A shared denominator used to add or subtract fractions, found after rewriting the subtraction as addition of the opposite.

Worked example

At 5 p.m. the temperature on a mountain was

-4\frac{1}{2}

degrees Celsius. By midnight it had reached

-11\frac{1}{4}

degrees Celsius. (a) What was the change in temperature? (b) How many degrees apart were the two readings?

Part (a): change means final minus initial.

Write the subtraction:

-11\frac{1}{4} - \left(-4\frac{1}{2}\right)

.

Convert to improper fractions, keeping the negatives attached to the whole quantity:

-11\frac{1}{4} = -\frac{45}{4}

and

-4\frac{1}{2} = -\frac{9}{2}

.

Rewrite subtraction as adding the opposite. The opposite of

-\frac{9}{2}

+\frac{9}{2}

, so the expression becomes

-\frac{45}{4} + \frac{9}{2}

.

Use a common denominator of 4:

\frac{9}{2} = \frac{18}{4}

, giving

-\frac{45}{4} + \frac{18}{4}

.

Different signs, so subtract absolute values and keep the sign of the number farther from zero:

45 - 18 = 27

, and

-\frac{45}{4}

is farther from zero, so the result is

-\frac{27}{4} = -6\frac{3}{4}

.

The change was

-6\frac{3}{4}

degrees Celsius — a drop of

6\frac{3}{4}

degrees. The negative sign is correct because the temperature fell.

Part (b): distance is the absolute value of the difference.

\left|-11\frac{1}{4} - \left(-4\frac{1}{2}\right)\right| = \left|-\frac{27}{4}\right| = \frac{27}{4} = 6\frac{3}{4}

The readings were

6\frac{3}{4}

degrees apart. Same size as part (a), but stated as a positive distance with no sign, since "apart" already tells you there is no direction involved.

Check: both temperatures are below zero, and

11\frac{1}{4} - 4\frac{1}{2} = 6\frac{3}{4}

, matching the distance found from the number line.

Practice questions

Which expression is equivalent to

-6 - (-2.5)

$-6 + (-2.5)$
$-6 + 2.5$
$6 + 2.5$
$6 - 2.5$

Answer: $-6 + 2.5$

Applying

a - b = a + (-b)

, keep the first number as

-6

, change subtraction to addition, and replace

-2.5

with its opposite,

+2.5

. That gives

-6 + 2.5 = -3.5

. The first choice changes the operation but forgets to flip the second sign, and the last two choices wrongly change the sign of the first number, which the rule never does.

A submarine is at

-260

feet relative to sea level. A second submarine is at

-95

feet. Find the vertical distance between them, and explain why your answer is positive.

Answer: 165 feet. Distance $= |-260 - (-95)| = |-260 + 95| = |-165| = 165$ feet.

Rewrite the subtraction as adding the opposite: subtracting

-95

becomes adding

95

, so

-260 + 95

. The signs differ, so subtract absolute values,

260 - 95 = 165

, and keep the negative because

-260

is farther from zero: the difference is

-165

. Taking the absolute value gives

165

. The answer is positive because distance measures how far apart two points are, not which one is deeper; subtracting in the other order gives

|-95 - (-260)| = |165| = 165

, the same number.

Evaluate

\frac{2}{5} - \frac{7}{10}

and state whether the result is greater or less than zero.

Answer: $-\frac{3}{10}$ , which is less than zero.

Rewrite as

\frac{2}{5} + \left(-\frac{7}{10}\right)

. Use tenths:

\frac{2}{5} = \frac{4}{10}

, so the sum is

\frac{4}{10} + \left(-\frac{10}{10}\right)

... more precisely

\frac{4}{10} + \left(-\frac{7}{10}\right)

. The signs differ, so subtract absolute values:

7 - 4 = 3

, and keep the negative since

-\frac{7}{10}

is farther from zero. The result is

-\frac{3}{10}

, less than zero, which makes sense because you started with a smaller positive number and subtracted a larger one.

FAQ

Why does subtracting a negative number make the answer bigger?: Because subtracting means moving in the opposite direction from the number you subtract. Subtracting $5$ moves you 5 units left, so subtracting $-5$ moves you 5 units right. The rule $a - b = a + (-b)$ captures this: the opposite of a negative is positive, so $3 - (-5) = 3 + 5 = 8$ . Real check: if the temperature is 3 degrees and you find it is 5 degrees warmer than yesterday's $-5$ , that fits.
Is $|a - b|$ the same as $|a| - |b|$ ?: No, and assuming they are equal is one of the most common errors in this lesson. Try $a = -7$ and $b = 4$ : $|-7 - 4| = 11$ , but $|-7| - |4| = 3$ . Absolute value bars act like parentheses, so finish all the arithmetic inside them before taking the absolute value.
When should my answer be negative and when should it be positive?: Look at what the question asks. "What was the change?" or "Find the difference" wants a signed answer, so a decrease stays negative. "How far apart?", "How much did it drop?", or "What is the distance?" wants a positive number, because the words already say the direction. Writing "negative 27 degrees colder" double-counts the direction.
Do I have to rewrite subtraction as addition every single time?: Not forever, but do it in writing while you are learning. The rewriting step is what prevents sign errors, especially with double negatives and mixed numbers. Later, when you work with the order of operations and with expressions containing variables, being fluent at turning every subtraction into addition of the opposite makes long expressions much easier to reorder and simplify.

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