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# The math lie - are Real numbers Real

## on Jun 09, 2007

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Should we be so quick to use real numbers? Are they "Real" at all?

Should we be so quick to use real numbers? Are they "Real" at all?

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## The math lie - are Real numbers RealPresentation Transcript

• The Math Lie
• Are real numbers real?
• What are real numbers?
• Let’s take a ride down history’s lane
• In the beginning, there was 1
• 1 gave birth to 2 (1+1), 3 (1+1+1), … to infinity
• And we got what is called “The Natural Numbers” (N), or positive integers
• They’re call “Natural” for a reason
• They’re very intuitive, and have been around for as long as there have been man
• God gave us the integers, all else is the work of man
• L. Kronecker
“ ”
• But, they aren’t perfect
• You can formulate questions with natural numbers, that have no answer IN natural numbers
• 1+? = 1
• So the zero came into being
• And we get what is called N+
• N+ = 0,1,2…
• But there were still unanswerable questions
• 1+? = 0
• So once again, new numbers were added - negative numbers
• -x was define to be the number that answers
• X+? = 0
• This gave us the Z, the integers
• But unanswerable questions kept popping up
• 3*? = 1
• So we gave birth to the “Rational” numbers - Noted Q
• Rational = represent a ratio between two integers
• The rational numbers had to fit the known image of the world
• And indeed 3 (integer) = (rational)
• Rational numbers seem to solve any question they can represent
• A little hard work, but no conceptual problem
• But, life can’t be that good
• Around 400BC, the Greeks have found that isn’t rational
• Which means that this question has no answer in the rational domain
• ?*? = 2
• So, a new set of numbers were introduced - irrational-numbers,
• Irrational just means it can’t be expressed as a ratio, but we’ll see it turns out to have a deeper meaning
• Actually, the trip doesn’t end here, because there’re still unanswerable questions
• ?*? = -1
• But let’s focus on the irrationals
• Rational + Irrational numbers
• =
• Real numbers
• They are casually presented to every junior-high pupil as a natural extension of rational numbers
• As it turns out, the irrational numbers are the really important ones
• e, π, cos, sin, …
• And there are infinitely more irrational numbers then rational ones
• So, Real numbers are pretty much the corner stone of modern math
• And we all use then quite regularly
• Good thing they work
• Do they?
• Shouldn’t we stop and ask?
•
•
• That’s pretty impressive
• Any specific rational problem can be solved and narrowed down to a answer
• Lets what the “Real” numbers can do
•
•
• Something’s fishy…
• It seems we’re quite helpless in the face of real numbers
•
•
• Now, that’s just cheating
• It’s not that it never works…
• But it almost never does
• Wait, does it ever work?
• Is , really?
• What is anyway?
• Is it 1.141?
• No, that’s only the beginning…
• Is an unending beast
• No one has ever held in its hand
• And no one ever will
• So what do we mean by ?
• If we’ll ever be able to hold an unending beast, and multiply it by another unending beast (a process which will never end), we will get the exact result 2
• Anyone convinced?
• But maybe the problem lays with us
• We’re idiots
• The real smart professors must have the answer…
• Well, actually…
• The whole thing is pretty young
• A wide consent on the nature of real numbers was only achieved around 1920+, when Cantor’s theory of sets grew popular
• This means that for ~2300 years, people had no idea what they’re talking about
• Not even the smart professors
• (It doesn’t mean that they know what they’re talking about NOW. It just means most of them agree on the nonsense they’re saying)
• Real numbers are extremely unnatural
• You have to go through 2 months of university calculus before you’re (kinda) convinced that
• But still, if you draw a 1X1 square, the diagonal just IS .
• And if you want to know a area of a circle, π is the only way to go
• These are not some invented pure mathematical entities
• They’re really “out there” in nature
• We can’t help but deal with them
• And that’s OK
• But let’s not kid ourselves
• Real numbers are not real
• And shouldn’t be presented casually
• They are irrational, in the fullest sense of the word