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While enjoying my coffee and bagel this morning, I've been reacquainting myself with different kinds of numbers, and am entertained once again by a particular discussion topic of how common each of these kinds of numbers are. Did you know that most of the numbers we like to use are VANISHINGLY RARE among all the numbers there are?
Let's say I could draw an infinitely precise line segment on a dartboard, and that you could throw an infinitely sharp dart so that it lands exactly (you're REALLY good at this) on the line segment at some point. We've agreed between ourselves how we are going to measure the position of the dart on the line - and even how to draw a precise perpendicular from the dart to the line if you are having a terrible day and you missed it!
Let us number equally spaced divisions on the line segment so that it goes from 0 to some agreed-upon integer. Let's choose the number 10.
Since your dart and our ability to measure its position are infinitely precise (but your ability to aim for a specific number is somehow not!), let's imagine your dart could have landed on any position on the line segment with equal probability. As far as I know, no point on the line is more likely than any other, because you want to make our game more challenging! My part in the game is to try to guess what kind of number your dart found: Integer, rational, algebraic, or transcendental. I plan to win the game by guessing it lands on a transcendental number, one that is not integer, rational, or even algebraic. Here is why I think I will win.
What are the odds your dart landed on an integer? Well, those odds seem pretty small. There are eleven marks on the line, labeled 0, 1, 2, ... 10. You would have had to land exactly on one of those marks, with infinite precision. If we had a measuring device installed on the line, and you had landed on 3 exactly, the device would have said 3.0000000... with zeroes going on forever. What are the odds of that?
So how many integer points are on the line? Eleven. How many other points are on the line? All the rest of them! That's a little bit of a jokey answer, but it is not far from what I want to show here: There are an infinite number of points on the line, since it is continuous, between any two of the precise number markers 0 through 10, and thus an infinite number of points that are not integers. When we consider the ratio of the number of integers to that of non-integers on the line, we consider 1/N as N approaches positive infinity (the quantity of non-integer numbers), and that number gets closer and closer to zero as N gets closer and closer to infinity. This is what I meant by my earlier term VANISHINGLY RARE. The integers are VANISHINGLY RARE among all the other numbers.
So yes, it's more likely that your dart landed on a place between two numbers... a FRACTION of the distance between them, you might say.
Good point about points! The set of fractions, or rational numbers, between 0 and 10, or between any two consecutive numbers on our line, is an infinite set. Any finite segment of our line, in fact, will have infinitely many rational numbers on it. And again, the ratio of 9 integers to an infinite quantity of rational numbers makes those integers VANISHINGLY RARE.
To explain the mathematics behind the rest of what I was considering would take too long. If you're interested, you can look up the concepts yourself. I want to keep this light and entertaining and save you headaches, especially if you're not inclined to go as deep as I wanted to go today.
The integers are a subset of rational numbers. I just showed that integers are VANISHINGLY RARE among rational numbers.
The rational numbers are a subset of algebraic numbers, like the square root of negative one, or five plus the square root of two, or the Golden Ratio. They all satisfy integer polynomial equations. Rational numbers are, it has been shown (albeit not here), VANISHINGLY RARE among all algebraic numbers.
In fact, thinking of our dart board, look at it this way: Your dart, to have landed on a rational number, would have had to land on a number that the infinitely precise measuring device would show as having a digit, followed by a decimal point, followed by a finite number of other digits, and then a sequence of other digits, again finite, that would repeat over and over FOREVER, exactly. The chance of your hitting a number like that is just as small as you think it is: very unlikely.
So integers are VANISHINGLY RARE among the rational numbers. And rationals are VANISHINGLY RARE among the algebraic numbers, even those that are on the real number line... even those just between 0 and 10.
But mathematicians have gone further. There are numbers that are not solutions of any integer polynomial. Two of the numbers most studied by mathematicians, e and pi, are such numbers. We call them transcendental numbers.
So let's look at your dartboard toss, and now let's forget the fact that you're so precise at throwing your dart. It could have landed anywhere above or below the line. Let's consider the whole dartboard to be a bounded region of the Complex Plane, the places your dart could have landed corresponding to complex numbers. Now what are the probabilities?
Integers are VANISHINGLY RARE among the rationals.
Rational numbers are VANISHINGLY RARE compared to the set of algebraic numbers on the real line.
So how many integer points are on the line? Eleven. How many other points are on the line? All the rest of them! That's a little bit of a jokey answer, but it is not far from what I want to show here: There are an infinite number of points on the line, since it is continuous, between any two of the precise number markers 0 through 10, and thus an infinite number of points that are not integers. When we consider the ratio of the number of integers to that of non-integers on the line, we consider 1/N as N approaches positive infinity (the quantity of non-integer numbers), and that number gets closer and closer to zero as N gets closer and closer to infinity. This is what I meant by my earlier term VANISHINGLY RARE. The integers are VANISHINGLY RARE among all the other numbers.
So yes, it's more likely that your dart landed on a place between two numbers... a FRACTION of the distance between them, you might say.
Good point about points! The set of fractions, or rational numbers, between 0 and 10, or between any two consecutive numbers on our line, is an infinite set. Any finite segment of our line, in fact, will have infinitely many rational numbers on it. And again, the ratio of 9 integers to an infinite quantity of rational numbers makes those integers VANISHINGLY RARE.
To explain the mathematics behind the rest of what I was considering would take too long. If you're interested, you can look up the concepts yourself. I want to keep this light and entertaining and save you headaches, especially if you're not inclined to go as deep as I wanted to go today.
The integers are a subset of rational numbers. I just showed that integers are VANISHINGLY RARE among rational numbers.
The rational numbers are a subset of algebraic numbers, like the square root of negative one, or five plus the square root of two, or the Golden Ratio. They all satisfy integer polynomial equations. Rational numbers are, it has been shown (albeit not here), VANISHINGLY RARE among all algebraic numbers.
In fact, thinking of our dart board, look at it this way: Your dart, to have landed on a rational number, would have had to land on a number that the infinitely precise measuring device would show as having a digit, followed by a decimal point, followed by a finite number of other digits, and then a sequence of other digits, again finite, that would repeat over and over FOREVER, exactly. The chance of your hitting a number like that is just as small as you think it is: very unlikely.
So integers are VANISHINGLY RARE among the rational numbers. And rationals are VANISHINGLY RARE among the algebraic numbers, even those that are on the real number line... even those just between 0 and 10.
But mathematicians have gone further. There are numbers that are not solutions of any integer polynomial. Two of the numbers most studied by mathematicians, e and pi, are such numbers. We call them transcendental numbers.
So let's look at your dartboard toss, and now let's forget the fact that you're so precise at throwing your dart. It could have landed anywhere above or below the line. Let's consider the whole dartboard to be a bounded region of the Complex Plane, the places your dart could have landed corresponding to complex numbers. Now what are the probabilities?
Integers are VANISHINGLY RARE among the rationals.
Rational numbers are VANISHINGLY RARE compared to the set of algebraic numbers on the real line.
But even more, the numbers on the real line - ALL these types of numbers - are VANISHINGLY RARE compared to the set of numbers not on the real line: the complex numbers. Let's face it, you would have had to be a DAMN good darts player to land your infinitely precise dart anywhere on that infinitely thin line. Infinitely good, in fact.
Hey, maybe you are. I'd not put it past you. My understanding of your talents is VANISHINGLY SMALL.
