Algebra: A Game of Numbers
Numbers are like rabbits! When you think you've seen 'em all, up pops another one.
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Ordinary algebra
is as simple as "one plus one is two."Algebra is just a game of numbers -- a game of adding and subtracting and multiplying and dividing numbers. If you can do these things, you can do algebra.
There is one slight difference, though -- when you do all the adding and subtracting, multiplying and dividing, you rarely see the numbers themselves. You work mostly with letters of the alphabet. But the numbers are there, represented by the letters. The letters are symbols, or forms. It's a kind of simulation. An example of what I call Representation Theory. The numbers are hiding behind the letters.
So don't be fooled -- even though you're moving letters around when you do ordinary algebra, you're really adding, subtracting, multiplying, or dividing numbers. No matter how you slice it, one plus one is still two. Numbers won't let you down. You can count on them.
Click here if you might like to compare number systems with statement systems, in what is called Boolean Algebra.
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Of course numbers count! We make them count. One, two, three, skiddoo!
I'm talking about the counting numbers, of course. They count because that's how use them. The counting numbers are what we also call natural numbers. We write them in their normal order from small to large as follows:
1, 2, 3, 4, 5, 6, 7, ...
Notice the numbers increase by one as you move from left to right. You can't tell by looking at the names, of course. I mean, there's nothing in the name, 3, that says it's one more than two. You have to know by convention that 3 is the number that follows 2 and is larger by 1, which means that 3 = 2 + 1. And it's the same for all the other numbers.
Notice, too, there's a smallest number in the list, namely 1 -- but no largest number. No matter how many things you count, there's always room in the natural number system for one more. You can always count one more, then another, and another, indefinitely, so there are many names to memorize. Yes, memorize!
An important thing about numbers is their order. With natural numbers there's no problem with order. Given any number, you can always tell which number comes before it and which comes after it. You only have to add or subtract 1 to get it. So, for example, 732 follows 731 and precedes 733. No sweat! But these are just names. And you have to remember them.
I'm sure you understand these properties of numbers; you probably just take them for granted. All I'm really doing is drawing your attention to them. I'm confident of this because you deal with numbers every day, like when you're buying groceries or paying the rent or paying for new clothes, or whatever. You know whether or not you have enough for the groceries or clothes, and you know there's no limit to the amount of money you could have.
But there's more to numbers than just the system of natural numbers. Indeed, there are several other systems -- several more layers. Oh, and by the way, if you'd like to count the different number systems, feel free to say that the natural numbers are the first, and simplest. They were also the first to be invented. (Or is that copyrighted?)
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When you count -- one, two, three, four -- you never say positive one, positive two, positive three, and so on. It doesn't make sense to do that. In fact, it's silly. The positive or negative part of the number business doesn't enter the natural number picture at all. Only when you jump to integers, do the signs make sense. With the introduction of signs, you go beyond the natural numbers. That's because integers are defined as positive and negative whole numbers. So, if you're still counting, integers are the second of our number systems.
But what precisely are the integers? Easy enough to say they're all the natural numbers with plus or minus signs attached, and with zero thrown in as well. What other properties do they have?
We take plus and minus numbers pretty much for granted now, but it was a big step going from the naturals to the integers. It was an amazing jump, really, because it meant coming up with the idea of negative numbers. Who had ever heard of a negative something!
What an insight!
It also meant conjuring up the idea of zero. Who ever heard of such a thing as zero! Indeed, the "number" zero, or 0, was the key that connected the positive numbers to the negatives and put them in their proper order. We can now write them as the ordered series:
... -7, -6, -5, -4, -3, -2, -1, 0, + 1, + 2, + 3, +4, +5, +6, +7, ...
The integers are ordered in the sense that each one in the list is one unit bigger than the previous one in the list, going from left to right. In this sense integers are like the naturals. Given any number, like 45, you can always say which integer (44) precedes it and which integer (46) follows it. On the negative side you can say that -6, for instance, is one unit smaller than -5 and one unit larger than -7. You only need to subtract or add 1 to get the correct number. Before zero came along, though, you couldn't do this, because (1 - 1) isn't -1.
There are differences though. For one thing, there isn't a smallest number in the integers. Amazing! Just when you think you've got the smallest rabbit (non-rabbit?), here comes a smaller one!
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We've only just begun to fight -- over the different kinds of number systems, of course. But we're not really fighting. Rather, we're dividing and conquering -- ideas, that is.
On a more rational note, let's talk about fractions, or what we refer to as rational numbers, or the rationals. Call the rational system the third in our list of number systems, if you like.
Remember fractions? Divide one integer by another, and that's what you get -- a fraction. So, for instance, divide three by four and get the fraction:
3/4.
In general, a fraction is the ratio a/b, where a and b are integers.
That's it! Simple as that! You can think of 3/4 as the result of parceling three things into four parts.
Trouble is, this might lead you to think fractions can't get smaller than zero or bigger than one. After all, a fraction of something is just a part of that something, right? Now the smallest part can only be no part of it at all, isn't that so? And doesn't that mean it must be zero? While the biggest part can only be the whole thing, and that must be one, wouldn't you say?
Well, yes. Except that you can put any integer in as a numerator and any integer in as a denominator, including negative integers. So you'd better think of rationals as ratios of two integers. The numerator of a rational can even be a large negative integer, and the denominator can be any small positive integer, like in the following fraction:
-56479/2,
which is a large negative number and therefore much smaller than zero. So parceling doesn't do justice to fractions.
Like integers, the fractions -- or rationals -- can be positive or negative. Also, there's no smallest fraction and no biggest fraction. Rational numbers are also ordered. Given any two rationals, you can always tell which is smaller. For some pairings it's very easy. For instance, 1/5 is clearly smaller than 3/5. And 21/32 is smaller than 23/32. For others, though, it takes a bit more work to be able to say. For example, which of 2134/26137 and 3467/38180 is the smaller? You probably can't tell without a bit of thought! But it's easy enough to decide when you know how to divide.
Try the division yourself to see which is the smaller.
So, given any two numbers, you can always say which is greater than (or less than) the other -- you only need to compare their decimal equivalents.
But suppose you turn it around. Instead of being given two numbers to compare, suppose you are only given one rational number and you have to find the very next rational number in the list. Is there one that's just larger? Indeed, is there one that's just smaller?
Say the number is 23/47. ... No, wait a minute, let's use an easier example! Let the number be 1/2. Well, right away you know that the rational number, 1/1, is definitely larger, but is it next? 5/8 is also larger, but is it next? What about 9/16? Or maybe 17/32? Or even 65/128? Or possibly 10, 001/20,000? Or, etcetera, etcetera.
Ha! You can try more numbers 'till you're blue in the face, but you won't find the very next one, because there isn't one. Unlike with natural numbers or integers, given any number, no matter how far you go, there's always a slightly smaller value that's still bigger than the number. Isn't that a crock?
What's more, there is also no number that comes just before a number, like before 1/2, for example. Here, again, you can approach 1/2 from below and get closer and closer to it, but no matter how close you get, there's always another number that's closer yet.
Things get even funnier when you consider ordered integers and their limits. For instance, look at the fraction series:
1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ...
which goes from left to right. This series of rationals has the integer, 1, in the numerator of each rational and a progressively increasing positive integer in the denominator.
As the denominator gets bigger and bigger, the fraction gets smaller and smaller, and you can see it gets closer and closer to zero, which is to say it has the limit of zero. But the number can never really be zero. There is no integer for the denominator that makes the fraction zero. The fraction may get smaller and smaller, and for practical purposes may be considered to be zero. But it's still only a very small rational number, and never zero.
This isn't the only series of rational numbers that approaches but never reaches a limit -- either a smallest or largest value. There are, in fact, as many of these series as you can count.
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Real numbers are something else, again! I mean, really! Because now you have to deal with something called an irrational number. The real number system -- system number 4, if you're still counting -- consists of just that, the system you get by adding the irrational numbers to the rationals. (Keep in mind that each of the number systems can be considered separately, so you have to be clear about your reference when you raise questions about numbers.) If you know an irrational number when you see one, you'll know the reals when you see them.
So, what are irrational numbers?
Not a trivial question! But we can say they are used as a basis of measurement. The answer can be easier to see if we're careful about defining rational numbers. We got a hint of the possibility when I mentioned that rational numbers -- or fractions -- can be compared to each other by comparing their decimal equivalents.
The fact is, rational numbers are numbers that can be expressed as decimal numbers having a special attribute. The attribute is that the decimal of a rational number has a closed form. In other words, when you can divide the numerator of a fraction by the denominator and the division comes to an end without a remainder, the fraction is a rational number. For example, 1/2 is the decimal .5 exactly. And 3/4 is the decimal .75. In either case nothing is left over. No remainder! Not a crumb!
All fractions have this property, and all decimals with this property are fractions, or rational numbers.
But irrational numbers are different. With irrational numbers, there is no closure on their decimal equivalents, so there is no representative fraction. Take the square root of 2, for example. You can carry out the square root process until dooms day, but you'll never reach the point when there is no remainder. In other words, the decimal form never comes to an end. The square root of 2 is an irrational number.
Another peculiarity of rational and irrational numbers is the amount of space they fill, or don't fill. (It's called their measure.) If you record them on a line, an odd thing occurs, something that's hard to believe.
Let's start with the rationals and record them on a line. (You might know that the real numbers are represented by a line -- a dimension of space.) First lay out all the correspondents of the integers. (These are fractions you get by dividing the integers by unity -- by one. For example, divide the integer, 43, by 1, and you get the rational number, 43/1, which we write simply, and hopefully without confusion, as 43.)
Next, record all the fractions having the denominator, 2. Then those with denominator, 3. And so on. And so on. And so on. We would never actually be able to record all of the fractions, because it would take more time than there is in the universe, but we can imagine doing it.
Even so, i.e., even with all the recordings, ... and here's where you have to hold your breath, ... the line will always be completely empty! That's worth another exclamation mark! It's as if you throw stone after stone after stone into a bucket and always have an empty bucket. Putting down all the rational numbers doesn't even make a dent in the line. Because of this, we say the measure of the rational numbers is zero. They don't fill any space whatsoever!
Wow!
But doesn't the line represent the real numbers? ... In fact, it does. So what does that say about the irrational numbers?
It says that the irrationals, alone, fill the line. If the reals consist of the rational numbers together with the irrational numbers, and the reals fill the line, but the rationals don't even make a dent, the measure of the irrational numbers has to be the length of the line itself! They have to pack the whole line! Whew! That's one huge mess of irrational numbers! If your teeth were the real numbers, they would all be irrational, and rational numbers wouldn't even show up in a cavity! That's how many more irrationals there are than rationals! And look at how many rationals there are!
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You think real numbers are spooky! Well, you ain't seen nothin', yet! Just get a look at complex numbers!
Ever hear of imaginary numbers? That's what you have to contend with when you work with complex numbers. Not only that, but complex numbers are two-dimensional, at least in their simplest form. You need a flat plane to represent them -- unlike the real numbers, which only need a line, or a space of one dimension. But even that is awesome!
As it happens with complex numbers, one of the dimensions is, in fact, the dimension of real numbers. So that part of complex numbers is at least familiar. But there's still another part that isn't real. You might say it's unreal! That part is the other dimension, and it's imaginary!
Here's where it gets spooky. It's not imaginary because you're dreaming, or anything like that. It's imaginary because it has that as its truth value, looking at it as a logical system of numbers. What's new about complex numbers is their key element, namely the square root of minus one. How does that grab you!
Remember what it means to be the square root of something? The square root of 4, say?
Remember that the square root of a number, x -- oops, here's another bit of algebra! -- is the number you multiply by itself to get x. So the square root of 9, for example, is 3. And the square root of 25 is 5. But what in the world is the square root of -1? What number can you possibly multiply by itself to get -1? Well. The only answer you can give is that there is none. That is, there is no such real number. This is to say there is no answer if your reference is the Real number system, just as there would be no answer to the question: 'What is the number between one and two?' if your reference was the natural number system. Indeed, to get an answer you have to jump to the Complex number system, in which case you get the square root of -1, which is purely imaginary!
But it's real enough when it comes to solving algebraic equations!
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