kitchen table math, the sequel: Multiplying Fractions with Prof. Wu

Tuesday, April 29, 2008

Multiplying Fractions with Prof. Wu

This is the 6th in a 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, Part IV , and Part V .

We are now progressing rapidly through operations on fractions! In this post, we discuss multiplication of fractions.

We recall our definition for a fraction k/l. k/l was defined as the concatenation of k pieces when the unit length is divided into l parts. That is, the fraction k/l is the same as saying the fraction k/l of the number 1.

Generalizing this, we can define a fraction of a number as follows:

k/l of a number m is the sum of k l-length pieces of m, that is, when m is divided into l equal parts, and we concatenate k of those parts.

But a fraction is just a number, as indicated by its length on the number line (or its location on the number line), so the definition of k/l of a number m can is true whether m is a whole number or a fraction. If m is a whole number, then k/l of that whole number is defined as above. If m is a fraction, say m = a/b, then k/l of that fraction means: we take the length [0,a/b], partition it into l pieces, and then concatenate k of them.

We now define the product, or the multiplication of two fractions as:
k/l * m/n = k/l of a segment of length m/n.

This fits our intuition, if we are comfortable with multiplication: 1/3 of 6 is 2 = 6/3, 2/3rds of 9.5 is 2/3 * 9.5 = (2* 9.5 )/3 = 19/3 = 6 and 1/3. But we still need a rule for how to multiply such products.

The product rule for fractions is: k/l * m/n = k*m/l*n.

Now, this should make sense: the denominator is telling us how many partitions we need to make, and the numerator tells us how many of those partitions we concatenate together. When we multiply fractions k/l * m/n, we simply partition m/n into l pieces again, and concatenate k of them. If we remember order of operations, the rule should also become clear: m/n was when we partitioned 1 into n pieces, and concatenated m of them, and (a/b)/c is the same as a/(b * c): partitioning first into n and then each of them into l is the same as partitioning into n*l = l*n same sized pieces in the first place. Likewise, d*(a/(b*c)) is the same as (d*a)/(b*c): counting up d pieces of a pieces of size (b*c) is the same as counting d*a pieces of size b*c.

More formally, though, we can prove this rule as follows. We begin by going to our definition: k/l of m/n means we divide the length m/n into l parts, and concatenate k of them. Partitioning the segment m/n into l pieces is done as following: using equivalent fractions and association, m/n = l*m/l*n = l*(m/l*n). That is, m/n is the concatenation of l pieces, each of size m/ln. In other words, dividing m/n into l parts means creating parts each of size m/ln. Concatenating k of them means k*(m/ln) = km/ln.

Such a simple formula helps clarify the multiplication algorithm of decimals. Recall that the formula is:
* Multiply two numbers as if they are whole numbers, ignoring the decimal points,
* Count the total number of nonzero digits to the right of the decimal places in the two numbers, call it P
* Put the decimal point in position so that the resultant product has P nonzero digits to the right of the decimal place.

This follow naturally as a consequence of the powers of ten in the denominator:

1.25 * 0.0067 = 125/100 * 67/10000 = (125 * 67)/ 1000000 = 8375/1000000 = 0.008375

See, we’re getting the hang of these fractions! Onto division next time.

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