OK. Lets have a go at imaginary numbers. These are NOT straightforwards. The name is also very annoying because, they are very real indeed, and I will try to satisfy you that that is the case in due course.
First of all what the hell are they? We have already seen four different types of numbers. We have seen whole numbers such as one, two, three, ten and so on. These are obvious in our every day world. How many pens are on the desk? Three. How many cows are in that field? Twenty. They are used to count separate and distinct objects. Looked at formally, the \(x\) in the following equation is a whole number:
\[x+5=7\]
We can see without too much effort that \(x\) is two. Two is obviously a whole number so we are quite happy with that as a value of \(x\) that makes the equation work.
When we looked at the opposite of addition, we found a different type of number, a negative number. We agreed that saying you had negative five cows in a field did not make sense. The concept was so odd that for a long time mathematicians refused to accept that these numbers really existed. Nowadays we are perfectly happy with the concept (unless your bank account is very, very, negative, in which case you will be perfectly unhappy with the concept).
So what about a slightly different equation which works with a negative value of x?
\[x+5=2\]
To work, we need to set \(x\) equal to negative three. \(x\) cannot be a positive number, because there are no positive numbers that are five smaller than two. Again, we are quite happy with this, but weirdly even just a few hundred years ago the best mathematicians in the world would have said that there was NO ANSWER to that problem.
OK, next we considered fractions. We agreed that while you could have half a cow, it would not be very pleasant to look at. More fundamentally we realised that fractions are ratios between two whole numbers. So if I have five apples, and my friend has ten apples then I have half as many apples as my friend. That is a statement about the ratio of our apple collections to each other. What would this look like stated as an algebra question?
\[3\cdot x=2\]
This is just a little bit trickier because it is saying three what's are two? Or, what is two divided by three? We are quite happy with the answer two thirds. And in general people have always been comfortable with this idea. After all, you are just comparing two whole numbers.
OK, moving on we then learned about irrational numbers. These are numbers that cannot be written as one whole number divided by another whole number. We satisfied ourselves that the number which you multiply by itself to get two is one of these numbers. The equation which has one of these as an answer looks like this:
\[x^2-2=0\]
For the hard of thinking, you add two to both sides and then take the square root of both sides getting \(x\) equal to the square root of two. The ancient greeks really did not like this. They felt that all numbers should be rational, and they were really disturbed to find out that was not the case. it's a bit more abstract today, but we are generally not bothered by the idea that there are some numbers which would just go on for ever if you tried to write them out.
Now then. What is the answer to the following puzzle. What number, if you multiply it by itself, and then add one, gives zero? Or algebraically:
\[x^2+1=0\]
It is very similar to the square root of two equation just above isn't it? So what do we do? We subtract one from each side, getting:
\[x^2=-1\]
And we then take the square root of each side:
\[x=\sqrt{-1}\]
So the answer is the number, that if you multiply it by itself, makes negative one. OK, so what would that be then? One multiplied by itself is one, so is negative one multiplied by itself negative one? We need to think about multiplying negative things to get an answer to that.
We said that multiplying is just a special type of addition. So that three multiplied by four is:
\[3+3+3+3=12\]
Notice that there are four threes there. So what would negative three multiplied by four look like? Well, we would just add together four negative threes. That would look like this:
\[(-3)+(-3)+(-3)+(-3)=-12\]
Remember that adding a negative is the same as subtracting a positive. And also remember that you do stuff in brackets first. So while you have plus signs in between each set of brackets, the fact that there the numbers INSIDE the brackets are negative means that you end up subtracting. Didn't we say though that it doesn't matter which way round you multiply things? So what does four multiplied by negative three look like as addition? Well, sticking with our definitions, it is four added together negative three times. How do you add something a negative amount of times? Remember that a negative something is the same as the something subtracted from zero. So, for positive multiplication you add up a group of things, but for negative multiplication you subtract your number from zero the same amount of times you are supposed to negatively multiply it by. So it looks like this:
\[(0)-(4)-(4)-(4)=-12\]
(The zero looks a bit odd there, and I suppose it could be implied in the same way that positive one is implied to be zero plus one. So we could have written the four threes above as zero plus the four threes.)
As we would expect that is also negative twelve. So we have now looked at a positive multiplied by a positive (where both the number and the sign between the numbers are positive). That gives a positive result. We have looked at a negative multiplied by a positive (the numbers are negative but the sign between them is positive). And we just looked at a positive multiplied by a negative (the numbers are positive but the sign in between them changes to a subtraction instead of addition sign). What about the last option, multiplying a negative number, a negative amount of times? What does that look like as an addition?
Well, you will be multiplying a negative number, so the numbers IN the brackets are going to be negative. And we are going to be doing it a negative amount of times so the numbers BETWEEN the brackets will also be negative. The sum looks like this:
\[(0)-(-3)-(-3)-(-3)-(-3)=12\]
Because we are duplicating a negative number a negative amount of times we end up with two negative signs. Adding a negative is the same as subtracting the number, so subtracting a negative is the same as adding the number. Sound weird? Not really, if I lend you £10, then you owe me £10 (lets call that negative £10). If I then subtract, or cancel, the debt I have effectively gifted you £10. So I turn a negative obligation (you have to give me £10) into a positive benefit (I have given you £10).
So what you actually get when you subtract all those negative numbers is a positive number. In my minds eye, I see the two minus signs combine into a plus sign, with one of them rotating through ninety degrees. So for every two minus signs you create a plus.
So what about our hypothesis that if you multiply negative one by itself negative one times you get negative one? That would mean that you subtract (-) negative (-) one (1) from zero once (1). That would look like this:
\[(0)-(-1)\]
The two minus signs combine to form a plus, and you get positive one. So the square root of negative one cannot be negative one. It cannot be positive one either, because two positive numbers multiplied together always give a positive answer. So if the answer cannot be negative and cannot be positive, then what the hell is it? All the numbers that we know about - all of the numbers above, are either less than zero or greater than zero (or zero itself, I grant you). So how can a number be neither positive or negative?
Oh dear. It appears that we are stumped. And indeed for a long time people treated equations which produced answers that were the square roots of negative numbers in the same way as they treated equations which gave negative numbers themselves as answers. In other words they ignored them.
We do not do that any more though. Instead we say that there IS a number which is the square root of negative one, or a number which if you multiply it by itself gives negative one. We have a symbol for that number, and the symbol is \(i\). What we say, and just work with me on this, is that each number has a 'real' part which is a multiple of one, and an 'imaginary' part which is a multiple of \(i\). The number that is actually the square root of negative one is a number with no real part (or technically a real part multiplied by zero), and an imaginary part which is \(i\) multiplied by one. We say that \(i\) is not positive or negative in the sense of being more or less than zero. Instead we say that it has a whole positive and negative spectrum all to itself. So the following equation makes perfect sense:
\[i-2i=-i\]
This all sounds a bit abstract. What would such a number look like, and how does it relate to the other numbers that we are familiar with? I'll try and make it more visual next time.
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