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The math lie - are Real numbers Real

The math lie - are Real numbers Real



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 Real The math lie - are Real numbers Real Presentation 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
      • Though we had to look beyond addition to find them
      • 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
      • No question is asked
      • 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
      • And we should be asking hard questions about them
      • So, start filling the blanks
      • _______?