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.
Monday, 25 April 2011
Monday, 18 April 2011
The Opposite of Multiplication
Following in from addition and subtraction, the opposite of multiplication is division. This is the process of taking a group of things and asking, if I have to separate these things into a certain number of equally sized groups, how many will be in each group. So if you have six apples and three hungry people, you would want to know how many apples each hungry person could have if they all had to get the same amount. This is the process of dividing six by thee. Each hungry person will get two apples.
Like subtraction and addition, we can treat the act of division as a special type of multiplication. But my feeling is that this gets hopelessly circular very quickly. The logic goes that if you divide a number by another number that is the same as multiplying the first number by the inverse of the second number.
Well, what the hell is an inverse? Good question. An inverse is the size of the individual pieces you get if you split the unit number (for us the number one remember) up into the same number of equal pieces as the original number. If you consider the apples and hungry people example, we are dividing six by three. The inverse of three is a third, because if you split the number one up into three equal pieces, each piece is a third in size. So instead of asking how many ones are in three, you ask the INVERSE of the question, which is how big is each piece if one is split up into three equal pieces.
What you then do, according to the multiplication theory of division, is multiply six by one third. Remember that multiplication is replicating one collection of things so that you end up with a second number of those collections. I did not say that you had to use WHOLE numbers.
So what is a whole number, and more importantly, what is NOT a whole number? As with negative numbers and addition, multiplying by an inverse forces us to consider another new type of number. A whole number is a number that is made up of ones and ones alone. It could have one one in it, or two, or three or fifteen billion, or negative twenty two, but it is only made up of ones. Another name for this type of number is an integer. If it is a positive integer (more than zero) then it is a natural number (but still an integer).
A number which is NOT a whole number is made up of PARTS OF ones. It may be made of some whole ones as well, but it will have parts of one in there too. So, the quantity two and a half is made up of two ones, and a half of one. The time a quarter past three is three hours (past midnight or midday) and a quarter part of an hour. The question is are these quantities or times really numbers? We quickly find ourselves back at the same kind of philosophical problem as we met at the negative number stage. Is it meaningless to take about half a kilogram of flour? No. Is it meaningless to talk about half an egg? Maybe - after all you would never go into a shop and try to buy three and a half eggs.
Well the answer, as you may have expected, is that we are quite comfortable now with the idea of these parts of one being numbers in their own right. They are just not whole numbers. They are also much less abstract than negative numbers. If you have half a cow, you can at least see and smell it, albeit the experience may not be a pleasant one.
So this part, or fraction, of one is a number is its own right. You can then multiply another number by that number. What does it mean to multiply a number by a fraction? Well remember that multiplying a group of things by a number means you end up with that number of groups of things. So all that happens if you multiply by a fraction is that you end up with a fraction of the original group. So if you have a group of twenty things, and you multiply that by a half, you end up with ten things. It still does not matter what order you do this in, so if you gather up twenty halves, and put them all together, each one finds a friend and you end up with ten whole things. In that example, of course, one half is the INVERSE of two, because there are two equal halves in one, rather than two ones in two.
So why is all this circular? Because we represent fractions as one number divided by another. So we show a half as one divided by two. So we are back to division already. Brilliant. So to be able to multiply by an inverse instead of dividing the first thing you have to do is divide to get the value of the inverse. Seemingly pointless. However it will be surprisingly useful when we come to try to multiply fractions together. Lets have a look at that kind of thing before we look at our last opposite operation.
Like subtraction and addition, we can treat the act of division as a special type of multiplication. But my feeling is that this gets hopelessly circular very quickly. The logic goes that if you divide a number by another number that is the same as multiplying the first number by the inverse of the second number.
Well, what the hell is an inverse? Good question. An inverse is the size of the individual pieces you get if you split the unit number (for us the number one remember) up into the same number of equal pieces as the original number. If you consider the apples and hungry people example, we are dividing six by three. The inverse of three is a third, because if you split the number one up into three equal pieces, each piece is a third in size. So instead of asking how many ones are in three, you ask the INVERSE of the question, which is how big is each piece if one is split up into three equal pieces.
What you then do, according to the multiplication theory of division, is multiply six by one third. Remember that multiplication is replicating one collection of things so that you end up with a second number of those collections. I did not say that you had to use WHOLE numbers.
So what is a whole number, and more importantly, what is NOT a whole number? As with negative numbers and addition, multiplying by an inverse forces us to consider another new type of number. A whole number is a number that is made up of ones and ones alone. It could have one one in it, or two, or three or fifteen billion, or negative twenty two, but it is only made up of ones. Another name for this type of number is an integer. If it is a positive integer (more than zero) then it is a natural number (but still an integer).
A number which is NOT a whole number is made up of PARTS OF ones. It may be made of some whole ones as well, but it will have parts of one in there too. So, the quantity two and a half is made up of two ones, and a half of one. The time a quarter past three is three hours (past midnight or midday) and a quarter part of an hour. The question is are these quantities or times really numbers? We quickly find ourselves back at the same kind of philosophical problem as we met at the negative number stage. Is it meaningless to take about half a kilogram of flour? No. Is it meaningless to talk about half an egg? Maybe - after all you would never go into a shop and try to buy three and a half eggs.
Well the answer, as you may have expected, is that we are quite comfortable now with the idea of these parts of one being numbers in their own right. They are just not whole numbers. They are also much less abstract than negative numbers. If you have half a cow, you can at least see and smell it, albeit the experience may not be a pleasant one.
So this part, or fraction, of one is a number is its own right. You can then multiply another number by that number. What does it mean to multiply a number by a fraction? Well remember that multiplying a group of things by a number means you end up with that number of groups of things. So all that happens if you multiply by a fraction is that you end up with a fraction of the original group. So if you have a group of twenty things, and you multiply that by a half, you end up with ten things. It still does not matter what order you do this in, so if you gather up twenty halves, and put them all together, each one finds a friend and you end up with ten whole things. In that example, of course, one half is the INVERSE of two, because there are two equal halves in one, rather than two ones in two.
So why is all this circular? Because we represent fractions as one number divided by another. So we show a half as one divided by two. So we are back to division already. Brilliant. So to be able to multiply by an inverse instead of dividing the first thing you have to do is divide to get the value of the inverse. Seemingly pointless. However it will be surprisingly useful when we come to try to multiply fractions together. Lets have a look at that kind of thing before we look at our last opposite operation.
Monday, 11 April 2011
The Opposite of Addition
Last time we looked at three mathematical operations, addition (combining two groups of things into one big thing and counting how many things you have in the big thing), multiplication (a special type of addition where you add together a certain number of equal groups of things) and raising to the power (a special type of multiplication where you multiply a number by itself a certain number of times).
Each of these operations has its opposite, and I want to say a few words about those. These do not arise in the formula that we are looking at, but they do crop up in some of the explanations of the individual components of the formula.
Firstly the opposite of addition is subtraction. You start with some number of things, remove a certain amount of them, and count how many are left. You have subtracted the certain amount from the number of things.
What you can also do is just consider this to be a special type of addition where you add a negative number to your starting number. At this point we encounter our first real philosophical problem. What is a negative number? We have been imagining lots of "things" in our discussions of addition, subtraction, and raising to the power. These things could have been anything - carrots, tractors, stamps, people, bacteria - it doesn't really matter. But you could picture the thing in your minds eye - you could touch it, look at it - it was real. We call these numbers of things, NATURAL numbers. But what can it possibly mean to talk about a negative tractor? What colour would it be?
Well, this is a thorny problem that ancient mathematicians dealt with by ignoring it and hoping it would go away. For instance the ancient Egyptian mathematicians said there was no sensible answer to the question "what number, if you multiply it by four and then add twenty, equals zero". Today we would say that the answer is negative five, because four lots of negative five make negative twenty, and twenty added to negative twenty makes zero. In the middle ages, Hindu and Muslim mathematicians worked out how to use negative numbers, primarily as a reflection of a debt. Meanwhile the western world was trying to work out how to stop the barbarians from clubbing them over the head and being eaten. As recently as the Eighteenth century (when America got its independence) mathematicians would ignore negative answers to problems because they were meaningless.
Nowadays we are relatively comfortable with the idea of negative numbers. If you gift me your only tractor for my birthday, then you have zero tractors. If, however, I borrow your only tractor you have a negative tractor because I am due to return your tractor to you.
So if you have five of something, and you add negative three somethings to it you end up with two of the somethings left.
There is an important difference between ACTUAL subtraction and adding a negative number. Remember that with addition, it does not matter which way round you do it. So, if you have four and you add negative three to that you get one. If you have negative three and add four to it you get one again. Just as we would have expected. BUT if you have four and SUBTRACT three you get one, WHILE if you have three and SUBTRACT four you get negative one. So watch with this - if subtracting you need to get the numbers in the right order, but if you just add using negative numbers it does not matter.
We use the symbol \(-\) to show the action of subtracting, and we put the same symbol to the left of a number to show that it is negative: \(-5\).
Each of these operations has its opposite, and I want to say a few words about those. These do not arise in the formula that we are looking at, but they do crop up in some of the explanations of the individual components of the formula.
Firstly the opposite of addition is subtraction. You start with some number of things, remove a certain amount of them, and count how many are left. You have subtracted the certain amount from the number of things.
What you can also do is just consider this to be a special type of addition where you add a negative number to your starting number. At this point we encounter our first real philosophical problem. What is a negative number? We have been imagining lots of "things" in our discussions of addition, subtraction, and raising to the power. These things could have been anything - carrots, tractors, stamps, people, bacteria - it doesn't really matter. But you could picture the thing in your minds eye - you could touch it, look at it - it was real. We call these numbers of things, NATURAL numbers. But what can it possibly mean to talk about a negative tractor? What colour would it be?
Well, this is a thorny problem that ancient mathematicians dealt with by ignoring it and hoping it would go away. For instance the ancient Egyptian mathematicians said there was no sensible answer to the question "what number, if you multiply it by four and then add twenty, equals zero". Today we would say that the answer is negative five, because four lots of negative five make negative twenty, and twenty added to negative twenty makes zero. In the middle ages, Hindu and Muslim mathematicians worked out how to use negative numbers, primarily as a reflection of a debt. Meanwhile the western world was trying to work out how to stop the barbarians from clubbing them over the head and being eaten. As recently as the Eighteenth century (when America got its independence) mathematicians would ignore negative answers to problems because they were meaningless.
Nowadays we are relatively comfortable with the idea of negative numbers. If you gift me your only tractor for my birthday, then you have zero tractors. If, however, I borrow your only tractor you have a negative tractor because I am due to return your tractor to you.
So if you have five of something, and you add negative three somethings to it you end up with two of the somethings left.
There is an important difference between ACTUAL subtraction and adding a negative number. Remember that with addition, it does not matter which way round you do it. So, if you have four and you add negative three to that you get one. If you have negative three and add four to it you get one again. Just as we would have expected. BUT if you have four and SUBTRACT three you get one, WHILE if you have three and SUBTRACT four you get negative one. So watch with this - if subtracting you need to get the numbers in the right order, but if you just add using negative numbers it does not matter.
We use the symbol \(-\) to show the action of subtracting, and we put the same symbol to the left of a number to show that it is negative: \(-5\).
Monday, 4 April 2011
Order of Operations, Part One
OK, before we go any further we need to think about the order that the operations we looked at are done in. What do we mean? Well one plus two can only be done two ways. You can add one to two, or you can add two to one. Either way it makes no difference.
Think about it, if you have a table with one apple towards the left hand side of the table and two to the right hand side, then you have three apples on the table. If you do not touch the apples, but instead walk around to the other side of the table, you still have three apples on the table, despite the fact that now you are looking at two on the left and one on the right. Changing your perspective doesn't change the number of apples on the table.
OK, so what about three multiplied by two plus one. Lets look at that. It sounds simple. Surely the answer is seven, because three multiplied by two is six, which if you add one is seven. But hang on. What if you metaphorically walk around to the other side of the table. Then you are looking at one plus two multiplied by three. That is surely nine, because one plus two is three, and three multiplied by three is nine.
Hmm. We have already seen it doesn't matter what order you multiply things in, or what order you add things in, but apparently it DOES matter what order you do both in. We had better agree on what we mean when we say "three multiplied by two plus one". Do we mean three groups of two objects, the total of which we add one to? Or do we mean two added to one three times?
Well, by convention we mean the first option. We do multiplications first. So you first look at which numbers need multiplied together, do that, and then do any additions. I say, by convention but it turns out if you accept a certain set of axioms (stuff you just accept to be true to get on with the argument, like Hitler was a bad guy, or Jar Jar Binks is an idiot), then you can actually prove that it has to be multiplication first. Once can read about this here, but when one understands that this was only proven 150 years ago, and for the two millennia before that mathematicians just had to get by with "its convention", then one can be quite comfortable not bothering with this proof.
What about the third operation - exponents? Now we are talking about what three to the power of two plus one amounts to. Is that three to the power of two (nine) plus one (ten)? Or is it one plus two (three), to which power you then raise three (twenty seven)? Conventionally we do the exponential bit first. So the answer is ten not twenty seven. Actually it is a bit easier with exponents than addition or multiplication, because the symbols give you the clues. With the superscript notation you would write \(3^2+1\) for the first example and \(3^{2+1}\) for the second. So the actual notation is different which was not the case with addition and multiplication. A final issue which will be important is what and exponent to an exponent is. Huh? Well, what about \(2^{2^{2}}\)? Actually thats quite easy if you just think it through. THe first \(2^2\) is just \(2\cdot 2\). And then the second, tiny, power of two multplies that by itself as well. So you get \((2\cdot 2)\cdot(2\cdot 2)\). And that is just \(2\cdot 2\cdot 2\cdot 2\). Which is just \(2^4\). THat's a general rule that works quite well. All it says is that if you have a number to an exponent, then you end up with the exponent number of the numbers multiplied together. If you then duplicate that group by however many times as the second, smaller, exponent, then you are really just adding more and more of the number being multiplied. This means that you can just multiply the two exponents together to get one big exponent.
If you really want to do something out of order then you put brackets round it. So if we said three added to one multiplied by two, and we actually meant take three things, add one to them to make four and then double it, then you would say (three plus one) multiplied by two. That would look like \((3+1)\cdot 2\). Anything in brackets we do first. Even before exponents. So the system goes brackets -> exponents -> multiplication -> addition.
One more thing about brackets. Some of the stuff we are going to look at has stuff in a bracket multiplied by stuff in ANOTHER bracket. We may not want that, and we need a way to get rid of the brackets. How do we do that? Back to the fruit. Let's say that you have baskets of apples and oranges. You have the same number of apples and the same number of oranges in each basket. We are saying that you are a greengrocer with OCD, but just run with it. Let's also say that you have red baskets and blue baskets. It doesn't matter what colour the baskets are, they still have the same number of apples and the same number of oranges in them.
So how do we work out how many pieces of fruit you have in total? Firstly, you could just tip all the fruit out into a big pile and count them. Sensible enough approach, because it is going to get you the answer eventually. Lets call that the brute force approach. It may not be wise though because, having OCD, you are going to have to put all the fruit back into the baskets afterwards. A more efficient approach would be to count up how many apples and oranges you have in each basket. You then count up how many red baskets, and how many blue baskets you have. You can then ask, how many apples in blue baskets do I have, how many oranges in blue baskets do I have, how many apples in red baskets do I have and how many oranges in red baskets do I have? You would write that down like this:
\[\begin{multline}
blue baskets\cdot apples + blue baskets\cdot oranges \\
+ red baskets\cdot apples + red baskets\cdot oranges
\end{multline}
\]
That is four multiplications and three additions, which is quite a lot. If you think about it, we are just interested in the total numbers of bits of fruit, we are not bothered about the differences between apples and oranges. So we could just say, how many bits of fruit in the blue baskets and how many in the red baskets? The total number of bits of fruit in each basket is just the number of apples plus the number of oranges:
\[(apples + oranges)\]
So we could replace our previous question with this:
\[blue baskets\cdot (apples + oranges) + red baskets\cdot (apples + oranges)\]
That is still going to give us the same answer, which is the total number of bits of fruit. In fact, going further we could say, we don't need to know about how many bits of fruit are in red baskets and how many are in blue baskets - we just want the total number of bits of fruit in ALL the baskets. To do that we would need to work out the total number of baskets we have, which is:
\[(blue baskets + red baskets)\]
You would then just multiply that with the statement of the total number of bits of fruit in each basket:
\[(blue baskets + red baskets)\cdot (apples + oranges)\]
All you have to do now is two additions and one multiplication, which is much easier. The baskets you have and the total number of bits of fruit in them have not changed at all in this process. We have actually gone through this process in reverse. What we wanted to find out was how to multiply two brackets together. What we have done though is end up with two brackets. The good news is that to get the answer we wanted, we just run that process in reverse. So if you ever have two items in one bracket, and two in another bracket, and the brackets have to be multiplied, you can see that you just need to take the first item in the first bracket and multiply it by each of the items in the second bracket and add the answers. Then you take the second item in the first bracket and multiply it by each of the items in the second backet, and add the answers. Then you add both sets of answers, and hey presto, the brackets have gone.
Think about it, if you have a table with one apple towards the left hand side of the table and two to the right hand side, then you have three apples on the table. If you do not touch the apples, but instead walk around to the other side of the table, you still have three apples on the table, despite the fact that now you are looking at two on the left and one on the right. Changing your perspective doesn't change the number of apples on the table.
OK, so what about three multiplied by two plus one. Lets look at that. It sounds simple. Surely the answer is seven, because three multiplied by two is six, which if you add one is seven. But hang on. What if you metaphorically walk around to the other side of the table. Then you are looking at one plus two multiplied by three. That is surely nine, because one plus two is three, and three multiplied by three is nine.
Hmm. We have already seen it doesn't matter what order you multiply things in, or what order you add things in, but apparently it DOES matter what order you do both in. We had better agree on what we mean when we say "three multiplied by two plus one". Do we mean three groups of two objects, the total of which we add one to? Or do we mean two added to one three times?
Well, by convention we mean the first option. We do multiplications first. So you first look at which numbers need multiplied together, do that, and then do any additions. I say, by convention but it turns out if you accept a certain set of axioms (stuff you just accept to be true to get on with the argument, like Hitler was a bad guy, or Jar Jar Binks is an idiot), then you can actually prove that it has to be multiplication first. Once can read about this here, but when one understands that this was only proven 150 years ago, and for the two millennia before that mathematicians just had to get by with "its convention", then one can be quite comfortable not bothering with this proof.
What about the third operation - exponents? Now we are talking about what three to the power of two plus one amounts to. Is that three to the power of two (nine) plus one (ten)? Or is it one plus two (three), to which power you then raise three (twenty seven)? Conventionally we do the exponential bit first. So the answer is ten not twenty seven. Actually it is a bit easier with exponents than addition or multiplication, because the symbols give you the clues. With the superscript notation you would write \(3^2+1\) for the first example and \(3^{2+1}\) for the second. So the actual notation is different which was not the case with addition and multiplication. A final issue which will be important is what and exponent to an exponent is. Huh? Well, what about \(2^{2^{2}}\)? Actually thats quite easy if you just think it through. THe first \(2^2\) is just \(2\cdot 2\). And then the second, tiny, power of two multplies that by itself as well. So you get \((2\cdot 2)\cdot(2\cdot 2)\). And that is just \(2\cdot 2\cdot 2\cdot 2\). Which is just \(2^4\). THat's a general rule that works quite well. All it says is that if you have a number to an exponent, then you end up with the exponent number of the numbers multiplied together. If you then duplicate that group by however many times as the second, smaller, exponent, then you are really just adding more and more of the number being multiplied. This means that you can just multiply the two exponents together to get one big exponent.
If you really want to do something out of order then you put brackets round it. So if we said three added to one multiplied by two, and we actually meant take three things, add one to them to make four and then double it, then you would say (three plus one) multiplied by two. That would look like \((3+1)\cdot 2\). Anything in brackets we do first. Even before exponents. So the system goes brackets -> exponents -> multiplication -> addition.
One more thing about brackets. Some of the stuff we are going to look at has stuff in a bracket multiplied by stuff in ANOTHER bracket. We may not want that, and we need a way to get rid of the brackets. How do we do that? Back to the fruit. Let's say that you have baskets of apples and oranges. You have the same number of apples and the same number of oranges in each basket. We are saying that you are a greengrocer with OCD, but just run with it. Let's also say that you have red baskets and blue baskets. It doesn't matter what colour the baskets are, they still have the same number of apples and the same number of oranges in them.
So how do we work out how many pieces of fruit you have in total? Firstly, you could just tip all the fruit out into a big pile and count them. Sensible enough approach, because it is going to get you the answer eventually. Lets call that the brute force approach. It may not be wise though because, having OCD, you are going to have to put all the fruit back into the baskets afterwards. A more efficient approach would be to count up how many apples and oranges you have in each basket. You then count up how many red baskets, and how many blue baskets you have. You can then ask, how many apples in blue baskets do I have, how many oranges in blue baskets do I have, how many apples in red baskets do I have and how many oranges in red baskets do I have? You would write that down like this:
\[\begin{multline}
blue baskets\cdot apples + blue baskets\cdot oranges \\
+ red baskets\cdot apples + red baskets\cdot oranges
\end{multline}
\]
That is four multiplications and three additions, which is quite a lot. If you think about it, we are just interested in the total numbers of bits of fruit, we are not bothered about the differences between apples and oranges. So we could just say, how many bits of fruit in the blue baskets and how many in the red baskets? The total number of bits of fruit in each basket is just the number of apples plus the number of oranges:
\[(apples + oranges)\]
So we could replace our previous question with this:
\[blue baskets\cdot (apples + oranges) + red baskets\cdot (apples + oranges)\]
That is still going to give us the same answer, which is the total number of bits of fruit. In fact, going further we could say, we don't need to know about how many bits of fruit are in red baskets and how many are in blue baskets - we just want the total number of bits of fruit in ALL the baskets. To do that we would need to work out the total number of baskets we have, which is:
\[(blue baskets + red baskets)\]
You would then just multiply that with the statement of the total number of bits of fruit in each basket:
\[(blue baskets + red baskets)\cdot (apples + oranges)\]
All you have to do now is two additions and one multiplication, which is much easier. The baskets you have and the total number of bits of fruit in them have not changed at all in this process. We have actually gone through this process in reverse. What we wanted to find out was how to multiply two brackets together. What we have done though is end up with two brackets. The good news is that to get the answer we wanted, we just run that process in reverse. So if you ever have two items in one bracket, and two in another bracket, and the brackets have to be multiplied, you can see that you just need to take the first item in the first bracket and multiply it by each of the items in the second bracket and add the answers. Then you take the second item in the first bracket and multiply it by each of the items in the second backet, and add the answers. Then you add both sets of answers, and hey presto, the brackets have gone.
Monday, 28 March 2011
What are Mathematical Operations?
For our restricted purposes here, a mathematical operation is some way of combining two numbers to get a third number. The third number you get will depend on the type of operation you use. For consideration of our formula, we need to look at addition, multiplication and raising to the power.
Addition is pretty much the easiest thing you can do if you are given two numbers. If you are given three apples (for some reason explanations like this always involve fruit) and then four more apples, then how many apples do you have in total? The process you go through to answer that question is called addition. (The answer is seven apples).
So basically, if you have two or more groups of things, then to work out how many things there are in all the groups combined you add them together. Simples.
We represent the process of adding two numbers by putting the symbol [+] between them.
What is Multiplication?
Multiplication is a special type of addition. If you have two numbers, then if you multiply them together it means that you take the first number and duplicate it so that you end up with the second number of individual instances of the first number. You then add up the results. So 3 multiplied by 4 is 3+3+3+3 - notice there are 4 threes in that list. It doesn't matter which way round you do the operation. So 4+4+4 is the same as 3+3+3+3. That is pretty much all there is to multiplication for our purposes.
The representation of multiplying two numbers is a bit trickier than adding, where the symbol is [+]. We are first taught in school to use [x] to show we are multiplying. So three multiplied by four is 3x4. That's the symbol you will find on your calculator to multiply. This is fine, until we get to algebra. That is a subject for another day, but suffice to say a big part of it is using variables to represent unknown numbers. Now these variables could be represented by a pictogram of a gerbil, or a house, or a dragon or anything you like really. Guess what one of the most common symbols chosen to represent these variables is? Yes, [x]. So if you were multiplying [x] by three you would find yourself writing 3xx. Not ideal. In fact far from ideal. So, there are two other ways to symbolise multiplication. The first is to put a dot between the two numbers (or variables) like this \(3\cdot x\). The second is just to forget about a symbol altogether giving 3x. Obviously the latter option cannot be used for two numbers on their own because we would not be able to tell the difference between the number 34 and three times four.
What is raising to the power?
This is a special type of multiplication, almost in the same way as multiplication is a special type of addition. Again you are given two numbers. This time you gather up a second number of first numbers and then multiply them together rather than adding. So two to the power of two is 2x2 or 2+2. Two to the power of 3 is 2x2x2, or (2+2)+(2+2). The brackets there do not do anything mathematically, they are just there to show the two sets of (2+2) that are being added together. The change from power of two in the last example to power of three in this example just means multiple everything by two again. So given that two to the power of two is (2+2), two to the power of three just means double that, or (2+2)+(2+2). So you have two groups of two groups of twos.
In the same way three to the power of four is 3x3x3x3, or ((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3)). I have used double brackets to show that there are three groups of three groups of threes. If that looks a bit complicated, consider this:
three to the power of one = 3 (you only have one number so you cannot do anything with it, it just sits there)
three to the power of two = three copies of the last answer = 3+3+3
three to the power of three = three copies of the last answer = (3+3+3)+(3+3+3)+(3+3+3)
three to the power of four = three copies of the last answer = ((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3))
An important point to note here is that we are just adding up to get the answer because 'powering' a number is a type of multiplication, which in turn is just a type of addition.
One difference between powering and multiplication is that it DOES matter which number does what. So while 2x3 (3+3) = 3x2 (2+2+2), two to the power of three, 2x2x2=(2+2+2+2)=8, DOES NOT equal three to the power two, 3x3=(3+3+3)=9.
Using a computer keyboard the symbol for powering numbers is [^], so 2^3 is two to the power of three. Normally though we do not use that symbol, or indeed any symbol. Instead we just write the number to be powered first, then the value of the power second. To distinguish this from multiplying (where we also usually do not use a symbol) we write the second number in superscript (which means a bit smaller and higher up). So two to the power of three is 2^3 or \(2^3\). We also call the small number, the power to which we are raising, the exponent. From this we get phrases like "exponential increase".
If you raise the number to the power of two, that is known as 'squaring' the number. This is because to find the area of a square you multiply the length of the sides by itself. If you raise a number to the power of three that is known as "cubing" the number because, similarly, you would need to multiply together three lengths of the side of the cube to get the volume of it. We really stop using special numbers after squares and cubes because our brains do not work in the fourth dimension.
We need to say a quick word about something that is not immediately obvious. One plus zero, is one. If you have something and don't add anything you have the thing you started with. Two multiplied by zero is zero. If you want zero individual instances of two, then when you come to add them up you will get nothing. So what is three to the power of zero? Is it three, or is it zero? Well, it isn't three, because as we have seen that is three to the power of one. Can it be zero? Hmmm.
Lets look at three to the power of three. That is three multiplied together three times or three multiplied by three multiplied by three. What is three to the power of one? Three. And what is three to the power of two? Three multiplied by three. So what is (three to the power of one) MULTIPLIED by (three to the power of two)? Well that is (three) MULTIPLIED by (three multiplied by three). Or just three multiplied by three multiplied by three. So we can say that:
\[3^3=3^2\cdot3^1=3^{2+1}\]
This is going to work for whatever numbers we choose, because raising to the power is just telling us how many of the chosen numbers we are multiplying together. We still end up just doing a multiplication. We have seen that it doesn't matter which order we do our multiplication in - and certainly not if it is just the same number being multiplied together again and again. So if we have a much bigger number like five to the power of six, that is just six fives in a row getting multiplied together. You could also look at it as a group of four fives multiplied together and then multiplied by a group of two fives multiplied together. It really doesn't matter where you draw the lines.
\[5\cdot5\cdot5\cdot5\cdot5\cdot5=(5\cdot5\cdot5)(5\cdot5\cdot5)=(5\cdot5)(5\cdot5\cdot5\cdot5)\]
Good. Makes sense. So what is three to the power of zero? Well if we want our rule to hold, that you just add the exponents, then the answer has to be something that satisfies this formula:
\[3^3=3^{3+0}=3^3\cdot3^0\]
If we make three to the power zero equal to zero then that doesn't work. The bit at the end will break because you are multiplying something BY zero, which makes the whole thing zero. So what we want three to the power of zero to be is a number that DOES NOT change other numbers when it multiplies them. What number is that? One. One times anything is just the anything you started with. So:
\[3^3=3^{3+0}=3^3\cdot3^0=3^3\cdot1=3^3\]
We can safely conclude then that anything to the power of zero is the number one. Well, OK, maybe it wasn't that QUICK a word.
So, in our formula, we have addition when we add one:
\[+1\]
We have multiplication where we multiply \(\pi\) by \(i\) (we will come on to what these symbols mean later):
\[\pi i\]
And we then raise \(e\) to the power of \(\pi i\):
\[e^{\pi i}\]
Addition is pretty much the easiest thing you can do if you are given two numbers. If you are given three apples (for some reason explanations like this always involve fruit) and then four more apples, then how many apples do you have in total? The process you go through to answer that question is called addition. (The answer is seven apples).
So basically, if you have two or more groups of things, then to work out how many things there are in all the groups combined you add them together. Simples.
We represent the process of adding two numbers by putting the symbol [+] between them.
What is Multiplication?
Multiplication is a special type of addition. If you have two numbers, then if you multiply them together it means that you take the first number and duplicate it so that you end up with the second number of individual instances of the first number. You then add up the results. So 3 multiplied by 4 is 3+3+3+3 - notice there are 4 threes in that list. It doesn't matter which way round you do the operation. So 4+4+4 is the same as 3+3+3+3. That is pretty much all there is to multiplication for our purposes.
The representation of multiplying two numbers is a bit trickier than adding, where the symbol is [+]. We are first taught in school to use [x] to show we are multiplying. So three multiplied by four is 3x4. That's the symbol you will find on your calculator to multiply. This is fine, until we get to algebra. That is a subject for another day, but suffice to say a big part of it is using variables to represent unknown numbers. Now these variables could be represented by a pictogram of a gerbil, or a house, or a dragon or anything you like really. Guess what one of the most common symbols chosen to represent these variables is? Yes, [x]. So if you were multiplying [x] by three you would find yourself writing 3xx. Not ideal. In fact far from ideal. So, there are two other ways to symbolise multiplication. The first is to put a dot between the two numbers (or variables) like this \(3\cdot x\). The second is just to forget about a symbol altogether giving 3x. Obviously the latter option cannot be used for two numbers on their own because we would not be able to tell the difference between the number 34 and three times four.
What is raising to the power?
This is a special type of multiplication, almost in the same way as multiplication is a special type of addition. Again you are given two numbers. This time you gather up a second number of first numbers and then multiply them together rather than adding. So two to the power of two is 2x2 or 2+2. Two to the power of 3 is 2x2x2, or (2+2)+(2+2). The brackets there do not do anything mathematically, they are just there to show the two sets of (2+2) that are being added together. The change from power of two in the last example to power of three in this example just means multiple everything by two again. So given that two to the power of two is (2+2), two to the power of three just means double that, or (2+2)+(2+2). So you have two groups of two groups of twos.
In the same way three to the power of four is 3x3x3x3, or ((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3)). I have used double brackets to show that there are three groups of three groups of threes. If that looks a bit complicated, consider this:
three to the power of one = 3 (you only have one number so you cannot do anything with it, it just sits there)
three to the power of two = three copies of the last answer = 3+3+3
three to the power of three = three copies of the last answer = (3+3+3)+(3+3+3)+(3+3+3)
three to the power of four = three copies of the last answer = ((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3))+((3+3+3)+(3+3+3)+(3+3+3))
An important point to note here is that we are just adding up to get the answer because 'powering' a number is a type of multiplication, which in turn is just a type of addition.
One difference between powering and multiplication is that it DOES matter which number does what. So while 2x3 (3+3) = 3x2 (2+2+2), two to the power of three, 2x2x2=(2+2+2+2)=8, DOES NOT equal three to the power two, 3x3=(3+3+3)=9.
Using a computer keyboard the symbol for powering numbers is [^], so 2^3 is two to the power of three. Normally though we do not use that symbol, or indeed any symbol. Instead we just write the number to be powered first, then the value of the power second. To distinguish this from multiplying (where we also usually do not use a symbol) we write the second number in superscript (which means a bit smaller and higher up). So two to the power of three is 2^3 or \(2^3\). We also call the small number, the power to which we are raising, the exponent. From this we get phrases like "exponential increase".
If you raise the number to the power of two, that is known as 'squaring' the number. This is because to find the area of a square you multiply the length of the sides by itself. If you raise a number to the power of three that is known as "cubing" the number because, similarly, you would need to multiply together three lengths of the side of the cube to get the volume of it. We really stop using special numbers after squares and cubes because our brains do not work in the fourth dimension.
We need to say a quick word about something that is not immediately obvious. One plus zero, is one. If you have something and don't add anything you have the thing you started with. Two multiplied by zero is zero. If you want zero individual instances of two, then when you come to add them up you will get nothing. So what is three to the power of zero? Is it three, or is it zero? Well, it isn't three, because as we have seen that is three to the power of one. Can it be zero? Hmmm.
Lets look at three to the power of three. That is three multiplied together three times or three multiplied by three multiplied by three. What is three to the power of one? Three. And what is three to the power of two? Three multiplied by three. So what is (three to the power of one) MULTIPLIED by (three to the power of two)? Well that is (three) MULTIPLIED by (three multiplied by three). Or just three multiplied by three multiplied by three. So we can say that:
\[3^3=3^2\cdot3^1=3^{2+1}\]
This is going to work for whatever numbers we choose, because raising to the power is just telling us how many of the chosen numbers we are multiplying together. We still end up just doing a multiplication. We have seen that it doesn't matter which order we do our multiplication in - and certainly not if it is just the same number being multiplied together again and again. So if we have a much bigger number like five to the power of six, that is just six fives in a row getting multiplied together. You could also look at it as a group of four fives multiplied together and then multiplied by a group of two fives multiplied together. It really doesn't matter where you draw the lines.
\[5\cdot5\cdot5\cdot5\cdot5\cdot5=(5\cdot5\cdot5)(5\cdot5\cdot5)=(5\cdot5)(5\cdot5\cdot5\cdot5)\]
Good. Makes sense. So what is three to the power of zero? Well if we want our rule to hold, that you just add the exponents, then the answer has to be something that satisfies this formula:
\[3^3=3^{3+0}=3^3\cdot3^0\]
If we make three to the power zero equal to zero then that doesn't work. The bit at the end will break because you are multiplying something BY zero, which makes the whole thing zero. So what we want three to the power of zero to be is a number that DOES NOT change other numbers when it multiplies them. What number is that? One. One times anything is just the anything you started with. So:
\[3^3=3^{3+0}=3^3\cdot3^0=3^3\cdot1=3^3\]
We can safely conclude then that anything to the power of zero is the number one. Well, OK, maybe it wasn't that QUICK a word.
So, in our formula, we have addition when we add one:
\[+1\]
We have multiplication where we multiply \(\pi\) by \(i\) (we will come on to what these symbols mean later):
\[\pi i\]
And we then raise \(e\) to the power of \(\pi i\):
\[e^{\pi i}\]
Monday, 21 March 2011
What is Zero?
If you look at Roman Numerals, i.e. I for 1, II for 2, V for 5, VIII for 8, X for 10 and so on, you should spot two things. One thing is easy to spot, and the other is quite hard.
It is easy to spot that the complexity of Roman Numerals is not obviously connected to their size. For instance 37 is XXXVII, while 52 is LII. It is quite hard to spot that this is because their system does not place varying values on the symbols (I, X, L, V etc) depending on where they fall in the sequence. Instead the value is always the same for each symbol, X always is 10 and L is always 50.
The whole number being written is calculated by applying a sequence of rules, which basically says, read from left to right adding each symbol's value unless the symbol to the right of it is greater than it, in which case you deduct its value from that symbol. So, XI is 10+1=11 and IX is 10-1 which is 9. Taken a little further, 1998 is MCMXCVIII which is M(1000)+CM(1000-100)+XC(100-10)+V(5)+I(1)+I(1)+I(1). What is important to note is that the two [C] symbols BOTH mean 100 despite them falling in different positions.
Our system of numerals is different. We also use different symbols to represent the constituent parts of big numbers. So far, so Roman. What is different is that the symbols have no relation to each other but their value depends on where in the sequence they fall. In one sense we could be said to read our sequence of symbols from right to left. The right most symbol (if we are dealing with whole numbers only) tells us how many one's are in the total number. The next rightmost symbol tells us how many ten's, the next how many hundred's and so on. So if we see 123, we quickly add 3 one's, two ten's and one hundred.
If it seems odd that we read the number from right to left, remember it is done subconsciously. If read from left to right, one can not know what the first symbol in the number represents (be it millions, hundreds of thousands or so on) until you have counted how MANY symbols are in the number. So logically you start at the right, where you KNOW the symbol means one's, and then add up moving from right to left. In reality though, it doesn't really matter which direction you read the symbols in, as long as you know what value each symbol position has in advance. This is unlike Roman Numerals, where the end result would be very different if you started at one end rather than the other - see XI and IX for example. Also, note that if you have the number 144, the two [4] symbols mean different things. The middle one means 4x10 or 40, and the right most one means 4x1 or 4. So the symbols increase in relative value the further left you go because they are multiplied by larger and larger sums. So the complexity of our numbers is directly proportionate to their numerical size. 237 is a bigger sequence of symbols than 84 and so on. The Romans were no thickies, built loads of stuff, and conquered the majority of their world (just don't tell the Chinese), so why were they using this cumbersome system where thirty seven takes longer to write than fifty two, and both are longer than one thousand and one?
The answer is rather profound. If we want to write the number one hundred and nine, then we start with the symbol [9] at the right, followed by the symbol [0] to show there are no tens, followed by the symbol [1] to show there is one hundred: 109. The Romans could not do this because they had no symbol which meant nothing. They had I, V, X, L, C, D, and M, each of which represents a specific number, but they had nothing which represented zero, because they did not consider that to BE a number. For us it would be like asking "What is the wavelength of black light?" (I do not mean the purpley ultraviolet stuff that shows up washing powder residue and dandruff with equal aplomb). Black light does not have a wavelength because there are no waves, it is the absence of light. Or it would be like asking "How does the sound of a violin not playing differ from the sound of a trumpet not playing?". It is a meaningless question. The absence of something does not need to be measured, and we do not need a symbol to represent it.
You may say that they could represent nothing by just not writing anything, which makes a sort of sense, but the problem is then distinguishing between 10 and 100, or 11, 101 and 1001. No, for a system of numbers based on symbols having different values depending on their position you need a clear symbol to represent "none of this value needed". A Roman would be puzzled by this; it would be like making up a shopping list with one amphora of wine, three buckets of milk, two handfuls of berries, no apples, and five bags of nuts. Why do you even bother to mention the apples if you want none of them?
So we are forced to have the symbol [0] because of the simpler way we write out numbers. But just the fact that we got [0] makes us think differently about the concept. Once it is a symbol, is it a number? I do not propose to try to answer that, but I will say that we can sometimes use it like a number. We can add it to other numbers (we get the number we started with), we can multiply other numbers by it (we get zero every time), we can even multiply other numbers by themselves [0] times - meaning raise them to the power 0 - (oddly we get one and not zero every time). We cannot divide numbers by zero. That breaks things.
For our present purposes then, [0] is as fundamental as [1].
It is easy to spot that the complexity of Roman Numerals is not obviously connected to their size. For instance 37 is XXXVII, while 52 is LII. It is quite hard to spot that this is because their system does not place varying values on the symbols (I, X, L, V etc) depending on where they fall in the sequence. Instead the value is always the same for each symbol, X always is 10 and L is always 50.
The whole number being written is calculated by applying a sequence of rules, which basically says, read from left to right adding each symbol's value unless the symbol to the right of it is greater than it, in which case you deduct its value from that symbol. So, XI is 10+1=11 and IX is 10-1 which is 9. Taken a little further, 1998 is MCMXCVIII which is M(1000)+CM(1000-100)+XC(100-10)+V(5)+I(1)+I(1)+I(1). What is important to note is that the two [C] symbols BOTH mean 100 despite them falling in different positions.
Our system of numerals is different. We also use different symbols to represent the constituent parts of big numbers. So far, so Roman. What is different is that the symbols have no relation to each other but their value depends on where in the sequence they fall. In one sense we could be said to read our sequence of symbols from right to left. The right most symbol (if we are dealing with whole numbers only) tells us how many one's are in the total number. The next rightmost symbol tells us how many ten's, the next how many hundred's and so on. So if we see 123, we quickly add 3 one's, two ten's and one hundred.
If it seems odd that we read the number from right to left, remember it is done subconsciously. If read from left to right, one can not know what the first symbol in the number represents (be it millions, hundreds of thousands or so on) until you have counted how MANY symbols are in the number. So logically you start at the right, where you KNOW the symbol means one's, and then add up moving from right to left. In reality though, it doesn't really matter which direction you read the symbols in, as long as you know what value each symbol position has in advance. This is unlike Roman Numerals, where the end result would be very different if you started at one end rather than the other - see XI and IX for example. Also, note that if you have the number 144, the two [4] symbols mean different things. The middle one means 4x10 or 40, and the right most one means 4x1 or 4. So the symbols increase in relative value the further left you go because they are multiplied by larger and larger sums. So the complexity of our numbers is directly proportionate to their numerical size. 237 is a bigger sequence of symbols than 84 and so on. The Romans were no thickies, built loads of stuff, and conquered the majority of their world (just don't tell the Chinese), so why were they using this cumbersome system where thirty seven takes longer to write than fifty two, and both are longer than one thousand and one?
The answer is rather profound. If we want to write the number one hundred and nine, then we start with the symbol [9] at the right, followed by the symbol [0] to show there are no tens, followed by the symbol [1] to show there is one hundred: 109. The Romans could not do this because they had no symbol which meant nothing. They had I, V, X, L, C, D, and M, each of which represents a specific number, but they had nothing which represented zero, because they did not consider that to BE a number. For us it would be like asking "What is the wavelength of black light?" (I do not mean the purpley ultraviolet stuff that shows up washing powder residue and dandruff with equal aplomb). Black light does not have a wavelength because there are no waves, it is the absence of light. Or it would be like asking "How does the sound of a violin not playing differ from the sound of a trumpet not playing?". It is a meaningless question. The absence of something does not need to be measured, and we do not need a symbol to represent it.
You may say that they could represent nothing by just not writing anything, which makes a sort of sense, but the problem is then distinguishing between 10 and 100, or 11, 101 and 1001. No, for a system of numbers based on symbols having different values depending on their position you need a clear symbol to represent "none of this value needed". A Roman would be puzzled by this; it would be like making up a shopping list with one amphora of wine, three buckets of milk, two handfuls of berries, no apples, and five bags of nuts. Why do you even bother to mention the apples if you want none of them?
So we are forced to have the symbol [0] because of the simpler way we write out numbers. But just the fact that we got [0] makes us think differently about the concept. Once it is a symbol, is it a number? I do not propose to try to answer that, but I will say that we can sometimes use it like a number. We can add it to other numbers (we get the number we started with), we can multiply other numbers by it (we get zero every time), we can even multiply other numbers by themselves [0] times - meaning raise them to the power 0 - (oddly we get one and not zero every time). We cannot divide numbers by zero. That breaks things.
For our present purposes then, [0] is as fundamental as [1].
Monday, 14 March 2011
What is One?
Let's start at the very beginning. What is '1'? The obvious answer is that it is the first number. But is it? (See '0' next). Or is it even a number? Y'see every other number you can count on your fingers, toes, abacus etc etc, is made up of lots of '1's. So there is a school of thought that '1' isn't actually a number, but is instead the building blocks of all numbers.
One can now wander off down a long and winding path that is a career in Number Theory. Note that that is a sensible academic discipline which attracts seriously minded professional people dedicated to deep study and contemplation. Numerology is NOT to be confused with Number Theory. THAT is a nutty mystical system (in the loosest possible sense) of superstitions which attracts people with serious mental health issues dedicated to deep confusion and picking lucky lottery numbers.
Let us not wander down that path though. For our purposes here, let us instead contemplate 1 as a unit of something. The concept of a unit of something is a tricky one, and one which was never satisfactorily explained to me in school. For length, for instance, you have to decide what "units" you are working in - which is not quite the same as saying [imperial] or [metric]. No, the units will be miles or kilometres, yards or metres, inches or centimetre and so on. You need to get this bit right so that if you measure length 2 and add a length 3, what you get is a total length of 5. If you add 2 metres to 3 miles you do not get 5 of anything, you get a mess, and your probe fails to enter orbit. As I said though, this is not just imperial versus metric. If you add 2 metres to 3 kilometres, again you do not end up with 5 of anything.
So when you are deciding on which units you are using to measure you are really deciding an absolute reference length. ALL other lengths you are going to be measuring are then multiples (or maybe fractions) of that reference length. That means that you can do normal maths on these lengths - addition or multiplication for instance - and you will get sensible answers WHICH ARE themselves multiples or fractions of the SAME reference length.
I remember being very confused in school trying to answer the question "How many square centimetres are in a cubic centimetre?". I kept imagining pieces of paper cut into squares a centimetre on each side and then stacked one on top of the other, and being confused because I could not work out how many would fit to a cubic centimetre. I now know that I was confused about this because I was dealing with two completely different units. A cubic centimetre is a unit in its own right, and is not some multiple of units of square centimetres. There is no exchange rate. A cubic centimetre is a unit of volume and represents a cube (funny that) with sides which are one centimetre in length. A square centimetre is a unit of area and is a square with sides which are one centimetre in length. No one had properly explained the concept of units to me.
So 1 can be seen as the ultimate unit. Even centimetres can be broken down into smaller divisions. For instance a centimetre is actually one hundredth of a metre, which in turn is actually the distance travelled by light in a vacuum in roughly one three hundred thousandth of a second. But 1 does not need to be broken down any more, nor can it be. It is the absolute basic unit, from which everything else is referenced. So when I say I am adding 83 to 32, I actually MEAN I am adding 83 ones to 32 ones.
At a very basic level, and I LIKE very basic levels, 1 is the basic unit which every other thing is measured relative to. It therefore represents the difference between something and nothing. Which takes us neatly on to...
One can now wander off down a long and winding path that is a career in Number Theory. Note that that is a sensible academic discipline which attracts seriously minded professional people dedicated to deep study and contemplation. Numerology is NOT to be confused with Number Theory. THAT is a nutty mystical system (in the loosest possible sense) of superstitions which attracts people with serious mental health issues dedicated to deep confusion and picking lucky lottery numbers.
Let us not wander down that path though. For our purposes here, let us instead contemplate 1 as a unit of something. The concept of a unit of something is a tricky one, and one which was never satisfactorily explained to me in school. For length, for instance, you have to decide what "units" you are working in - which is not quite the same as saying [imperial] or [metric]. No, the units will be miles or kilometres, yards or metres, inches or centimetre and so on. You need to get this bit right so that if you measure length 2 and add a length 3, what you get is a total length of 5. If you add 2 metres to 3 miles you do not get 5 of anything, you get a mess, and your probe fails to enter orbit. As I said though, this is not just imperial versus metric. If you add 2 metres to 3 kilometres, again you do not end up with 5 of anything.
So when you are deciding on which units you are using to measure you are really deciding an absolute reference length. ALL other lengths you are going to be measuring are then multiples (or maybe fractions) of that reference length. That means that you can do normal maths on these lengths - addition or multiplication for instance - and you will get sensible answers WHICH ARE themselves multiples or fractions of the SAME reference length.
I remember being very confused in school trying to answer the question "How many square centimetres are in a cubic centimetre?". I kept imagining pieces of paper cut into squares a centimetre on each side and then stacked one on top of the other, and being confused because I could not work out how many would fit to a cubic centimetre. I now know that I was confused about this because I was dealing with two completely different units. A cubic centimetre is a unit in its own right, and is not some multiple of units of square centimetres. There is no exchange rate. A cubic centimetre is a unit of volume and represents a cube (funny that) with sides which are one centimetre in length. A square centimetre is a unit of area and is a square with sides which are one centimetre in length. No one had properly explained the concept of units to me.
So 1 can be seen as the ultimate unit. Even centimetres can be broken down into smaller divisions. For instance a centimetre is actually one hundredth of a metre, which in turn is actually the distance travelled by light in a vacuum in roughly one three hundred thousandth of a second. But 1 does not need to be broken down any more, nor can it be. It is the absolute basic unit, from which everything else is referenced. So when I say I am adding 83 to 32, I actually MEAN I am adding 83 ones to 32 ones.
At a very basic level, and I LIKE very basic levels, 1 is the basic unit which every other thing is measured relative to. It therefore represents the difference between something and nothing. Which takes us neatly on to...
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