Decimals

A decimal number is a number which, obviously enough, includes a decimal point. Every number can be written in it’s decimal form.

We shall discuss three different types of decimal numbers in this article, Exact Decimals, Recurring Decimals and Non-Recurring Decimals.


Exact Decimal

An exact decimal, as the name suggests, is one that you can write down all the digits to, ie. it is exact. These are the simplest forms or decimals and examples include 0.5, 0.1 and 0.045.

Exact decimals are also known as Termination Decimals.


Recurring Decimal

For something to recur it means to happen again and again. Therefore a recurring decimal is one that goes on forever, but some of the digits eventually repeat themselves over and over again.

Often one of the first kinds of recurring decimals we meet in mathematics is \frac{1}{3}. This can be written as 0.3333333… where the 3s go on for ever.

Another example is 0.345656565656…. With this example it is interesting to note two things. The first two digits after the decimal point, (3 and 4) do not repeat. Secondly, it is possible for more that one digit to recur, in this example it’s the digits 5 and 6, but it could be any number of digits.

In order to write recurring decimals in a neater way, mathematicians decided to write the decimal with dots over the recurring digits, for example:

\frac{1}{3} = 0.333333 = 0.\dot{3}

and
0.22103103103103 = 0.22\dot{1}0\dot{3} with the dots in this example over the first and last repeating digit.


Non-Recurring Decimal

There are decimals that go on and on forever, but don’t have any digits recurring. The most common example of a non-recurring decimal is pi which is 3.1415926535897932385…

Here’s a website with lots of interesting information about pi.

Fractions and Decimals

In decimal form, a rational number is either an exact or a recurring decimal. The opposite to this is also true, both exact and recurring decimals can be always be written as fractions.

For example, 0.345 = \frac{345}{1000} = \frac{69}{200}.

How can we tell if a fraction will be an exact or recurring decimal?

This is actually a pretty easy thing to do. Fractions with denominators which the only prime factors are 2 and 5 will be exact decimals, any others will be recurring decimals.

Before we take a look at a couple of examples make sure you have read up on prime factor decomposition.

Examples

Are \frac{3}{16} and \frac{2}{3} exact or recurring decimals?

The denominator of \frac{3}{16} is 16, which can be written as 2 x 2 x 2 x 2. As you can see the only prime factors of 16 are 2, so this is therefore an exact decimal, (0.1875).

Let’s take a look at \frac{2}{15}. If we break down the denominator into it’s prime factors we can see that 15 = 3 x 5. Because there is a 3 in the prime factor decomposition then we know that \frac{2}{15} is a recurring decimal, 0.1\dot{3} to be exact.

Rounding Numbers

If you work out an answer in an exam, you are often asked to ’round’ this answer. This is often because the actual answer is far too long and clumsy to work with, 0.35 is a lot easier to deal with than 0.3499857467385 for example. There are two ways in which you can round a number, either to a certain amount of decimal places or to a certain number of significant figures, we shall discuss both in this article.

Decimal Places

Write the number 0.234765 correct to 4 decimal places.

To do this we count 1 more digit past the decimal place that it asks for, so in the example above we’ll count to the 5th digit, 6. If this digit is 0 – 4 then we’ll leave the number as it is, correct to 4 decimal places. If however, the 5th digit is 5 – 9, then we add 1 to the previous digit. As the 5th digit in our example is 6, then the answer rounded to 4 decimal places would be 0.2348.

This may sound a little confusing but the reasoning behind it is actually quite straightforward.

Take the numbers 0.12341 and 0.12348. We can see that the first 1 is a lot closer to 0.12340, whereas the second is a lot closer to 0.12350, it’s because of this reason that you choose whether to round up or down.

Significant Figures

This is very similar to rounding to a certain amount of decimal places, but this time – as the name sort of suggests – we round to a certain amount of significant figures. But what are these significant figures?

Significant figures are basically any digit that is not a leading or trailing zero. For example, 2.5903 has 5 significant figures, 2, 5, 9, 0 and 3. The number 0.00345 however, only has 3 significant figures, (3, 4 and 5), as the leading digits are not counted. Let’s have another example.

Round 0.0004857533 to 4 significant figures.

To start this off we look for the 5th significant figure, which in this case is a 5. As you will remember from above, if the digit is between 5 and 9 then we round the previous one up, so 0.0004857533 becomes 0.0004858.

0.00256023164, rounded off to 5 decimal places (d.p.) is 0.00256 . You write down the 5 numbers after the decimal point. To round the number to 5 significant figures, you write down 5 numbers. However, you do not count any zeros at the beginning. So to 5 s.f. (significant figures), the number is 0.0025602 (5 numbers after the first non-zero number appears).

Note – If you get asked to round a number such as 3.995 to 2 decimal places, is the answer 4.00 or 4? Well technically both of these numbers are the same, however ask the question asked for 2 decimal places it is good practice to include the two zeros after the decimal point.

Converting a Recurring Decimal to a Fraction

We already know that every single recurring decimal can be written as a fraction. Most modern calculators can do this, but using algebra we can see how it’s done.

Example – Convert 0.4848484848.. into a fraction in it’s lowest form:

First we let,

x = 0.4848484848..

Now what we want to do is to move the first repeating section to the left hand side of the decimal point. With this example the repeated section is ..48.., so multiplying our first expression by 10 will achieve this.

100x = 48.4848484848..

Now we subtract our first equation from our second one, giving us:

99x = 48Note that even though the decimal repeats forever, you have eliminated this problem by subtracting 1 infinitely repeating section from another.

This is just now a basic equation to solve, we simply divide both sides by 9 to get

x = \frac{48}{99}

Remember that the questions asked for the fraction in it’s simplest form, so dividing the numerator and denominator by 3 gives us:

0.4848484848.. = \frac{16}{33}

This method applies for any recurring decimal, not matter how long or complicated.