The term "real number" confuses many who are not immersed in mathematics. For these, to whom 1, 2, 3 and the other counting numbers seem most real, the identification of the real numbers as all infinite decimals (i.e., all numbers representable by points on a number line) seems at first to go beyond intuition. But, upon further reflection, the idea of a number as "real" iff it can represent a distance on a line to the right or left of a central origin, 0, indeed seems reasonable.
Professor Fred Richman of Florida Atlantic University takes on the questions of computability and enumerability of the real numbers in his poem, "Dialogue Between Machine and Man": Dialogue between Machine and Man by Fred Richman
Hey man! Do you have time, would you agree,
To chat about real numbers now with me?
My friend, you've never seen a real real number.
You cut them off before they're halfway done.
Nor twenty decimal places, nor a thousand,
Are adequate to hold a single one.
I think you underestimate me, man.
My software package is the best in town.
See here's a little program that computes
The first n places of the number pi.
You choose the n, and if you have the time
You'll get as much of pi as you can stomach.
Such strings of digits are my cup of tea,
I mind not that they go on endlessly.
You've barely scratched the surface none the less.
The set of numbers you can calculate
With programs, like the one you wrote for pi,
Can be enumerated one by one,
And Cantor showed that, given such a list,
There is at least one number that is missed.
Insanity! What number has been seen
In all the world that I can't calculate?
Nor can you list the programs that compute
Each digit in a number's decimal string.
A child of ten can write the code to list
The programs my compiler will accept.
But here the programs must consist of those
That endless strings of digits do produce.
Write such a code, man, and the pigs will fly.
There's no such code. Look here, I'll show you why.
I know why, Mac, you just use Cantor's proof
To show that there is no recursive list
That itemizes all recursive functions.
The problem is that every list you know
Is general recursive---that's your world.
It's a paradox you'll never understand,
That I can count your numbers one by one
Despite your proof that it cannot be done.
To chat about real numbers now with me?
My friend, you've never seen a real real number.
You cut them off before they're halfway done.
Nor twenty decimal places, nor a thousand,
Are adequate to hold a single one.
I think you underestimate me, man.
My software package is the best in town.
See here's a little program that computes
The first n places of the number pi.
You choose the n, and if you have the time
You'll get as much of pi as you can stomach.
Such strings of digits are my cup of tea,
I mind not that they go on endlessly.
You've barely scratched the surface none the less.
The set of numbers you can calculate
With programs, like the one you wrote for pi,
Can be enumerated one by one,
And Cantor showed that, given such a list,
There is at least one number that is missed.
Insanity! What number has been seen
In all the world that I can't calculate?
Nor can you list the programs that compute
Each digit in a number's decimal string.
A child of ten can write the code to list
The programs my compiler will accept.
But here the programs must consist of those
That endless strings of digits do produce.
Write such a code, man, and the pigs will fly.
There's no such code. Look here, I'll show you why.
I know why, Mac, you just use Cantor's proof
To show that there is no recursive list
That itemizes all recursive functions.
The problem is that every list you know
Is general recursive---that's your world.
It's a paradox you'll never understand,
That I can count your numbers one by one
Despite your proof that it cannot be done.
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Hi JoAnne,
ReplyDeleteI recently published a post that asks the question "what are real numbers?" Your explanation is a much simpler and more succinct version of my entire post! For something that seems so simple now, we forget how strange this concept was when we first learned about it.