Always room for one more
Density of rational numbers
Does it stop?
Between 1 and 2 there is the rational number 32. Between 1 and 32 there is another, 54. Does this ever stop?
How many rational numbers lie between 1 and 2?
What this lesson covers
The idea
Between any two rational numbers there is always another rational number, such as their average, so there are infinitely many rational numbers between any two.
Does it stop?
Between 1 and 2 there is the rational number 3/2. Between 1 and 3/2 there is another, 5/4. Does this ever stop?
How many rational numbers lie between 1 and 2?
- None
- Only 3/2
- Infinitely many
Average and zoom
Find the average of the two end numbers. That is a rational number between them. Then zoom into one gap and do it again.
Dense
Rational numbers are dense: between any two rational numbers there is always another, for example their average. So there are infinitely many rational numbers between any two.
Between 1 and 3/2 we found their average: add them and divide by 2. The result is 5/4. No matter how close two rational numbers are, taking their average gives one more between them.
Notes
Rational numbers are dense: between any two rational numbers there is always another, such as their average, so there are infinitely many between any two.
Check yourself
What is the average of 1 and 3/2, a rational number between them?
Which rational number lies between 2/5 and 3/5?
How many rational numbers lie between two rational numbers that are very, very close?
The average of 0 and 1 is 1/2. The average of 0 and 1/2 is 1/4. The average of 0 and 1/4 has what denominator?
Answer: 8
The average of 0 and 1/4 is 1/8, so the denominator is 8. The next ones are 1/16, 1/32, and so on without end.
- 5/4 — correct. Add the two numbers: 1 + 3/2 is 5/2. Divide by 2 and you get 5/4.
- 7/4. 7/4 is bigger than 3/2. The average must lie between 1 and 3/2.
- 2. 2 is bigger than 3/2. The average must lie between 1 and 3/2.
- 3/10. 3/10 = 0.3 is smaller than 2/5 = 0.4.
- 1/2 — correct. Add 2/5 and 3/5 to get 1, then divide by 2: the average is 1/2. Check in decimals: 0.4, then 0.5, then 0.6.
- 7/10. 7/10 = 0.7 is bigger than 3/5 = 0.6.
- None, there is no room. There is always room: take the average.
- Exactly one. After the average there is another average, and another, and another.
- Infinitely many — correct. Yes. Taking the average again and again never runs out of numbers.