Jugami Pre-Algebra

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Integers

Below zero, under the sea, and in the red: numbers with a direction.

Integers are the whole numbers, their opposites, and zero: … −3, −2, −1, 0, 1, 2, 3 … They let us describe temperatures below zero, places below sea level, and money we owe.

This page shows how to place integers on a number line, compare them, and add, subtract, multiply and divide them, and why the sign rules work. Then try the practice problems and the Integer Climb game.

Jump to practicePlay Integer Climb
A number line from negative 3 to 3 with arrows showing that −3 and 3 are each 3 units from 0, so they are opposites −3−2−101233 units3 units−3 and 3 are opposites

What is an integer?

An integer is a whole number with no fraction or decimal part. There are three kinds:

  • Positive integers: 1, 2, 3, … (greater than zero, to the right of 0 on the number line)
  • Negative integers: −1, −2, −3, … (less than zero, to the left of 0)
  • Zero, which is neither positive nor negative

Two numbers the same distance from 0 on opposite sides are opposites: 4 and −4 are opposites, and the opposite of 0 is 0. The distance a number is from 0 is its absolute value, written with bars: |−4| = 4 and |4| = 4. A distance is never negative, so an absolute value is never negative.

A number line from −6 to 6. The points −4 and 4 are both 4 units from 0, so the absolute value of each is 4 −6−5−4−3−2−10123456|−4| = 4|4| = 4← negativepositive →0
Opposites: −4 and 4 are both 4 units from 0, so |−4| = |4| = 4.

Example 1: Opposites and absolute value

What is the opposite of −9, and what is |−9|?

  1. −9 is 9 units to the left of 0.
  2. Its opposite is the same distance on the other side: 9.
  3. Its distance from 0 is 9, so |−9| = 9.

Comparing and ordering integers

On a number line, numbers get bigger as you move right. So any positive number is greater than any negative number, and among negative numbers, the one closer to 0 is greater. −2 is greater than −7, even though 7 is greater than 2: owing $2 is better than owing $7.

Example 2: Comparing negatives

Which is greater, −7 or −2?

  1. Picture the number line: −2 is 2 units left of 0, −7 is 7 units left of 0.
  2. −2 is farther right, so −2 > −7 (or −7 < −2).

Example 3: Ordering

Put −4, 3, 0, −1, 2 in order from least to greatest.

  1. Negatives first, farthest left first: −4, then −1.
  2. Then 0, then the positives: 2, 3.
  3. Order: −4, −1, 0, 2, 3.

Adding integers

On a number line: start at the first number. Adding a positive number moves you right; adding a negative number moves you left.

A number line from −5 to 5. Start at −3 and jump 5 units right to land on 2, showing −3 + 5 = 2 −5−4−3−2−1012345+5 (move right 5)startend−3 + 5 = 2
Start at −3 and move 5 units right: −3 + 5 = 2.

Example 4: Adding on a number line

Find −3 + 5.

  1. Start at −3.
  2. Adding +5 means move 5 units right: −2, −1, 0, 1, 2.
  3. You land on 2, so −3 + 5 = 2.

With zero pairs: use a counter for each +1 and a counter for each −1. One positive and one negative counter together make a zero pair, because 1 + (−1) = 0. Remove the zero pairs; whatever is left is the answer.

Counters: 3 positive counters above 5 negative counters. Three positive-negative pairs are circled as zero pairs, leaving 2 negative counters, so 3 + (−5) = −2 +−+−+−−−3 zero pairs = 0left: −23 + (−5) = −2
3 + (−5): three zero pairs cancel, leaving two negative counters, −2.

Example 5: Adding with zero pairs

Find 3 + (−5).

  1. Lay out 3 positive counters and 5 negative counters.
  2. Match them up: 3 zero pairs, each worth 0.
  3. 2 negative counters are left, so 3 + (−5) = −2.

The shortcut both models show

Same signs: add the absolute values and keep the sign. (−6) + (−7) = −13.
Different signs: subtract the smaller absolute value from the larger, and use the sign of the number with the larger absolute value. 3 + (−5): 5 − 3 = 2, and −5 wins, so −2.

Subtracting integers: add the opposite

Subtracting a number gives the same answer as adding its opposite: a − b = a + (−b). So 5 − 8 = 5 + (−8) = −3, and −4 − (−6) = −4 + 6 = 2.

Why? With counters, to take away −6 from −4 you first need 6 negative counters to remove. Add 6 zero pairs (that adds 0, so nothing changes), then take away the 6 negatives. You are left with the 6 positives you added, and those cancel the original −4, leaving 2: exactly what adding 6 does.

A number line from −5 to 5. Subtracting −6 is the same as adding 6: start at −4 and jump 6 units right to 2 −5−4−3−2−1012345+6startend−4 − (−6) = −4 + 6 = 2
Subtracting −6 gives the same result as adding 6: −4 − (−6) = −4 + 6 = 2.

Example 6: Subtracting a negative

Find −4 − (−6).

  1. Change subtraction to adding the opposite: −4 + 6.
  2. Different signs: 6 − 4 = 2, and 6 has the larger absolute value, so the answer is positive.
  3. −4 − (−6) = 2.

Multiplying and dividing: the sign rules

Multiply or divide the absolute values as usual, then decide the sign:

  • Same signs (both positive or both negative): the answer is positive. 6 × 4 = 24 and (−6) × (−4) = 24.
  • Different signs: the answer is negative. (−6) × 4 = −24 and 36 ÷ (−9) = −4.

Division follows the same rules because it undoes multiplication: −36 ÷ 9 = −4 because 9 × (−4) = −36. As always, you cannot divide by 0.

Example 7: Multiplying and dividing

Find (−6) × (−4) and −36 ÷ 9.

  1. 6 × 4 = 24. Both factors are negative (same signs), so (−6) × (−4) = 24.
  2. 36 ÷ 9 = 4. The signs are different, so −36 ÷ 9 = −4.
  3. Check: 9 × (−4) = −36. ✓

How do we know negative × negative is positive?

It isn’t just a rule to memorize. Here are two reasons.

1. Follow the pattern. Multiply −2 by 3, 2, 1, 0. Each time the first number goes down by 1, the answer goes up by 2. If the pattern keeps going, (−1) × (−2) = 2 and (−2) × (−2) = 4.

A pattern: 3 times −2 is −6, 2 times −2 is −4, 1 times −2 is −2, 0 times −2 is 0. Each answer goes up by 2, so −1 times −2 is 2, −2 times −2 is 4, and −3 times −2 is 6 Multiply by −2 and watch the answers3 × (−2) =−62 × (−2) =−4+21 × (−2) =−2+20 × (−2) =0+2−1 × (−2) =2+2−2 × (−2) =4+2−3 × (−2) =6+2
As the first factor drops by 1, the product rises by 2, so it keeps rising past 0 into the positives.

2. Use the distributive property. We know 3 + (−3) = 0, so (−2) × (3 + (−3)) = (−2) × 0 = 0. Distributing gives (−2) × 3 + (−2) × (−3) = 0, which is −6 + (−2) × (−3) = 0. The only number that adds to −6 to make 0 is 6, so (−2) × (−3) must equal 6.

Integers in real life

  • Temperature: −8°F means 8 degrees below zero.
  • Elevation: sea level is 0. Badwater Basin in Death Valley is 282 feet below sea level (−282 ft), and the top of Denali in Alaska is 20,310 feet above it.
  • Money: a deposit of $25 is +25, and a $40 purchase is −40. A balance of −15 means you owe $15.
A thermometer marked from −10 to 10 degrees. The temperature starts at −8°F and rises 15 degrees to 7°F −10°−5°0°5°10°6 a.m.: −8°Fnoon: 7°Frose 15°
Rising 15 degrees from −8°F: −8 + 15 = 7.

Example 8: Temperature and elevation

At 6 a.m. it was −8°F. By noon it rose 15 degrees. What was the noon temperature? And how much higher is Denali’s top (20,310 ft) than Badwater Basin (−282 ft)?

  1. Rising means adding: −8 + 15. Different signs, 15 − 8 = 7, positive: 7°F.
  2. The difference in height is a subtraction: 20,310 − (−282).
  3. Add the opposite: 20,310 + 282 = 20,592 feet.

History: red rods and black rods

Some of the oldest known rules for negative numbers come from China. The Nine Chapters on the Mathematical Art, a book of problems put together over centuries and finished by around the 1st century CE, gives rules for adding and subtracting positive and negative numbers while solving systems of equations. Calculators of the time used small counting rods on a board: red rods for positive numbers and black rods for negative numbers, with the black cancelling the red, just like zero pairs.

In India in 628 CE, the mathematician Brahmagupta wrote the Brāhmasphuṭasiddhānta, which treats zero as a number and states rules for “fortunes” (positive numbers) and “debts” (negative numbers), such as “a debt subtracted from zero is a fortune” and “the product or quotient of two debts is a fortune.” That last one is our rule that negative × negative (and negative ÷ negative) is positive.

Even so, many European mathematicians refused to accept negative numbers until the 1700s and 1800s.

Did you know?

The ancient Chinese rods used red for positive, but modern bookkeepers do the opposite: being “in the red” means owing money, a negative balance.

Practice problems

Try each one, then open it to check your answer.

1. Order from least to greatest: 1, −3, 4, −8, 0

−8, −3, 0, 1, 4. The negatives go first, the one farthest from 0 first.

2. Which is greater, |−15| or |12|?

|−15| = 15 and |12| = 12, so |−15| is greater.

3. −9 + 4

Different signs: 9 − 4 = 5, and −9 has the larger absolute value, so −5. On a number line: start at −9, move 4 right.

4. −6 + (−7)

Same signs: 6 + 7 = 13, keep the negative sign: −13. With counters, 13 negatives and no zero pairs.

5. 5 − 12

Add the opposite: 5 + (−12). 12 − 5 = 7, and −12 wins: −7.

6. −3 − (−10)

Add the opposite: −3 + 10 = 7.

7. (−7) × (−8) and (−45) ÷ 5

7 × 8 = 56 with same signs: 56. 45 ÷ 5 = 9 with different signs: −9.

8. It was 6°C at noon and −5°C at midnight. What was the change in temperature?

Change = end − start = −5 − 6 = −5 + (−6) = −11. The temperature fell 11 degrees.

Game: Integer Climb

Ten mixed problems: add, subtract, multiply and divide integers. Type a whole number like 7 or -7 and press Enter. One point per correct answer. Your best score is saved only on this device. About 3 minutes.

Talk About It

  • Is −10 greater or less than −1? Explain using a number line and using money.
  • Why does adding a zero pair never change a number? How does that help explain subtraction?
  • Can the absolute value of a number ever be negative? Why or why not?
  • Your friend says “two negatives always make a positive.” Is that true for −3 + (−4)? When is it true?

Sources

Credits: All diagrams on this page are original SVG drawings made for Jugami. Temperatures and money amounts in the examples are made up for practice.