Grand Mathemagican: Indeed, I have deduced from our past meetings that there is a high probability that you will have many questions for me on any given day.
Apprentice: Umm, ok sure. Anyway, I was studying under the tutelage of the Algebra Oracle, and she mentioned something called imaginary numbers, something about the square root of negative one. She didn’t go into much detail, so I was left wondering, what is the point of imaginary numbers. If they’re imaginary, why should I care to learn about them?
Grand Mathemagican: Ah, yes, I have heard this question before, yet I never tire of answering it; for in answering this question, deep truths about the nature of mathematics are revealed. So let us begin. What is a number anyway? Lets start with something simple, what is three? Can you show me a three?
Apprentice: But there are three of them!
G.M.: Indeed, but without the shells what does three mean?
Apprentice: Well, it could be three of something else. Three birds, or three arrows. Three tasty meals.
G.M.: Your stomach can wait til after this lesson! Either way, you have got it right. In our real life experience numbers like three must be attached to some object to have meaning. By itself, three is just an abstract idea, that happens to represent a quantity.
G.M.:These are all symbols that we use to represent this quantity of 3. Symbols such as these allow us to strip away the objects from the numbers, which makes it easier to solve problems. Say there were 297 birds and 160 arrows. How many more arrows do you need to maximize your number of tasty meals?
Apprentice: Well, 297 minus 160 is 137, so I need 137 more arrows.
G.M.: Good. You could always have just counted, but by using numerals and a subtraction algorithm, you have solved the problem much faster. Further, consider the assumptions we made in posing the problem. You took for granted that 297 means two hundreds, 9 tens, 7 ones; that is, you used the base 10 place value system to make the calculation. There certainly are other ways to represent numbers, and the combination of digits 297 could mean a completely different quantity in another context. Nevertheless, this base 10 system, which seems very simple to us, and which makes calculation very easy, took thousands of years to develop and gain traction. Imagine trying to do that calculation with Roman numerals.
Apprentice: Hmm, I never really thought about numbers that way. Still, that was easy, and I asked you about imaginary numbers, not simple whole numbers.
G.M.: Yes you did, but in order to talk about imaginary numbers, we first must define what a real number is, and whole numbers are part of the set of real numbers. And you must realize that while these simple whole numbers are easy to imagine and apply to real life, they are still an abstraction, albeit a simple one. What other kinds of numbers has the Algebra Oracle told you about?
Apprentice: We learned a lot about the integers: positive, negative, and zero whole numbers. She also told us about rational numbers and irrational numbers.
G.M.: Ok. So the integers include negative numbers. Tell me, when have you used such negative numbers?
Apprentice: Well after playing a game of poker with the Oracle of Chance, he said that I had -20 shells.
G.M.: So, the introduction of negative numbers lets us solve new problems involving differences. Hapless Apprentice has 100 shells. He loses 120 shells in a game of chance. How many shells does he have now? The set of whole numbers can’t handle a problem like this, so we need to add negative numbers when differences are important, such as when you owe someone shells. But these negative numbers aren’t too hard to imagine or apply. What about rational numbers?
Apprentice: Rational numbers are just fractions, right?
G.M.: Right. A more precise definition is that rational numbers are all numbers that can be written as a ratio of integers. We use various symbols to represent these rational numbers, including fractions and decimals. And once again, by opening up the rational numbers, we can solve even more problems. Going back to the Oracle of Chance, I’m sure he has regaled you with lists of odds for various card games. Say I have a deck of ten cards, numbered 1-10. What is the chance that I will get a nine?
Apprentice: There’s 10 possibilities and 1 nine, so you have 0.1 chance of getting a nine.
G.M.: So what does that mean? I’m either going to get a nine or not, I can’t get 1/10 of a nine.
Apprentice: It means that if you draw ten cards, replacing after each draw, you will draw one nine.
|Number of nines||Probability|
G.M.: Oh-ho! Does it now? Actually, given the experiment you just described, I only have a 38.74% chance of getting exactly one nine. So it’s more likely that I will not get exactly one nine. I could get 0 nines, 2 nines, 3 nines, or even 10 nines! Nevertheless, of all the individual possibilities, I have the highest probability of getting exactly one nine. If you want to know more about this calculation, ask the Oracle of Chance about the Binomial Probability Distribution.
Apprentice: Wow, my head hurts. But it’s very unlikely that you would get 10 nines.
G.M.: Exactly, but still possible. The point I’m trying to make here is that rational numbers can be more abstract and harder to apply than the integers we talked about before. But that doesn’t make them any less useful. Would you bet on getting 10 nines in our little game?
Apprentice: Of course not! Well, maybe if the payoff was really high.
G.M.: Indeed, the payoff would have to be 10 billion to one. Either way, you get the point, rational numbers in the form of probabilities can be useful for making decisions. Rational numbers are also necessary to construct any type of efficient measurement system. Ok, what’s next. Now it starts getting fun, tell me about these irrational numbers.
G.M.: That’s a good start. The Oracle of Calculation has spent centuries working out the decimal expansion of , and yet he still has not finished, and most of the other oracles think he is a bit, well, irrational. Yet despite the difficulty in approximating , it is a fairly easy quantity to visualise. It can be seen as a the length of a side of a square of area 2, or as the length of the hypotenuse of an isosceles right triangle with legs of length 1.
Apprentice: Ah, yes, the Oracle of Metrics has shown these to me.
G.M.: And we have approximated to enough decimal places to allow us to build a square of area approximately 2. But we could never build a square with an area of exactly 2. So while these irrational numbers are easy to visualize, they are more difficult to apply, and they are certainly more abstract than the rational numbers or integers.
Apprentice: And they allow us to solve new problems in geometry and algebra. The Algebra Oracle gave me a bunch of problems involving right triangles and irrational numbers.
G.M.: Exactly. You have worked with irrational numbers that arise from irreducible roots, such as . These, along with the integers and rationals, form the set of algebraic numbers. There are also other irrational numbers, such as and e, called transcendental numbers. It’s not hard to visualize as half the circumference of a unit circle, and it has immediate applications in geometry and trigonometry. e doesn’t have an obvious geometrical interpretation, yet it is absolutely necessary for analyzing exponential growth and decay, and it shows up in a very interesting way when looking at the imaginary numbers we are supposedly talking about.
G.M.: Patience, my dear apprentice. We’re almost there. So all these numbers we’ve talked about so far, the sets of whole numbers, integers, rationals and irrationals are combined to form the set of so-called Real Numbers.
G.M.: But oh my, where has the time gone! I hear the others gathering for debate, and I believe you have duties to attend to. I suppose we’ll have to finish this discussion later.
Apprentice: Oh, come on, you didn’t answer my question yet.
G.M.: Well that just means you will have to come back if you want find out how the story ends. Now run along, and ponder what we have discussed. I’ll see you later this afternoon?
Apprentice: You bet, old man. And you better have some answers!
G.M.: Hah! We’ll see about that!
Click here for the next installment of A World of Pure Imagination! If you have questions that you would like the Grand Mathemagician, or one of the Oracles to address, send them to firstname.lastname@example.org.