Operations on Fractions

In measurements, fractions appear whenever units are not small enough to express quantities in integers. For example, five quarter-dollars will buy you exactly as mush as a dollar and a quarter. One and a half dollar stands for exactly the same quantity as three half-dollars or six quarter-dollars. So, an alternative to using fractions is to decrease the unit in use:

dollar → half-dollar → quarter-dollar.

Now, this approach is doomed to failure.

Indeed, this is impractical to keep adjusting units of measurements to measurements all the time. (How do you divide 100 quarters between three friends? Can you share one half of 5 quarter-thirds?) Units of measurement are only valuable if employed universally. For this, they must be agreed upon and kept fixed.

Fractions are unavoidable and sooner or later we all have to learn to work with fractions. The mathematical usage of the word fraction has a very clear everyday connotation as a part of a bigger object. It would be unthinkable nowadays to just introduce fractions as a pair of numbers and postulate their basic properties as did the famous German mathematician Edmund Landau some 70 years ago. Still, to express fractions one needs a pair of numbers with a meaning and intuition attached to them.

When one divides six tarts between six kids, each gets a tart. More ingenuity is needed to divide 1 tart between six kids: each gets only 1 sixth of the whole. The customary shorthand for this is 1/6 to emphasize its origin: 1 sixth part of something is obtained by dividing 1 whole into 6 parts. The notation 1/6 implies division. However, it does not reflect on the fact that division into parts may and does often imply modification of the unit: instead of 1 whole we get 6 sixth so that only 1 sixth goes to every kid. How many would two kids get? 2 sixths - an integer number of new (and smaller) units.

Let's call the latter aspect of fractions u-property: p/q means "p q^th". The d-property is introduced to remind of the implied division: p/q also means "p divided by q". The u-property is almost always overlooked although it is more basic of the two. They combine to explain the reduction algorithm: 2/6 = 1/3. Multiplication table may serve as a convenient tool for such an explanation. Consider two lines of the table (base 10):

0 1 2 3 4 5 6 7 8 9

3 0 3 6 9 12 15 18 21 24 27

5 0 5 10 15 20 25 30 35 40 45

	0	1	2	3	4	5	6	7	8	9
3	0	3	6	9	12	15	18	21	24	27
5	0	5	10	15	20	25	30	35	40	45

with the following interpretation: 3/5 = 6/10 = 9/15 = ... Dividing the whole into 5 parts and taking 3 of them is the same as dividing it into 10 parts and taking 6 parts, and so on. An applet illustrates this idea. Move the mouse slowly and watch equal fractions light up.

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Bearing in mind the u-property reduces addition and subtraction of fractions to that of integers. 1 fifth + 2 fifths = 3 fifths. In math notations: 1/5 + 2/5 = 3/5. What about more general situations: 2/5 + 3/7? Response: as we only know to add or subtract quantities expressed in the same units, we should first describe "2 fifth" and "3 seventh" in common units. Do not add apples and oranges. 2/5 + 3/7 = 14/35 + 15/35 = 29/35. Whether or not the result is in lowest terms is absolutely irrelevant at this point.

A combination of u- and d-properties is also useful to justify the multiplication algorithm: "2/5 times 3/7" is the same as "2 fifth times 3 seventh" which is the same as "6 fifth-seventh", where fifth-seventh is a unit of measurement equal to the fifth part of 1 seventh, i.e. 1/7 divided by 5 which is 1/35^th of the whole.

Division is of course the last operation to discuss. The ubiquitous rule "Invert the Divisor and Multiply" remains a deep mystery to many a student. In my view, division of fractions is best explained in terms of the u-property. Consider a (perhaps contrived) situation: 12 tarts have been evenly divided into several groups so that each group contained 3 tarts. How many groups were there? Answer: 12 tarts divided by 3 tarts gives 4. It is again convenient to tackle quantities expressed in the same units. What is 2/5 divided by 3/7? First, let's write 2/5 = 2 fifths = 14 thirty fifths. Also, 3/7 = 3 seventh = 15 thirty fifths. Invoking the d-property now we see that 14 thirty fifths divided by 15 thirty fifths is 14/15. Therefore, 2/5 divided by 3/7 equals 14/15. In general,

Here are additional pages related to the definitions, properties of and operations on, fractions:

Fractions

References

P. Hilton and J. Pedersen, Fear No More, Addison-Wesley, 1983
E. Landau, Foundations of Analysis, Chelsea Pub Co, 1960 (First German edition - 1930.)
W. D. McKillip, et al, Mathematics Instruction in the Elementary Grades, Silver Burdett Co, 1978

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