Add and subtract rational numbers · lesson

Adding positive and negative numbers

💡 Same signs: add and keep the sign; different signs: subtract the sizes and keep the sign of the bigger one.

If both numbers have the same sign, add their sizes. The answer keeps that sign. So −3.5 + (−2.25) = −5.75.

If the signs are different, they pull against each other. Subtract the smaller size from the bigger size. The answer takes the sign of the number with the bigger size.

On a number line, adding a positive moves right. Adding a negative moves left.

🧠 Let's work it out

Add −12 + 5.

The signs are different, so subtract the sizes.
The sizes are 12 and 5, and 12 − 5 = 7.
12 is bigger, and it is negative, so the answer is −7.

What do you think comes next?

rational number: any number that can be written as a fraction, including negatives and decimals
absolute value: a number's size, or distance from 0, with no sign

🔗 Khan Academy: Adding and subtracting negative numbers · Khan Academy ↗

Subtracting means adding the opposite

💡 To subtract a number, add its opposite: minus a negative becomes plus a positive.

Every subtraction can be turned into an addition. Change the minus to plus, and change the sign of the second number.

So 5 − (−8) becomes 5 + 8 = 13. Taking away a debt makes you richer.

And −7 − 6 becomes −7 + (−6) = −13.

🧠 Let's work it out

Find −2 − (−10).

Change to adding the opposite: −2 + 10.
The signs differ, so subtract the sizes: 10 − 2 = 8.
10 is bigger and positive, so the answer is 8.

What do you think comes next?

opposite: the same number with the other sign, like 6 and −6

Negative fractions and decimals

💡 Use a common denominator for fractions or line up decimal points, then follow the same sign rules.

The sign rules don't change for fractions. First give the fractions the same bottom number. Then add or subtract the tops.

For decimals, line up the decimal points. Then use the sign rules.

🧠 Let's work it out

Find −3/5 − 1/10.

The common bottom is 10, so −3/5 = −6/10.
Subtract: −6/10 − 1/10 = −7/10.
Both pull the same way (down), so it stays negative: −7/10.

What do you think comes next?

common denominator: the same bottom number for two fractions
denominator: the bottom number of a fraction

Rising and falling

💡 Write the start as a signed number, add rises, and subtract drops.

Many stories start at a number and then change. Up, rise, and gain mean add. Down, drop, and fall mean subtract.

Start below zero? Write the start as a negative number.

🧠 Let's work it out

It is −6°F at dawn. It warms up 15 degrees. What is the new temperature?

Warming up means adding: −6 + 15.
The signs differ: 15 − 6 = 9, and 15 is bigger and positive, so it is 9°F.
Another example: a submarine at −200 feet that rises 70 feet is at −200 + 70 = −130.

What do you think comes next?