The wondrous and mysterious rational numbers

In the last post, I argued how the classical proof of the existence of irrational numbers is didactically flawed on several accounts. Here I present my own, purely arithmetical, approach.

It all starts with a thorough appreciation of rational numbers. Before introducing real numbers, my students are well-trained in fraction arithmetic and the various accompanying geometric pictures. Of course they know about decimal numbers already, and are eager to type in any fraction to find the "true" value. Students have an almost unhealthy obsession with decimal numbers, as if it told you more, or even everything there is to know about a number. This is not so!

This blog post presents a way of transitioning from $\mathbb Q$ to $\mathbb R$ that aims to break the "decimal fixation" and to illustrate the "incompleteness" of the rationals in a purely arithmetical, less convoluted way than is usually done (see earlier post here).

TLDR

One of the goals here is to appreciate the distinction between a number and its representations. To "calculate" a fraction is to switch from the fractional to the decimal representation of the same number. Observing what kinds of decimals can occur is a very educational example of a classification theorem, perfectly accessible without the use of algebra or higher formal reasoning. Both the fractional and decimal representations have their advantages and drawbacks, and it is insightful (if not indispensable) to understand how one can switch from a decimal back to a fraction.

Against this backdrop, the existence of irrational numbers (decimals that cannot be rewritten as a fraction) might appear almost trivial. This important moment in any student's mathematical career lends itself to reflect critically a few "intuitions" we believe to have about numbers. (I will explore this in a later post.)

Fractions as decimals: let's go explore!

Remember how I mentioned students's love for their calculators? We can harness it in the beginning (its limitations will become visible soon enough). Here are a few examples of simple fractions as decimals:

 

$\frac 12 = 0.5$

$\frac 13 = 0.333333\ldots = 0.\overline{3}$

$\frac 23 = 0.666666\ldots = 0.\overline{6}$

$\frac 14 = 0.25$

$\frac 34 = 0.75$

$\frac 15 = 0.2$

$\frac 16 = 0.1666666\ldots = 0.1\overline{6}$

$\frac 17 = 0.14285714285714\ldots = 0.\overline{142857}$

$\frac 18=0.125$

$\frac 38 = 0.375$

$\frac 78 = 0.875$

$\frac 19 = 0.111111\ldots = 0.\overline{1}$

$\frac 29 = 0.222222\ldots = 0.\overline{2}$

$\frac 1{10} = 0.1$

$\frac 1{11} = 0.09090909\ldots = 0.\overline{09}$

$\frac 1{12} = 0.08333333\ldots = 0.08\overline{3}$

A few things can be observed:

  • Some decimals are short and to the point.
  • Others have an infinite number of digits (students like to say: "they are infinite"), in a repeating pattern.
  • There can be a single digit that repeats, or a string of two or more digits repeating.
  • The repeating pattern (period) does not always start right after the decimal point.

The long period of $\frac 17$ sticks out like a sore thumb. It has a few crazy cousins further along:

 

$\frac 1{13} = 0.\overline{076923}$

$\frac 1{17} = 0.\overline{0588235294117647}$

$\frac 1{19} = 0.\overline{052631578947368421}$

There are also the weird

$\frac 1{81} = 0.\overline{012345679}$

(sic!) and the mysterious

$$\frac 1{9801} = 0.0001020304050607080910111213\ldots$$

It is also worthwhile to notice this pattern (for later):

$$\frac 19 = 0.\overline{1}$$

$$\frac 1{99} = 0.\overline{01}$$

$$\frac 1{999} = 0.\overline{001}$$

$$\frac 1{9999} = 0.\overline{0001}$$

Making sense of all this

Is there any order behind this plethora of behaviors? Let's do the division by hand in a few select cases, such as these:

The third example shows where the periodic behavior comes from, by looking at the remainders at each step and noticing that the repetition kicks in the moment the remainder 20 appears again. Is it guaranteed that the remainders will eventually repeat? A strong hint at the answer, and its justification, lies in the long division of $\frac 17$:

 
 

These examples show two insights:

  • The periodic repetition in the decimals stems from a repetition in the remainders.
  • The remainders are bound to repeat eventually (or become 0), because they come from a finite pool of possible remainders (the positive integers less than the numerator).

Classifying

While our observations were on a small number of arithmetical examples only, the reasoning about the calculations, i. e. the sense we made of their behavior, is entirely general, independent from the chosen numbers. I claim that this is enough to constitute a proof of the following theorem:

 
Theorem. The decimal representation of a rational number $\frac pq$ ($p$, $q$ integers) is either:
  • an integer,
  • a terminating decimal, or
  • a repeating decimal.
 

In particular, this tells us that the mysterious decimal representation of $\frac 1{9801}$, too, is bound to repeat eventually. (In fact, it repeats after having cycled through all possible pairs of digits, except for the second-to-last:

$$\frac 1{9801}=0.\overline{000102030405\dots9799}.)$$

Which representation is better?

Now comes the moment to address "decimal fixation" head-on. Why should we continue dealing with fractions when we now have decimals? The answer lies in comparing a few problems in either representation. Give the following "a" problems to one half of the class, and the "b" problems to the other, and have them work by hand:

Problem 1a. $\frac {26}{75}+\frac {77}{100}=?$

Problem 1b. $0.34\overline{6} + 0.77 = ?$

Problem 2a. $\frac{3}{8}\times\frac{4}{15} = ?$

Problem 2b. $0.375 \times 0.2\overline{6} = ?$

Problem 3a. Which is greater: $\frac{7}{11}$ or $\frac{13}{20}$?

Problem 3b. Which is greater: $0.\overline{63}$ or $0.65$?

By comparing the effort needed for each problem, it becomes apparent that the decimal representation is much better suited for adding and comparing, while multiplication is the strong suit of the fractional representation. This is why we math teachers insist on keeping the latter around (especially since multiplication is the most frequent operation in higher and applied math). So it turns out that the fractional representation can often give us more useful information about a number than its decimal representation, and vice versa.

Moonwalking back to fractions

At this point, one can already see what irrational numbers are about: they are decimals that fall in none of the three categories listed in the theorem. Or at least, such decimals certainly have no fractional representation. But is this the only restriction that applies? What about very complicated repeating decimals such as $0.987\overline{14916253649}$ or $0.\overline{1234567891011112\dots9989991000}$: do they also have a fractional representation? Or are they irrational?

While the existence of irrational numbers can already be deduced here, and the question of how to transition from a decimal to a fractional representation casts no doubt on this result, it is worth investigating to gain a thorough understanding of two deep concepts:

  • Fractions and decimals are but two different representationsnone is a truer form than the other. They both have their advantages and drawbacks. (This is why from now on we speak of rational numbers, rather than fractions.)
  • Indeed, irrationals are precisely the decimals that fit none of the aforementioned three categories.

So how do we represent a decimal as a fraction? A terminating decimal is easy: it is always an integer multiple of one of the "base-ten" fractions

$$0.1 = \frac 1{10},\quad 0.01 = \frac 1{100},\quad 0.001 = \frac 1{1000}, \quad\ldots$$

so that e. g.

$$3.285 = 3285\times 0.001 = 3285\times\frac 1{1000} = \frac{3285}{1000} = \frac{657}{200}.$$

The "base-ten" fractions (inverse powers of ten) are the building blocks of the terminating decimals.

Repeating decimals require a bit more work. But we can find the general method by looking at select examples alone:

$$0.\overline{2} = 2\times 0.\overline{1}=\frac 29$$

$$0.\overline{35} = 35\times 0.\overline{01} = \frac{35}{99}$$

$$0.\overline{123} = 123\times 0.\overline{001} = \frac{123}{999} = \frac{41}{333}$$

The primitive fractions $\frac 19$, $\frac1 {99}$, $\frac1 {999}$, … turn out to be the building blocks of the purely repeating decimals, just as $\frac 1{10}$, $\frac1 {100}$, $\frac1 {1000}$, … were the building blocks of the terminating decimals.

What about periods that don't start right after the decimal point? Let's do the easier case of just zeroes before the period first:

$$0.0\overline{4} = 4\times 0.0\overline{1} = 4\times\frac{0.\overline{1}}{10} = 4\times \frac 1{10}\times\frac 1{9} = \frac 4{90} = \frac{2}{45}$$

$$0.00\overline{234} = 234\times\frac {0.\overline{001}}{100} = 234\times \frac 1{100}\times\frac 1{999} = \frac{26}{11100}$$

The inverse powers of ten help their friends find the right place after the decimal point!

Lastly, decimal numbers with non-zero digits before the period can be split into their non-repeating and purely repeating parts:

$$1.1\overline{45} =  1.1 + 45\times 0.0\overline{01} = \frac{11}{10} + \frac{45}{990} = \frac{63}{55}.$$

We have thus established, by exhaustion, the converse of the classification theorem from above:

 
Theorem. Any number that is either:
  • an integer,
  • a terminating decimal, or
  • a repeating decimal
  • can be writte as the ratio $\frac pq$ of two integers $p$ and $q$.
 

What's missing?

Already in its first form, the classification theorem begs the question: what about decimals that fall in none of the three categories, in other words: that go on in some other fashion than a period? We can think of simple examples such as:

$$0.12345678910111212131415\ldots,$$

known as Champernowne's constant, or

$$0.101001000100001\ldots$$

or the Copeland-Erdös constant

$$0.23571113171923\ldots$$

(have student's guess the pattern in that one!). So, what about such numbers? They simply cannot have a fractional representation, and we call them irrational numbers. Tada!

A bit anticlimactic, I know. But I seriously cannot see what the big fuss is about the existence of irrational numbers. Once you do math in a decimal system rather than relying on proportions between integers (looking at you, Ancient Greeks), this is a trivial result.

Where is the wonder?

As I hope to have demonstrated, there is plenty of wonder and mystery already in the rational numbers. Some things have even been left unmentioned:

  • The always controversial $0.\overline{9} = 1$, with its various proofs.
  • The resulting insight that decimals are not a unique representation, neither for terminating ($0.25 = 0.250 = 0.24\overline{9} = 0.25\overline{0}=\ldots$) nor repeating decimals ($0.\overline{12}=0.1\overline{21} = 0.\overline{1212}=\ldots$), a slight nuisance that also riddles the fractional representation: $\frac 23=\frac 46=\frac{66}{99}=\ldots$
  • The fact that the rationals are dense on the number line (there is no "number right after 1").
  • The geometric meaning of repeating decimals (self-similarity in nested intervals).
  • Which fractions have a periodic decimal? Can we anticipate this before the long division?
  • Same question for the length of the period.
  • Positional representation in bases other than ten: In which bases does a rational produce a period, in which not? 

Plenty of wondrous questions left to explore! And that is still in the rationals alone. Now, after we have made the jump to real numbers, all kinds of other questions pop up beyond the mere existence of irrationals. These questions touch on what should "count" as a number (pardon the pun) in a deep and even philosophical way, and not without controversy. This is a worthwhile, but nevertheless very accessible topic for students with lots of occasions for reflection and discussion, without reliance on abstract notation. And a topic for another time.