A student of mine was shocked to hear me describe mathematics as “abstract.” He said “I’ve always though of mathematics as the most concrete of subjects with evertything conecting to the real world.” Naturally, he wasn’t a math major.
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.
This video comes from mathpickle.com. There are many other incredible math lesson ideas there.
- Addition is a binary operation. What that means is that it takes two numbers and does something with them to spit out a third number.
- Addition is commutative, which means that the order of the numbers doesn’t change the answer. For example 4 + 5 = 9 and 5 + 4 = 9.
- Addition is associative, which means that when adding three or more numbers together it doesn’t matter the order you perform the operation.
For example (1 + 2) + 3 = 6 and 1 + (2 + 3) = 6 so it makes sense to just write 1 + 2 + 3 = 6.
- Addition has an associated identity element, 0. It’s called an identity element because adding zero to anything is the same as letting it be itself. For example 0 + 8 = 8 + 0 = 8.
- Adding by one is the same thing as counting.
Some of you who have studied some abstract algebra may note that there is one property of addition that is usually mentioned at the same time as the identity element that is missing from my list. I am waiting until next week to go negative.
Here is a short Persian language film written, directed and produced by Babak Anvari, an Iranian born filmmaker living in London, UK.
Two & Two
Mathematical facts are eternal, but mathematical learning is shaped by the culture around it.
Last time we learned how to add like the ancients of Europe, Asia and Africa. This time you have been sent back to ancient Mesoamerica. Good news! This time instead of only rocks you have some ink and some bark! Lets learn how to add Mayan style!
First we need to learn the Mayan numeral system. It is remarkably elegant in its simplicity. There are three symbols that you need to know.
One is represented with a dot, five with a line and zero with a picture of an up-side-down turtle shell. With these three symbols the Mayans were able to represent all the whole numbers. For the numbers up to 19 they had something like a tally system.
But this system could get difficult to handle so instead of continuing to twenty with four lines, they used a place value system similar to the one we used today, only instead of base ten they worked in base twenty.
So twenty would be represented like this:
Lets start with forty-two plus ninety-three. Note that since the Mayans wrote their numbers in columns, we will do the addition across the page rather than down the page like we are used to doing with the Hindu-Arabic numerals.
So now you know how to add like a Mayan! I hope you have gained a deeper insight into the importance a good place-value system has in making elementary arithmetic possible.
So you’ve found yourself in the ancient world, no paper or pens in sight, and you need to add some three digit numbers together. Lets so say that you’ve been hit on the head and you have no memory of the Hindi numeral system and only know… say… Roman numerals. How do you do it? Start by picking up some pebbles.
So you have just shown that CLVII + CDXCVI = DCLIII.
By the way, if you type an expression in Roman numerals into Google, it will give you the answer in Roman numerals. I just discovered that.
Now you know how to add like an ancient. Way to go!