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The math lie - are Real numbers Real
 

The math lie - are Real numbers Real

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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 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
      • _______?