Today I’m going to skip ahead a little in the usual progression of how we learn mathematics. Usually addition would be followed by subtraction and then multiplication, to be followed by division, but I’m going to go straight to fractions. I can justify this by pointing out that fractions are actually significantly older than decimal numbers. Fractions were used around 1000 BC. in Egypt, while, as I mentioned in a previous post, the decimal system came over a millennium later.
First of all, fractions do not suck. Fractions are beautiful and are an excellent tool for calculation. They do have a bad reputation though. I’d like to take this time and present to you a defence of fractions.
The main thing to remember is: fractions are about multiplication and division. Before we go too far we should probably do a crash course in what makes up a fraction.
A fraction is made of two numbers, separated by a vinculum (or sometimes a solidus, as in “¾”). Either way most people just call it a “fraction bar” or even just a “line.” The number above the fraction bar is called the numerator. From Webster’s Revised Unabridged Dictionary of 1913:
Nu”mer*a”tor, n.[L. numerator: cf. F. numérateur.]
1.One who numbers.
2. (Math.) The term in a fraction which indicates the number of fractional units that are taken. In a vulgar fraction the numerator is written above a line; thus, in the fraction 5/9 (five ninths) 5 is the numerator. See Fraction.
So a numerator is a person that counts how many things you have.
The number under the fraction bar is called the denominator.
De*nom”i*na`tor, n.[Cf. F. dénominateur.]
1.One who, or that which, gives a name; origin or source of a name.
This opinion that Aram . . . was the father and denomination of the Syrians in general. Sir W. Raleigh.
2. (Arith.) That number placed below the line in vulgar fractions which shows into how many parts the integer or unit is divided. Thus, in fraction 3/5 (three fifths) 5 is the denominator, showing that the integer is divided into five parts; and the numerator, 3, shows how many parts are taken.
Thus, a denominator is a person that give a name to the things you have. Together a numerator and denominator work together to tell you how many things you have, and what those things are.
Usually, the first operation that we learn to do with fractions (apart from their function as a ratio, more on this at a later date) is addition. On the surface of things this makes sense, after all, addition is the easiest operation to do with whole numbers, so why shouldn’t we start with that? But therein lies the rub. You see: fractions are not whole numbers. Adding fractions (and consequently subtracting them) is a difficult chore, and not a very pleasant experience with its multiple steps and the importance of finding a common denominator, etc. Remember what I said at the beginning, fractions are about multiplication and division, so it doesn’t make sense to begin with addition.
Fractions are About Multiplication and Division
The denominator says what they are and the numerator say how many. So one can think of a fraction as a multiplication. Five sevenths is 5 × 1/7. Alternatively one can also think of a fraction as a numerator being split by the denominator. So five sevenths is how many cookies each person gets when you share them among 7 people, that is 5 ÷ 7. In a future post I’ll discuss this duality in greater depth.
Numerators multiply with numerators, denominators multiply with denominators, and everything is simple and elegant. This is the real beauty of fractions. Multiplication is a breeze. Compare the above to what it would be as decimal numbers. I’m not certain where I would even begin if I had to do it this way.
So to sum up, fractions don’t really suck, it’s only that people have unrealistic expectations.