Why Do Some Fractions Repeat as Decimals?

Why Some Rational Numbers are Non-Terminating Decimals

I’d like to explore the relationship between non-terminating, but repeating, decimals and their rational equivalents. The topic is worth exploration because it can help provide insight as to why something like 0.99999999999999999… is equal to 10, not just approximately 10, kind of. The moment you stop writing the 9s, though, it is no longer equal to ten. Through this exploration I hope you’ll see why these things occur.

Beyond that, I think it is very cool to explain something that has bugged me for years. (For the sake of simplicity, we will only consider proper fractions that are fully reduced.)

Why 1/3 is 0.3333333333 … repeating infinitely is easily enough seen by dividing 1 by three with long division. You end up with a loop of 10 divided by 3, which always has a remainder of one. But why does such a nice and easy rational number end up with a non-terminating decimal equivalent, when some other, ugly numbers, like 7/32 are terminating?

For the sake of clarity, to convert a fraction into a decimal you divide the numerator by the denominator. So, $\mathrm{¼}$ would be 0.25 as shown below:

$\frac{1}{4}=4\begin{array}{c}\hfill 0.25\\ \hfill \overline{)\begin{array}{l}1.00\\ \underset{\_}{-8}\\ 20\\ \underset{\_}{-20}\end{array}}\end{array}$

I am going to make a few points that will be pulled together to show why some rational numbers are non-terminating decimals.

Point One:

The first big thing to note is that, as you probably already know, 0.25 is the same as $\frac{25}{100}$ . This ratio is equal: $\frac{1}{4}=\frac{25}{100}$. Similarly, $0.157=\frac{157}{1,000}.$ However, $0.\overline{13}\ne \frac{13}{100}$.

Point Two:

Decimals are equivalent to fractions with denominators that are a power of ten. Consider the following decimal, 0.111111… repeating.

The first decimal, 0.1 is $\frac{1}{10}$. The second decimal, 0.11 is $\frac{1}{10}+\frac{1}{100}$, the third, 0.111 is $\frac{1}{10}+\frac{1}{100}+\frac{1}{1000}$, and so on.

Then $0.1111=\frac{1}{10}+\frac{1}{100}+\frac{1}{1,000}+\frac{1}{10,000}$ .

This means that decimals, rewritten as fractions, all have denominators that are an exact power of ten.

So, $0.1111=\frac{1}{10}+\frac{1}{100}+\frac{1}{1,000}+\frac{1}{10,000}$, which is $0.1111=\frac{1}{{10}^1}+\frac{1}{{10}^2}+\frac{1}{{10}^3}+\frac{1}{{10}^4}.$

Let’s see what that would look like for a terminating decimal, like $\frac{1}{8},$ which is 0.125.

$0.125=\frac{1}{10}+\frac{2}{100}+\frac{5}{1,000}$. If you’re unsure of this being true, I’ll get common denominators and we will see the sum is $\frac{125}{1,000}.$

 

$\frac{1}{10}\cdot \frac{100}{100}+\frac{2}{100}\frac{10}{10}+\frac{5}{1,000}$

$\frac{100}{1,000}+\frac{200}{1,000}+\frac{5}{1,000}=\frac{125}{1,000}.$

Decimals are fractions with a denominator that’s an exact power of ten.

Point Three:

Let us consider a different method of converting fractions into decimals, to shed light on why non-terminating decimals spring forth from some rational numbers.

Let us convert one-fifth into a decimal by setting up a ratio where the denominator must be a power of ten.

$\frac{1}{5}=\frac{x}{10}$

The power of ten is easy here because five divides into ten evenly.

Changing the denominator into ten by multiplying by two over two:

$\frac{2}{2}\cdot \frac{1}{5}=\frac{x}{10}$

Give us:

$\frac{2}{10}=\frac{x}{10}$, so $\frac{1}{5}=\frac{2}{10},\text{ or 0}\text{.2}$.

Let’s see how this works for fraction like $\frac{1}{8}.$

$\frac{1}{8}=\frac{x}{{10}^{?}}$

The smallest power of ten that eight divides into is 1,000. The smallest that 16 divides into is 10,000, and the smallest that 32 divides into is 100,000. We will explore how to see that in a moment, but let’s finish 1/8^th first.

$\frac{1}{8}=\frac{x}{1000}$

$\frac{125}{125}\cdot \frac{1}{8}=\frac{x}{1000}$

$\frac{1}{8}=\frac{125}{1000},\frac{1}{8}=0.125$.

If we look at this in terms of factors and exponents, we can see an interesting occurrence. The column on the left will be a denominator of two and the column on the right will be the smallest power of ten we can use, set up as a ratio. Then we’ll factor these and gain valuable, hopefully, insight.

 

To easily see how to make $\frac{1}{16}=\frac{1}{10,000}$, let’s look at the factored form.

 

$\frac{1}{2^4}=\frac{1}{{\left(2\cdot 5\right)}^4}$

Do you see that if to make 2⁴ = ${\left(2\cdot 5\right)}^4$, we would need to multiply it by 5⁴?

$\frac{5^4}{5^4}\cdot \frac{1}{2^4}=\frac{1}{{\left(2\cdot 5\right)}^4}$

Another way to see this by setting up an equation, for the denominators:

$2^{4}n=2^4\times 5^4$

Solving by dividing by n we arrive at:

$n=5^4.$

Then:

$\frac{1}{16}\cdot \frac{5^4}{5^4}=\frac{x}{10,000}$

$\frac{1}{16}=\frac{625}{10,000}$

 

 

Let’s consider how this would work for a number that’s not a perfect power of 2, like 20.

$\frac{1}{20}=\frac{x}{10}$

$\frac{1}{\left(2^2\cdot 5\right)}=\frac{x}{\left(2\cdot 5\right)}$

Keep in mind, we’re not just trying to get the denominators equal, we want them to be equal and a perfect power of 10. (So a denominator of 20 will not work, because decimals are all powers of ten.)

If you see the factors of 20 have 2², which tells us we are going to have a denominator of 10², 100.

$\frac{1}{\left(2^2\cdot 5\right)}\cdot \frac{5}{5}=\frac{x}{\left(2\cdot 5\right)}\cdot \begin{array}{c}\\ 2\cdot 5\end{array}$

$\frac{5}{100}=\frac{x}{100}$

And 1/20^th is 0.05, so we’re golden!

By using this new method, we can see that we must change the denominator of our original rational number into a power of ten in order for it to be written as a decimal (since decimals are all powers of ten).

The Pay-Off

Given these three points I will show you why some decimals are non-terminating. Short-story long here, perhaps, but the purpose is not for answer-getting, but to explore things mathematically and discover understanding in things previously mysterious.

Let’s take 1/3 for starters. To rewrite one-third as a decimal, accurately, without truncating the decimal expansion, we need to write multiply three by itself so it will be a power of 10.

$\frac{1}{3^n}=\frac{x}{{10}^n}$

This is impossible. The easiest reason why is that the powers of three end in, 1, 3, 7, 9 only. There is no power of three that ends in zero, and all powers of ten end in zero.

It turns out that all rational numbers, with denominators that have a prime factor other than 2 or 5 suffer such a fate. Seven ends in 1, 3, 7 or 9, only. Eleven only ends in 1.

Decimals compare a whole number to a power of ten. There is no way to multiply the prime numbers, other than 2 and 5, by a power that will result in a whole number.

Let us consider another method of explanation of why 3, for example, will never be a power of 10.

$3^x={10}^x$

Taking the logarithm of both sides gives us:

$\log 3^x=\log {10}^x$

$x\log 3=x\log 10$

Because log10 = 1,

$x\log 3=x$

Solving for x

$x\log 3-x=0$

$x\left(\log 3-1\right)=0$

Log3 $–$ 1 cannot equal zero because 10¹ is not 3, so x must be zero.

Plugging that value into our original equation we see that only power of 3 and 10 that yields equal value is zero.

$3^0={10}^0$

Conclusion:

I hope you really see the reason why rational numbers become non-terminating decimals is because of the nature of what decimals are. They are powers of 10. There are not any whole number multiples of prime numbers, other than two or five that produce a power of ten. So you cannot covert a seven into a power of ten, other than the power of zero.

Again, the purpose of this exploration was just that, to explore the math behind something we just gloss over and hopefully make connections and other such things.

This is not an attempt at a proof, but rather a justification and explanation. Let me know what works and what doesn’t work for you.

As always, thanks for reading.