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Numbers between numbers

Infinity and strange math · level 3, lesson 4 of 6

A question to start

What number comes right after 5? Easy: 6. What comes right after 99? 100. Every whole number has a next one.

Now a harder question. What fraction comes right after 0? Is it \(\frac{1}{10}\)? \(\frac{1}{100}\)? \(\frac{1}{1000000}\)? Whatever answer you give, a friend can name a smaller fraction that is still bigger than 0. So which one is "next"?

This lesson shows something surprising: among fractions, nothing is ever next to anything else. Between any two fractions, however close, there is always another one, and so there are infinitely many.

The idea

Fractions live on the number line

A number line is a straight line with numbers marked on it in order, smaller on the left and bigger on the right. Whole numbers sit at evenly spaced marks: 0, 1, 2, 3, ... But fractions live there too. The fraction \(\frac12\) sits exactly halfway between 0 and 1. The fraction \(\frac14\) sits halfway between 0 and \(\frac12\). And \(\frac34\) sits halfway between \(\frac12\) and 1.

Between two whole numbers next to each other, like 5 and 6, there is no whole number. That is why 6 is the next whole number after 5. Fractions are different, as we will see.

Halving forever: no next fraction after 0

Start at 1 and keep halving: \(\frac12\), \(\frac14\), \(\frac18\), \(\frac1{16}\), \(\frac1{32}\), ... Each one is half as far from 0 as the one before.

0 1 1/2 1/4 1/8 1/16 halve again and again: always another fraction, closer to 0

The marks crowd closer and closer to 0, but none of them ever reaches 0. Halving a fraction that is bigger than 0 gives a fraction that is still bigger than 0, just smaller.

Now here is the proof that there is no "next" fraction after 0. Suppose someone claims a fraction \(f\) is the next one after 0. Then \(f\) is bigger than 0. But half of \(f\) is also bigger than 0, and it is smaller than \(f\). So half of \(f\) sits between 0 and \(f\), and \(f\) was not next after all. Whatever fraction anyone names, the same trick knocks it out. So no fraction is the next one after 0.

This kind of argument is called a proof by contradiction: you pretend something is true, show that it leads to nonsense, and conclude that it must be false.

The average: always a number in between

The halving trick works next to 0. For any two fractions, a different trick works: the average. The average of two numbers is what you get by adding them and dividing by 2. It sits exactly halfway between them.

Why exactly halfway? Think of two children standing on a number line, one at 2 and one at 8. They walk towards each other at the same speed and meet in the middle, at 5. The average of 2 and 8 is \((2 + 8) \div 2 = 10 \div 2 = 5\). The distance from 2 to 5 is 3, and from 5 to 8 is also 3. So the average is the meeting point in the middle.

The same is true for fractions. Take \(\frac13\) and \(\frac12\). To add them, write them with the same bottom number (the denominator): \(\frac13 = \frac26\) and \(\frac12 = \frac36\). Then

\[\frac26 + \frac36 = \frac56, \qquad \frac56 \div 2 = \frac5{12}.\]

So \(\frac5{12}\) is exactly halfway between \(\frac13\) and \(\frac12\). Check with twelfths: \(\frac13 = \frac4{12}\) and \(\frac12 = \frac6{12}\), and \(\frac5{12}\) sits right between \(\frac4{12}\) and \(\frac6{12}\).

1/3 1/2 5/12 3/8 11/24 the average of two numbers sits exactly halfway between them and then halfway again, on either side, forever

And then you can do it again: between \(\frac13\) and \(\frac5{12}\) is their average \(\frac38\); between \(\frac5{12}\) and \(\frac12\) is \(\frac{11}{24}\). And again, and again, forever.

So there are infinitely many in every gap

Put the two ideas together.

  • Between any two different fractions, the average is another fraction in between.
  • Then between the first fraction and the average there is another one, and between those another, without end.

So between any two different fractions there are not just one or two others, but infinitely many. Even between \(\frac1{1000}\) and \(\frac2{1000}\), which are extremely close, there are infinitely many fractions.

The same holds for decimals that stop, like 0.7 and 0.8: in between are 0.75, 0.71, 0.701, 0.7001, ... A mathematician says that fractions are dense: packed in so tightly that no two of them are next to each other.

Another way: bigger denominators

There is a second way to find fractions in between, which does not need averages. Take \(\frac25\) and \(\frac35\). In fifths there is nothing between them. But change to tenths: \(\frac4{10}\) and \(\frac6{10}\), and now \(\frac5{10}\) is in between. Change to hundredths: \(\frac{40}{100}\) and \(\frac{60}{100}\), and now 19 fractions are in between: \(\frac{41}{100}, \frac{42}{100}, \ldots, \frac{59}{100}\). The finer the ruler, the more marks fit in the gap, and you can always use a finer ruler.

But dense does not mean "more"

You might think: "With infinitely many fractions squeezed into every tiny gap, there must be far more fractions than counting numbers." Remember lesson 3: for infinite groups, "more" only means "cannot be paired perfectly". Amazingly, the fractions can be paired perfectly with the counting numbers, by a clever way of listing them. You will see how in level 6. Being packed tightly and being "more" are different things.

Worked examples

1. A fraction between \(\frac14\) and \(\frac13\).

  • Same denominator: \(\frac14 = \frac3{12}\) and \(\frac13 = \frac4{12}\). Nothing in twelfths fits between 3 and 4.
  • Use the average: \(\frac3{12} + \frac4{12} = \frac7{12}\), and half of that is \(\frac7{24}\).
  • Check with 24ths: \(\frac14 = \frac6{24}\), \(\frac13 = \frac8{24}\), and \(\frac7{24}\) is between them.

2. Three decimals between 0.5 and 0.6.

  • Write them with an extra place: 0.50 and 0.60. Then 0.51, 0.52, ..., 0.59 are all in between.
  • Three of them: 0.52, 0.55, 0.58. (The average is \((0.5 + 0.6) \div 2 = 1.1 \div 2 = 0.55\).)

3. Knocking out a "next" fraction. A friend says \(\frac1{1000000}\) (one millionth) is the next fraction after 0.

  • Half of it is \(\frac1{2000000}\), one two-millionth.
  • It is bigger than 0 and smaller than one millionth, so it is in between.
  • So one millionth is not next. Whatever they try, halving finds a fraction in between.

4. On Mars? A Mars rover is driving towards a rock 1 metre away. Each minute it drives half of the distance that is left: \(\frac12\) m, then \(\frac14\) m, then \(\frac18\) m. After 1 minute it is \(\frac12\) m from the rock, after 2 minutes \(\frac14\) m, after 10 minutes \(\frac1{1024}\) m, which is less than a millimetre. It is always some fraction of a metre away, never at 0. (Real rovers drive at a steady speed and simply arrive! This is just a picture of halving.)

Try it

  1. Find the average of \(\frac14\) and \(\frac12\). Check that it is between them.
  2. Write three decimals between 2.3 and 2.4.
  3. Find a fraction between \(\frac37\) and \(\frac47\).
  4. A friend says \(\frac1{50}\) is the next fraction after 0. Find a fraction that proves them wrong.
  5. What is the next whole number after 41? Why does the halving trick not work for whole numbers?
  6. How many fractions with denominator 100 lie between \(\frac3{10}\) and \(\frac4{10}\)? Name the smallest and the largest.
  7. (Harder) Here is a strange trick: to find a fraction between \(\frac13\) and \(\frac12\), add the tops and add the bottoms: \(\frac{1 + 1}{3 + 2} = \frac25\). Check, using decimals, that \(\frac25\) really is between them. Then try the trick on \(\frac25\) and \(\frac12\).
Answers
  1. \(\frac14 = \frac28\) and \(\frac12 = \frac48\). Add: \(\frac28 + \frac48 = \frac68\), and half of that is \(\frac38\). In eighths, \(\frac28 < \frac38 < \frac48\), so it is between them.
  2. For example 2.31, 2.35, 2.39. (Any decimal that is 2.3 followed by more digits, not all of them 0, works.)
  3. In fourteenths: \(\frac37 = \frac6{14}\) and \(\frac47 = \frac8{14}\), so \(\frac7{14} = \frac12\) is between them.
  4. Half of \(\frac1{50}\) is \(\frac1{100}\). It is bigger than 0 and smaller than \(\frac1{50}\).
    1. Half of a whole number is often not a whole number (half of 1 is \(\frac12\)), and there is no whole number between 41 and 42 at all. Whole numbers have gaps; fractions do not.
  6. \(\frac3{10} = \frac{30}{100}\) and \(\frac4{10} = \frac{40}{100}\). In between are \(\frac{31}{100}\) up to \(\frac{39}{100}\): 9 fractions. Smallest \(\frac{31}{100}\), largest \(\frac{39}{100}\).
  7. \(\frac13 = 0.333\ldots\), \(\frac25 = 0.4\) and \(\frac12 = 0.5\), so \(\frac25\) is between them. For \(\frac25\) and \(\frac12\): \(\frac{2 + 1}{5 + 2} = \frac37\), which is about 0.43, and that is between 0.4 and 0.5. (This trick is called the mediant. It always lands in between when the fractions are positive, though it is not usually the halfway point.)

Common mistake

Thinking there is a smallest fraction bigger than 0. There is no smallest one: halve it and you get a smaller one. The fractions bigger than 0 get as close to 0 as you like without ever having a first one.

Adding fractions by adding tops and bottoms when you want the sum. \(\frac13 + \frac12\) is not \(\frac25\); it is \(\frac56\). The add-tops-and-bottoms trick in question 7 is a fun way to find a fraction in between, but it is not how you add.

Where it leads