Proof: log10 of 2 is irrational

I don’t know if this is an original idea, but regardless I’m quite proud of myself for coming up with it. I think it’s very clever. This will be a proof by contradiction.

Suppose, to the contrary, that log10 of 2 is rational. Then log10 2 = a/b where a and b are unknown integers. Recall from high school math that:

log10 2 = a/b
2 = 10^(a/b)
2 = (10^a)^(1/b)
2^b = 10^a

This latter equation, 2^b = 10^a, is logically equivalent to log10 2 = a/b, and also much nicer to look at. Very ho-hum so far.

The trick is to remember that all composite numbers have one and only one prime factorization. Because 2 is prime, 2^b is a prime factorization. But we also know that every number 10^a has 5 as a divisor. Because 5 is prime, it must represented in the prime factorization. But if it is true that some number is equal to 2^b, then 2^b must be that number’s prime factorization. This is a contradiction.

Quod erat demonstrandum. Criticisms welcome.


About Aeoli Pera

Maybe do this later?
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17 Responses to Proof: log10 of 2 is irrational

  1. Koanic says:

    Well, nobody is going to complain that your mathematical proof isn’t mathematical.

    • Aeoli Pera says:

      Eh? I think I heard something *whoosh* just now.

      • Aeoli you’re far too literal. I just want to grab your cheeks. Awlookatdaweedlepuppybubububub!

      • Koanic says:

        When I write a mathematical proof, it’s for some philosophical issue of huge social import. When you write a mathematical proof, it’s actually math, and I don’t read it. TT = pure thinker.

      • Koranic you were totally vaguebooking/feinting in order to exist in super-position within the conversation. An act which seems like magic to literal thinkers.

      • Aeoli Pera says:

        “Aeoli you’re far too literal. I just want to grab your cheeks. Awlookatdaweedlepuppybubububub!”


        Not a little baby! Grr..

      • Koanic says:

        I just assumed he would get it because he followed my proof and I had to argue with dictionaryless aspies over “mathematical”. Yes explaining it was rude but I didn’t anticipate having to do so.

      • Koanic says:

        I also don’t think there’s anything fundamentally contemptuous about the core observation, although my style of humor prefers to make contempt salient. Someone has to do math. Even TM tendency to construct imaginary bridges leads to islands of truth that are often otherwise undiscoverable. Since we are all fragments of the original Adamic mind, we must confederate to think fully. Mother Internet just got a threshgorithm upgrade.

      • Edenist whackjob says:

        “Even TM tendency to construct imaginary bridges leads to islands of truth that are often otherwise undiscoverable.”

        I like that. TTs will go “but how the heck are you connecting that with *that*?” The TM will go “it’s a hyperdimensional bridge with no formal weight to it, it works just fine!”

  2. Edenist whackjob says:

    Theory of why dicks have gotten so big in porn:

    1. Makes guys alienated. They can’t think “that’s me doing that” when watching a porn. Cuckolding.
    2. More humiliating for the girl. Misogyny.
    3. Puts the dicks in focus. Repeated stimuli of watching cock + horniness == guess the result.
    4. Pander to gay audience.
    5. Make viewers want to buy penis enlargement pills.
    6. Knowledge of PE is more widespread in porn. Watch clips of Rocco Siffredi 25 years ago and you see a distinct difference.

    This is creepy stuff. But this seems to be the blog for creepiness.

  3. Russell says:

    You smell funny and have bad fashion sense.

    Oh, not that type of criticism?

    Mea culpa.

  4. Russell says:

    On a more serious side, no, I haven’t seen it tackled quite like that.

    I have seen this line of thinking, though:

    Assume log10(2) = m/n for some positive integers m and n, with m/n in lowest terms and m m, while the right side is odd for all positive integers m. An even number cannot equal an odd number, so the assumption is false, and there are no such integers m and n. Hence, log10(2) is irrational.

    Shamelessly swiped from:

    • Aeoli Pera says:

      That’s way easier. Clearly it was too easy for a genius like me. Though there’s one line I’d steal…

      2^n = 10^m = (2*5)^m = (2^m)(5^m)
      2^n = (2^m)(5^m)

      Same basic argument. No distinct prime factorizations can be equal, so n = m = 0, which would mean log10 2 = 0/0.

  5. Pingback: The Spirit of Edenic Tolerance | Altrugenics

  6. Pingback: Daily Linkage – September 14, 2015 | The Dark Enlightenment

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