With division came fractions, in the form of an inverse to a number, and simply one number divided by another. It sounds really simple, but it is worth saying that one divided by three is a third. The mathematical notation version of that can look very similar:
\[1/3=\tfrac{1}{3}\]
Right, so how do we add together two fractions, say a half and a third? Well, to start with you need to be talking about the same type of things. Thirds and halves are not the same type of thing, apart from the fact they are both fractions. It is easy to add a third to a third, you get two thirds!
You can't directly add a half to a third though - what would you get? A hird? Or a Thralf? So we need to first of all make them the same thing. This may NOT be either halves or thirds, and it is going to change the numbers on top in each case.
A half is the same as two quarters, or six twelfths and so on. A third is also the same as two sixths or seven twenty-firsts, and so on. The game is to find the first type of thing (quarters, sixths, twenty firsts or whatever), that we can make both a half and a third fit into exactly.
So, first question, how many thirds in a half? One and a bit, because one third is less than a half, and two thirds is more than a half, so that is no good. What about quarters? How many quarters in a half - exactly two. Good. How many in a third? Again one and a bit for the same reason as halves. Fifths are also no good; there isn't an exact number of them in a half or a third.
So what about sixths? Well, there are two sixths in a third so good. And there are exactly three sixths in a half. Bingo. So our problem is now to add three sixths (a half) to two sixths (a third) and the answer is exactly five sixths.
So for addition, all you do is add the number of the type of thing you have together if it is all the same type of thing. If it isn't you need to make the type of things the same (which automatically changes the number of them you have) and then just add.
It is no coincidence that a sixth is half of a third by the way. The easiest way (although not necessarily the neatest) of converting two different types of things into the same type is just to multiply the types together. So two times three is six in our example.
For subtraction, you just do all that in reverse. Easy.
Multiplication seems a bit trickier, but is actually easier once we work out a shortcut. Lets say you have so many somethings or other and you want to multiply them by another amount of something else. Let's represent the 'so many' with the letter 'a' (just a placeholder, could be a picture of a pony or a guitar). Let's represent the somethings with 'b', the 'another amount' with 'c' and the 'something else's with 'd'. So what we get looks like this:
\[\frac{a}{b}\cdot\frac{c}{d}\]
So we are trying to multiply a number of b'ths by c number of d'ths. If that makes sense. This looks a bit daunting, so lets make it easier and say we are only going to find what one b'th of the second fraction is. Once we have worked out what one b'th is, we can just multiply that by a, and we will get the full answer we want. That would look like this:
\[\frac{1}{b}\cdot\frac{c}{d}\]
Now we are really just taking two numbers the 'c' and the 'b' and dividing 'c' by 'b'. Why are we doing that? Lets use some numbers to see what happens:
\[\frac{1}{2}\cdot\frac{4}{7}\]
So we are asking what is half of four sevenths? The obvious answer is two sevenths. Two sevenths plus another two sevenths is equal to four sevenths. So all we did was divide four by two and we got the answer. (Or you could say we multiplied four by a half or by the inverse of two ... it is all the same thing).
The problem is what happens if you cannot divide the 'c' number cleanly by the 'b' number. What about:
\[\frac{1}{2}\cdot\frac{5}{7}\]
That's going to leave two and a half on top of the right hand side fraction, which is a mess. How do we solve this? Just as with addition, we need to convert the top of the right hand number into something that the bottom of the left hand number can divide into. But, if we change either the top or bottom of a fraction, the other bit of it automatically changes as well. With addition the easiest way to get the numbers to fit together was to multiply them together. Same thing happens here. We need to multiply the 'c' number by the 'b' number. To maintain the ratio between the 'c' number and the 'd' number we also need to multiply the 'd' number by the 'b' number. So we get this in our example:
\[\frac{1}{2}\cdot\frac{5\cdot2}{7\cdot2}\]
\[\frac{1}{2}\cdot\frac{10}{14}\]
We can now happily divide ten by two to get five:
\[\frac{5}{14}\]
But hang on - the five is exactly the same as we started with. Why is that? Well remember that these are rational numbers. So if you double the number on the bottom without doubling the number on the top, the ratio is now half of what it used to be. If you triple the number on the bottom without changing the number on the top the ratio is now a third of the size it used to be. Remember I said that if you change either the top or bottom then the other part has to change automatically to remain the same? Well, here we do not WANT it to remain the same. So we can just multiply the bottom of the second fraction by the bottom of the first fraction and it will give us one b'th of the second fraction automatically.
Now we have one b'th of the second fraction, but we actually want 'a' number of b'th of the second fraction. Remember we simplified what we were going to do by ignoring the 'a'. Well that's easy to bring back in now. We know how big one b'th of the second fraction is, we just need 'a' number of them. So we just multiply the answer we got by 'a'. If you have one eighth and you multiply it by three you get three eighths. If you have two sevenths and you multiply it by two you get four sevenths. The eighth or seventh bit does not change, just the number of them. So all we do now is multiply the top number of the second fraction by 'a'. Remember the top number has not changed - we found out we could leave it as 'c'.
In summary then what we do is multiply the bottom of the second fraction by the bottom of the first, and then multiply the top of the second by the top of the first:
\[\frac{a}{b}\cdot\frac{c}{d}=a\cdot\frac{c}{b\cdot d}=\frac{a\cdot c}{b\cdot d}\]
What about division of fractions? This could be painful, really really painful. Think about this:
\[\frac{a}{b}/\frac{c}{d}=\frac{\frac{a}{b}}{\frac{c}{d}}\]
Eeeewwww. That looks messy. Remember our alternate rule about division? It is just multiplication by the inverse of the number you want to divide by? We know that the inverse of three is a third. Well the inverse of a third is three. So what is the inverse of two thirds? Do we (following what we did with fractions above) ask what is the inverse of one third (three) and then multiply that answer by two (getting six)? No, because we know what the inverse of six is (a sixth) and it isn't two thirds.
Remember that we said than an inverse was what happened when you asked a question like "how big is each piece if one is split up into so many equal pieces". That got us from three to the inverse of three being a third. We can also go in the other way. Starting with a third, we can get its inverse if we ask "what is one a third of?" The answer is obviously three, because there are three ones in three. How does this help with two thirds? The question we ask is "what number is one two thirds of?". The answer is one and a half, because there are three halves in one and a half, and two halves in one.
The question may be a little bit more tricky, like "what is the inverse of five eighths". So we ask "what number is one five eighths of?". Now just take one, and split it into five equal pieces. Each one will be a fifth in size. One is now the same as five fifths. The question is now "what number is (five fifths) five eighths of?" The answer has to be eight fifths. Five fifths is five eighths of eight fifths. Sounds confusing? Work backwards. Eight fifths are eight things. Five of them will be five eighths of the whole. But each one of the things is one fifth in size. Five fifths is also one. So the inverse of five eighths is eight fifths.
Hey there is a symmetry there! What about two thirds and one and a half? Well remember that one and a half is three halves. So the inverse of two thirds is three halves. That's symmetrical as well. The rule we have found is that the inverse of a fraction is just the fraction turned upside down. That fits with our earlier, simpler, inverse of whole numbers, because three is equal to three ones, flip the one and the three, and you get one third.
So, our horrendous expression:
\[\frac{a}{b}/\frac{c}{d}=\frac{\frac{a}{b}}{\frac{c}{d}}\]
can actually be written as:
\[\frac{a}{b}/\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}\]
because dividing by c number of d'ths is the same as multiplying by the inverse of c number of d'ths, which as we have just seen is d number of c'ths. Simple.
You can also do things with fractions as exponents, but I will leave that until next time.
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