Modules for Prerequisite Mathematics

Subsection 5.1.1 Fraction Concepts

A fraction is an expression with one number (or more complicated mathematical expression) written over another

The top piece of a fraction is called the numerator and the bottom piece is called the denomninator.

In mathematics, we use the fraction notation \(\frac{a}{b}\) for (at least) two related and intertwined ideas. First, fractions are the familiar parts that are taught in elementary mathematics: \(\frac{3}{5}\) is \(3\) parts, each part being one fifth of the whole. Second, as discussed in Section 1.1, fractions are also the most common division notation: instead of writing \(3 \div 5\text{,}\) we usually write \(\frac{3}{5}\text{.}\) So the fraction notation, in addition to meaning \(3\) parts with each part being a fifth of the whole, also means the result of dividing \(3\) by \(5\text{.}\)

These two uses are related and intertwined, of course. If we want to write numers in decimal notation, then the division \(3 \div 5 = \frac{3}{5} = 0.6\text{.}\) But a fifth of a whole is \(0.2\) and so \(3\) fifths is \(3 \times 0.2 = 0.6\text{.}\) So both concepts, in some sense, refer to the number \(0.6\text{;}\) indeed, \(\frac{3}{5}\) is just as good a way of writing this number as \(0.6\text{.}\)

In the following review of fraction arithmetic, we don't want to present the rules of fractions as arbitrary symbol manipulation. All the rules relate back to these two concepts: they are the rules for combining parts of a whole, as well as the rules for doing various arithmetic combinations with division.

Subsection 5.1.2 Some Notations and Comments

A fraction where the numerator is smaller than the denominator (\(\frac{4}{11}\)) is called a proper fraction. A fraction where the numerator is larger thatn the denominator (\(\frac{13}{11}\)) is called a improper fraction/. In much of elementary arithmetic, students are encouraged to write improper fractions as mixed fractions. Here is an example.

This is certainly a nice notation if we want a sense of the size of the number: it's easier to see that this is a bit more than \(2\) in mixed form. However, mixed fractions are more difficult to work with in most situation. In university mathematics, for the most part, we are usually quite happy to write improper fraction and we mostly avoid mixed fractions.

Sometimes a fraction might have a square root (or other root) in its denominator (\(\frac{1}{\sqrt{2}}\)). In many high-school situations, students are encourage to rationalize the denominator (\(\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)). For university mathematics, there is no need to do this. We are perfectly happy with square roots in denominators. (Indeed, I really have no idea where this high-school tendency to want rational denominators comes.)

Sometimes fractions have common factors in the numerator and denominator.

In this case, there is a common factor of \(5\text{.}\) We have a very important rule for manipulation fractions that applies here.

IF WE MUTLIPLY OR DIVIDE BOTH THE NUMERATOR AND DENOMINATOR OF A FRACTION BY ANYTHING (EXCEPT ZERO), THE FRACTION IS UNCHANGED.

A fraction where the numerator and denominator have no common fractors is said to be written in \(lowest terms\text{.}\)

Sometimes we don't want fractions in lowest terms. In the next section in particular, it will be useful to go the other way: to multiply both nuemrator and denominator by some number to change the fraction.

For another notational note, we can move negative signs around fraction.

All these of these things mean the same thing. In decimal form, they all correspond to the number \(-0.8\text{.}\) This kind of notational change, where we move this negative sign around fractions, is the kind of thing that is often done automatically and without comment in many calculation.

Division by \(1\) has no effect: dividing something into one part doesn't change it. Therefore, if we have a fraction with a denominator of \(1\text{,}\) we can simply write it as a whole number.

We can go the other way as well. Sometime we may want to write a whole number as a fraction, so we simply make its denominator \(1\text{.}\)

Finally, some of you may be familiar with the term cross-multiply from certain operations with fractions (or from other operations in algebra). I'm going to very intentionally avoid this term. In my experience, it has two many different meanings in different settings and will often just confuse students. Moreoever, it tends to be a mechanical term which hides the concepts behind the operation. If you remember the term and it serves you well, but all means continue to use it, but I'll avoid it in these notes.

Subsection 5.1.3 Addition and Subtraction

The addition and subtraction of fractions relies on having a common denominator. The priciple of the common denominator is that we want to add and subtract the same kind of things. In their simple form, \(\frac{4}{5}\) and \(\frac{7}{3}\) are not thing that can be added together: they are not comparable. Using common denominator puts them in a form where we can reasonable compare: instead of looking at thirds and fifths and being confursed, we can look at fifteenths and fifteenths and understand how to related them to each other.

Creating a common denominator uses the rule from the previous section: we can mulitply the numerator and denominator of any fraction by a non-zero number without changing the fraction. In our example, we can multiple the first fraction by \(3\) (the denominator of the other fraction).

We can multiplicy the numerator and denominator of the second fraction by \(5\) (the denominator of the first fraction).

Now both of the fractions are expressed in fifteenths. In this form they are comparables and can be added or subtracted. Here is addition.

Subtraction is similar.

In the example above, we made a common denominator out of the product of the two denominators. The first denominator was \(5\) and the second was \(3\text{.}\) We multiplied the first fraction by \(3\) and the second by \(5\text{,}\) so that the denominator of both is \(15\text{.}\)

The choice to multiply each fraction by the denominator of the other fraction will always work. It will always produce a common denominator which is the product of the two original denonimanors. However, sometimes there is a more efficient choice.

Consider this addition.

We could mutliply \(24 \times 36 = 864\) and use that as a common denominator; that would work. However, I'll multiply the first fraction by \(3\) and the second by \(2\text{.}\)

This produces a common denominator of \(72\text{,}\) which is much smaller and nicer to work with, compared with \(864\text{.}\)

You might have wondered how I came up with \(72\) as a common denominator. There are several answers. The intuitive answer is that I noticed that \(24\) amd \(36\) have common factors. More precisely, there are both multiples of \(12\text{.}\) So I wondered if there was another multiple of \(12\) which would work for a common denominator. By a little trial and error, I found that \(72\) was a multiple of \(12\) which I could use for a common demoninator. In practice, when I notice common factors in denominators, I'll often do this kind of trial and error to find a reasonable common denominator.

For those of you who like a more complete and formal answer, \(72\) is the least common multiple of \(24\) and \(36\text{.}\) I'm not going to review GCDs or LCM in these notes, but for those of you who remember that material, the LCM of the denominators is always the most efficient choice for a common denominator of fractions.

Subsection 5.1.4 Multiplication of Fractions

Addition and subtraction of fractions required common denominator, which is a fairly complicated multi-step procedure. Multiplication of fractions is quite a bit easier. The reason for this goes back to order of operations, as discussed in Subsection 1.1.3. Fraction are a way of writing division. Order of operation tells us that we can do multiplication and division in any order we wish without changing anything. Consider this multiplication.

In terms of arithmetic operations, this is all multiplication and division. We can read this expression in the following way: take \(4\text{,}\) divid by \(7\text{,}\) multiply by \(2\text{,}\) and then divide by \(5\text{.}\) By order of operations we can do those in any order. If we do the multiplication first, we get \(4 \times 2 = 8\text{,}\) after which we divide by \(5\) and then divide by \(7\text{.}\) If we divide by two number consecutively, we are dividing parts into parts. Dividing by \(5\) is making fifths. Then dividing by \(7\) is then dividing each fifth into sevents, resulting in \(5 \times 7 = 35\) total parts. All of this justifies the arithmetic:

To multiply two fractions, we simply multiply numerators and multiply denominator. In more general terms and writing multiplication as simply adjacent variables:

Subsection 5.1.5 Division

Division is likewise not as complicated as addition or subtraction with their common denominators. To get at the general rule, let's first consider the division of whole numbers. Say I wanted to think about \(4 \div 5\text{.}\) I could write this as a fraction \(\frac{4}{5}\text{.}\) Then, using the multiplication rules from the previous section, I could write this as the multiplication of fraction.

Using this, I can say that diving by \(5\) is the same thing as multiplying by \(\frac{1}{5}\text{.}\) This gives me a rule for fractions: to divide by a fraction is to multiply by its reciprocal. (A reciprocal of a fraction is the fraction with denominator and numerator interchanged.)

That gives me a rule which I can extend to any fractions. If I want to divide by a fraction, I can multiply by its reciprocal.

We can write this in general terms.

I could also have writen this division as fractions themselves. Consider the numerical example first.

This expression, with a fraction inside another fraction, is called a nested fraction. The rules for division can also be thought of as rules for dealing with nested fractions. Here is the rule in general form.