Fractions: An Example

We're up to 5 in our series of posts fleshing out the material written by Hung Hsi Wu in Critical Concepts for Understanding Fractions. See also Part I ,Part II, Part III, and Part IV.


Here, we go to a concrete example of all of the elements we've discussed: definitions of fractions, use of the number line, equivalent fractions, and some simple operations on fractions--comparing fractions and addition of fractions.

We consider Decimal Fractions. Decimal Fractions are a particular kind of fractions: fractions whose denominator is a power of 10: 10, 100, 1000, or any number represented by 10^n, where (for simplicity) n is a non negative integer.

here are some examples: 1489/100, 24/100000, 58900/10000

Decimal fractions have another convenient representation: as numbers that can be abbreviated into decimals: 14.89, 0.00024, 5.8900

Specifically, the above are finite decimals, though they are usually referred to simply as decimals. The number of digits to the right of the decimal point tells us the number of zeros in the denominator: 2 in 1489/100, 5 in 24/100000, 4 in 58900/10000. In this form, the convention is that zeros are added to the left of numbers as necessary to indicate the number of zeros in the denominator: in the case of 24/100000, we added three zeros to the left of 24: .00024.

The example 5.8900 shows that our decimal convention is not enough for explaining that we can remove the trailing zeros to the right of our number and decimal point. to show that, we need to invoke equivalent fractions:

for all whole numbers k, m, and n, where n and k are non-zero,
m/n = km/kn.

In this case, we can show that 5.8900 is the same as 5.89 by going back to the fraction form:

5.8900 = 58900/10000 = (589 * 100)/(100 * 100) = 589/100 = 5.89

This invocation of equivalent fractions works for in general, so any number of trailing zeros on the right can be shown to be equivalent to their absences:
12.700000 = 12.7 = 12.70 = .... etc.

After equivalent fractions, we moved on to comparing of fractions. Here, we show how to compare decimals.

Example: given 0.0082 and 0.013, which is bigger? We can compare these decimals easily by first converting them to fractions, giving them common denominators, and then comparing the numerators.

Converting to fractions, we see that we are comparing 82/10000 and 13/1000. To compare fractions, we then invoke again the rule that any two fractions may be represented by the same denominator:

m/n = ml/nl and k/l = nk/nl.

If m = 13 and n = 1000, then l = 10, and we have 13/1000 = 130/10000.Now we can compare 82/10000 and 130/10000 by comparing numerators. 130 is larger, so .013 > .0082.


Now, we move onto an application of fraction addition: to understand the algorithm for adding decimal fractions.

Example: 4.0451 + 7.28

Looking at our example and recalling what we know about fractions, the decimal fractions are just short hand for these numbers as fractions: 4.0451 = 40451/10000, and 7.28 = 728/100. We can then add them as we add any fractions, by giving them the same denominator. In this case, we call m = 728 and n = 100. If l = 100, then ml = 72800 and nl = 10000. Now, adding the whole numbers is the same as adding the numerator: we add

40451/10000 + 72800/10000 = (40451 + 72800)/10000 = (113251)/10000.

This leads us to an algorithm for adding decimals is:
1. line up the numbers by their decimal point
2. add the numbers just as you would "normal" whole numbers
3. put the decimal point in the number, based on where it "lined up" with the decimals in the additions.

When we line up the numbers by their decimal points, we are essentially rewriting the two decimals so they have the same number of digits to the right of the decimal point. This is the same as giving both of the decimal fractions the same denominator.

Then, we add the numbers, just as we would add the numerators.

Then, we determine the appropriate location of the decimal point just as we do by convention: we count the number of zeros in the denominator, and place the decimal point so that the number of digits to the right of the point matches the number of zeros in the denominator: 11.3251.

The point here is to stress that decimal fractions are really just fractions, and that we know how to perform elementary operations on fractions just as we do on whole numbers. Everything we understood about how to manipulate whole numbers on the number line mapped to fractions on the number line, and decimals are just a convenient notation for a certain type of fraction. The point is to underline how similar the processes are.

We'll get into more complications, like multiplication of decimals, later, after we advance a bit more with fractions themselves. Onto understanding fractions as division, then to multiplication and division operations!