What is a rational number?
A rational number is a number you can write as p/q. Here p and q are integers (whole numbers, with their negatives). And q is not 0.
- 3/4, −5/3 and 22/7 are rational.
- Every integer is rational: 7 = 7/1, −2 = −2/1, 0 = 0/1.
- 0.25 is rational, because 0.25 = 1/4.
The word comes from ratio. A ratio compares two numbers, just like p/q does.
One rational number has many names: 1/2 = 2/4 = 50/100. The simplest name is the one where p and q have no common factor.
Rational numbers on the number line
Follow two moves: Cut and Hop.
- Cut: split every unit (0 to 1, 1 to 2, …) into q equal parts.
- Hop: start at 0. Move p parts. Go right if the number is positive, left if it is negative.
Example: 7/3. Cut into thirds. Hop 7 thirds right. You land at 2 1/3 (just after 2).
For a mixed number like 2 1/3, you can also jump to 2 first, then cut only the piece from 2 to 3 into 3 parts and take 1.
Density: a rational between any two rationals
Claim: between any two different rational numbers a and b there is another rational number.
Proof: Say a < b. Look at m = (a + b)/2, the average.
- a and b are fractions, so a + b is a fraction. Half of a fraction is a fraction. So m is rational.
- m − a = (b − a)/2, which is more than 0. So m is bigger than a.
- b − m = (b − a)/2, which is also more than 0. So m is smaller than b.
So m lies between a and b. Now use a and m, and find a new average. We can do this again and again, so there are infinitely many rational numbers between any two. This is called the density of rational numbers. It means: there are no gaps you can find by only using fractions.
Finding many rationals at once
To find 5 rationals between 3/5 and 4/5, make the denominators bigger. Multiply top and bottom by 6 (one more than 5): 3/5 = 18/30 and 4/5 = 24/30. Now 19/30, 20/30, 21/30, 22/30, 23/30 are five rationals in between.
Rule of thumb: if you need n numbers, multiply by n + 1.
With decimals it is also easy: between 0.3 and 0.4 lie 0.31, 0.32, 0.35, 0.399 and so on.
Decimal expansions: terminating or repeating
Divide p by q with long division. Only two things can happen.
- The remainder becomes 0. The decimal ends. This is a terminating decimal. Example: 3/8 = 0.375.
- The remainder never becomes 0. When dividing by q, the remainder can only be 1, 2, …, q − 1. So after at most q − 1 steps a remainder must come back. From then on the same digits come again. This is a non-terminating repeating (recurring) decimal. Example: 1/7 = 0.142857142857… written 0.142857.
Quick test: write the fraction in lowest terms. If the denominator has only 2s and 5s as prime factors (like 8 = 2×2×2, 20 = 2×2×5), the decimal ends. Otherwise it repeats.
The reverse is true too: every terminating or repeating decimal is a rational number (see the next lesson for 0.4747… = 47/99).
Try it: build your own number line
Take a strip of paper and a ruler. Mark 0 and 1 exactly 12 cm apart. Fold to find 1/2. Fold again for 1/4 and 3/4. Now use the ruler: every 4 cm is 1/3. Where do 1/3 and 1/4 sit? Which is bigger? Then check in the 3D free-play step with p = 1, q = 3 and p = 1, q = 4.
Predict, then check: before moving the sliders, guess whether 5/12 ends or repeats. (12 = 2×2×3, so it repeats.)
Key formulas and definitions
- Rational number = p/q, p and q integers, q ≠ 0
- Between a and b: (a + b)/2 is rational and lies between them
- n rationals between a/q and b/q: multiply top and bottom by (n + 1)
- Lowest-terms denominator has only 2s and 5s ⇒ decimal ends
- Otherwise ⇒ decimal repeats (at most q − 1 digits in the block)
Worked examples
1. Show 5/4 on the number line.
Step 1 (Cut): split each unit into 4 parts. Step 2 (Hop): from 0, hop 5 quarters to the right. 4 quarters reach 1, one more quarter goes past it. So 5/4 = 1 1/4, a quarter of the way from 1 to 2.
2. Show −2/3 on the number line.
Cut each unit into 3 parts. From 0 hop 2 parts to the left. The point is two-thirds of the way from 0 to −1.
3. Find one rational number between 1/3 and 1/2.
Average: (1/3 + 1/2) ÷ 2 = (2/6 + 3/6) ÷ 2 = (5/6) ÷ 2 = 5/12. Check: 1/3 = 4/12 and 1/2 = 6/12, and 5/12 is between them.
4. Find five rational numbers between 3/5 and 4/5.
We need 5, so multiply top and bottom by 6. 3/5 = 18/30, 4/5 = 24/30. Five rationals: 19/30, 20/30 (= 2/3), 21/30, 22/30, 23/30.
5. Write 7/8 as a decimal. Does it end?
70 ÷ 8 = 8 remainder 6. 60 ÷ 8 = 7 remainder 4. 40 ÷ 8 = 5 remainder 0. Stop. 7/8 = 0.875. It ends, because 8 = 2×2×2 has only 2s.
6. Write 10/3 and 1/7 as decimals.
10/3: 10 ÷ 3 = 3 remainder 1, then 10 ÷ 3 = 3 remainder 1 again and again, so 10/3 = 3.333… = 3.3̄. 1/7: remainders go 3, 2, 6, 4, 5, 1 and then repeat, so 1/7 = 0.142857142857… with block 142857.
7. Without dividing, say which end: 13/40, 7/12, 21/35.
13/40: 40 = 2×2×2×5, ends (0.325). 7/12: 12 = 2×2×3 has a 3, repeats (0.58333…). 21/35: first reduce to 3/5; 5 only, ends (0.6). Always reduce first!
8. Find three rationals between 0.1 and 0.11.
Write 0.1 = 0.100 and 0.11 = 0.110. In between: 0.101, 0.105, 0.109 (and many more). As fractions: 101/1000, 105/1000, 109/1000.
Common mistakes
- Thinking there is no rational number between 1/2 and 3/4 because 'nothing comes between 2 and 3'. Rewrite with a bigger denominator (10/20 and 15/20) and many appear.
- Hopping the wrong way for a negative number. −2/3 goes LEFT of 0.
- Deciding 'ends or repeats' before reducing the fraction. 21/35 looks like it has a 7 below, but it is really 3/5.
- Saying q can be 0. p/0 has no meaning, so q ≠ 0 is part of the definition.