Rational numbers

Rational numbers are integers, fractions and decimals, positive or negative, like −3, −½ and 2.75. The integer sign rules work for all of them.

The idea: filling the gaps between integers

Between −1 and 0 there are no integers, but there are lots of numbers: −0.5, −¼, −0.9 and more. These are negative fractions and decimals.

A rational number is any number that can be written as one integer over another (not zero). So −3 = −3/1, −0.5 = −1/2 and 2.75 = 11/4 are all rational.

See it: pictures to draw with your child

Copy these onto paper or a whiteboard. A quick sketch makes the idea concrete before any written method.

−3−2−10123−2.5−¾1.5
Rational numbers between the integers: −2.5, −¾ and 1.5.
−1−0.9−0.8−0.7−0.6−0.5−0.4−0.3−0.2−0.10−0.8−0.3
Zoom in between −1 and 0: −0.3 is further right, so −0.3 > −0.8.
−2.5 + 1.2different signs: 2.5 − 1.2 = 1.3bigger size is negative−1.3
The integer rule for different signs works for decimals too.

Rational numbers on the number line

−2.5 sits halfway between −3 and −2. −¾ sits three quarters of the way from 0 to −1.

The mirror rule still works: −2.5 is the opposite of 2.5, the same distance from 0 on the other side.

See decimals on a number line and fractions on a number line for the positive side.

Comparing negative fractions and decimals

The rule is the same as for integers: further right is greater. So −0.3 > −0.8, because −0.3 is closer to 0.

For fractions, change them to decimals or to a common denominator. −1/2 = −0.5 and −3/4 = −0.75, so −1/2 > −3/4.

Adding, subtracting, multiplying and dividing

Use the same sign rules as integers. −2.5 + 1.2: different signs, so 2.5 − 1.2 = 1.3, and the bigger size is negative: −1.3.

−0.5 × 6 = −3, because the signs are different. −1/2 ÷ (−1/4) = 2, because the signs are the same.

The integer rules page has the full chart.

Every rational number is a decimal that ends or repeats

Dividing turns a fraction into a decimal. −3/8 = −0.375 ends, and −1/3 = −0.333… repeats.

That is a test for rational numbers. Numbers like π, whose decimals never end or repeat, are not rational.

Key words

How the number families fit together

Counting numbers (1, 2, 3, …) sit inside the whole numbers (0, 1, 2, …). Whole numbers sit inside the integers (…, −1, 0, 1, …). Integers sit inside the rational numbers.

So 5 is a counting number, a whole number, an integer and a rational number. −5 is an integer and rational, but not a whole number. −5.5 is rational, but not an integer.

A Venn diagram of nested circles shows this well. Draw it with your child and place a few numbers in the right ring.

A tip for negative fractions

Put the minus sign in front of the whole fraction: −3/4. This is the same as (−3)/4 and 3/(−4), but it is clearer to read.

When adding negative fractions, find a common denominator first, then use the integer rules on the numerators.

Money is the best model

Amounts of money with cents are rational numbers. Owing $2.50 is −2.50, and earning $1.25 is +1.25.

Ask your child to track a pretend balance through a few spends and earnings. The sign rules for decimals feel natural when the numbers are dollars and cents.

Quiz: rational numbers

Add, multiply and compare negative decimals and fractions.

Where this fits: integers step by step

  1. What are integers
  2. Integers on a number line
  3. Opposite numbers
  4. Absolute value
  5. Comparing integers
  6. Negative numbers in real life
  7. Zero pairs
  8. Adding integers
  9. Subtracting integers
  10. Distance between integers
  11. Multiplying integers
  12. Dividing integers
  13. Integer rules
  14. Order of operations with integers
  15. The coordinate plane
  16. Rational numbers (this page)
  17. Integer word problems

What to say to your child

A short script for teaching it the first time. Pause after each line and let your child do the next step.

  1. “Is there an integer between −1 and 0?” (No.)
  2. “Is there any number between them?” (Yes, like −0.5.)
  3. “Which is greater, −0.3 or −0.8?” (−0.3, closer to 0.)
  4. “What is −0.5 × 4?” (−2.)
  5. “Same rules as integers?” (Yes.)

More worked examples

Example 1: compare

−2/3 and −0.7: −2/3 ≈ −0.67, which is further right, so −2/3 > −0.7.

Example 2: add

−1.6 + (−2.3) = −3.9.

Example 3: subtract

1.5 − 4.25 = 1.5 + (−4.25) = −2.75.

Example 4: divide

−3/4 ÷ 3 = −1/4.

Practice questions (answers below)

Ask these out loud or write them down. Check each answer, and ask “how did you get that?” for one or two.

  1. −0.4 + 1
  2. −0.4 − 1
  3. −2.5 × 2
  4. −1.2 × (−3)
  5. −4.8 ÷ 4
  6. Which is greater, −1/3 or −1/2?
  7. Write −0.25 as a fraction
  8. Write −5/8 as a decimal
  9. Is −7 rational?
  10. Stretch: −1/2 + 3/4 − 1
Show the answers
  1. 0.6
  2. −1.4
  3. −5
  4. 3.6
  5. −1.2
  6. −1/3
  7. −1/4
  8. −0.625
  9. Yes, −7/1
  10. −3/4

Common mistakes

Comparing digitsSays −0.8 > −0.3 because 8 > 3.Fix: for negatives, closer to 0 is greater: −0.3 > −0.8.
Forgetting the sign rulesWrites −0.5 × 4 = 2.Fix: different signs give a negative: −2.
Thinking only integers can be negativeDoesn’t place −½ on the line.Fix: every fraction and decimal has a negative opposite.

How to teach it at home

Money with cents

Use pretend money: owe $2.50, then earn $1.25. What is the balance?

Ruler number line

Mark 0 in the middle of a ruler and place halves and quarters on both sides.

Weather decimals

Some weather apps show −0.5°C. Order a week of readings.

By grade

More for each grade: Grade 7 math and the CBSE class 7 maths guide.

Worksheets and practice

Integers worksheet, page 1

Integers worksheet: printable PDF with an answer key. · Make a fresh test

Integer practice · integer calculator · integer number line · integers for kids · ratios

Last reviewed

Questions parents ask

What is a rational number?

Any number that can be written as one integer over another non-zero integer, like −3/4, 0.5 or −7.

Are integers rational numbers?

Yes. Every integer can be written over 1, so −5 = −5/1.

How do you compare negative fractions?

Change them to decimals or a common denominator. The one closer to 0 is greater.

Do integer rules work for decimals and fractions?

Yes. The same sign rules apply to all rational numbers.

What grade learns rational numbers?

Grades 6 and 7 under Common Core 6.NS.C and 7.NS.A. In India, class 7 and 8.