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Song by Tom Lehrer called New Maths

  1. Apr 7, 2005 #1
    Hi all.

    I look up and down this thread and see that the threads are rather intellectual but I have a question that is not so challenging.

    I have been listening to a song by Tom Lehrer called New Maths and although what he says in it is true he goes into Base 8. I followed his maths in Base 10 but 8 got me.

    So I simply want to know what is the best way to do sums in different bases. Base 10 is easy and we can all do sums with it but what if I had 342 - 173. In base 10 the answer is 169. Fine. But how would I go about solving it in base 9 or 8? What difference, in my mind, do I need to apply to make it easier to do?

    I have worked out that 169 in Base 9 is 207 and in Base 8 is 251 but I had to find the answer in Base 10 and then convert it and that just takes ages. There must be a simplier way.

    Cheers. :smile:

    The Bob (2004 ©)

    P.S. If anyway can find the thread about working out square roots without a calculator I would be grateful. :smile:

    P.S.S. And before you ask, this is not homework. This is my own curiousity.
     
  2. jcsd
  3. Apr 7, 2005 #2

    Hurkyl

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    The same way you would in base 10; you just need to use the addition table of the appropriate base.
     
  4. Apr 7, 2005 #3
  5. Apr 7, 2005 #4
    So in 342 - 173 in:

    Base 10 is 169
    Base 9 is 158
    Base 8 is 147??

    The Bob (2004 ©)

    P.S. If it is I will kick myself at the ease in which I did it. :grumpy:
     
  6. Apr 7, 2005 #5
    Cheers. I may turn the thread into a square root one if the bases are as easy as I fear they are. :smile:

    The Bob (2004 ©)

    P.S. I think I really have made a fool of myself.
     
  7. Apr 7, 2005 #6

    Hurkyl

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    Don't kick too hard!
     
  8. Apr 7, 2005 #7
    Cheers anyway. I will not be able to sit down much for the next week. :smile:

    The Bob (2004 ©)

    P.S. Anyone else thing Tom Lehrer is good?
     
  9. Apr 7, 2005 #8

    Alkatran

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    There's a simple way to convert numbers from one base to another.

    Let's say we want to go from 26 (base 7) to base 8
    First, convert to base 10
    26 -> 2*7^1 + 6*7^0 = 20
    Then to convert to bas 8 you do:
    1 > 20, false: 8 > 20, false, 64 > 20, TRUE
    20 / 8 = 2.4, first digit is 2, 20-2*8 = 4
    4 / 1 = 4, second digit is 4
    remainder is 0. We're done.

    24 in base 8 = 26 in base 7
     
  10. Apr 8, 2005 #9
    I am now wondering how you multiply and divide bases other than 10. I can't just be the same. I will try it out but please reply anyway.

    The Bob (2004 ©)
     
  11. Apr 8, 2005 #10
    You just need to write out the multiplication tables. The general pencil and paper multiplication algorithm you're used to is just a concise way of writing out the distribution of multiplication over addition axiom a*(b+c) = a*b + a*c where in your case a, b and c are multiples of powers of your base (This is why you "add a zero" on every line for multi-digit multiplications such as (a+b+c)*(e + f) = a*(e+f) + b*(e+f) + .... In an arbitrary base, you can multiply by the digit (the coefficient of the power of the base) then multiply by the power of the base as allowed by the commutative law.
    Ie., in base 10,
    45 * 6 = (4*101 + 5*100)*(6*100)
    = (5*6*[100]2) + (4*6*[100+1]) = blah blah...
    You can see where "carry the 1" is really "carry the base" and so forth.
     
    Last edited: Apr 8, 2005
  12. Apr 8, 2005 #11

    Alkatran

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    For any operation in another base you can:
    Convert the numbers to base 10
    Do the work
    Convert back

    Or learn the multiplication tables for that base :tongue2:
     
  13. Apr 8, 2005 #12
    Trouble is I see that as cheating (in my mind). I am happy to do that but I would rather understand what I am doing in a different base.

    So if it was 45 * 6 in base 9 it would be:

    45 * 6 = (4*91 + 5*90)*(6*90)
    = (5*6*[90]2) + (4*6*[90+1]) ???

    The Bob (2004 ©)

    P.S. Giving 246
     
    Last edited: Apr 8, 2005
  14. Apr 8, 2005 #13
    You might enjoy basic number theory. :smile:

    Almost. You wrote your answer in base 10. :wink: A full calculation using base 9 operations would look like the following:
    [​IMG]
     
  15. Apr 8, 2005 #14

    mathwonk

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    as usual i feel like the oldest active poster on the planet:

    tom lehrer was my section man for calc I ("math 11") in 1960. Boy was he funny!

    I still remember his example of proof by vacuous hypothesis: (forgive me if the politics grate against your own)

    "Every progressive republican wears green eyeshades!"

    i just thought: "dorothy, you're not in kansas anymore."
     
  16. Apr 9, 2005 #15
    Is there anyway in which I can do it like the way it was wirtten? Is it simply a case of converting everything or is there a way of doing it all in Base 9?

    I can see what has happened but it was all converting from Base 10 not done in Base 9.

    The Bob (2004 ©)
     
    Last edited: Apr 9, 2005
  17. Apr 9, 2005 #16
    My graphic is done all in base 9 (303 corresponds to 246 in base ten). The only mistake in your post was in treating the ith digit placeholders 9i as their base ten representation and actually evaluating them. That was unnecessary and gave you the base ten number 246.
    For example, in base ten, 246 = 2*102 + 4*101 + 6*100 is just another way of writing that sum. We don't actually evaluate the 10i's and add them together. We just say the 100 is the 1's place, the 101 is the tens place, and so on. In that case the notation 100 = 102, (1 followed by i zeros) = 10i is forced on us. Similarly in base 9, 92 would be represented by 100. The coefficient of 92 in a number would be the "81's place" if one wanted a familiar name.
    The series representation with symbols corresponding to each digits place was just a reason why the machinations in the graphic work. One shouldn't have to work with the these motivational definitions more than once. :smile:
     
    Last edited: Apr 9, 2005
  18. Apr 9, 2005 #17
    So I am just going to have to convert the end answer, right?

    45 * 6 = (4*91 + 5*90)*(6*90)
    = (5*6*[90]2) + (4*6*[90+1]) = 246 and then convert.

    If this is the case is there a way I can just get straight to the Base 9 answer? Or am I misunderstanding?

    The Bob (2004 ©)
     
  19. Apr 9, 2005 #18
    Why did you evaluate the placeholder powers of 9 into base 10 ? The coefficients and exponents are what we use to write number strings. The symbol 9 is only there to remind us what base we're in. I would write it like this:
    (5*6*90) + (4*6*91) =
    (3*91 + 3*90) + [(2*91 + 6*90)*91] =
    2*92 + 9*91 + 3*90 =
    3*92 + 3*90, or written as a number string 303.
    A cleaner way of writing all the above in base 9 is:
    45*6 = (40 + 5)*6 = 40*6 + 5*6 = 260 + 33 = 303.
    The only thing about the above is that you have to keep in mind that you're working in base 9, so when multiplying and adding, your cap is 9 for carrying 1's (instead of 10), and you should know your single-digit multiplication tables in base 9 (or at least convert easily from base 10).
    To check our answer, 3039 in base ten is 3*81 + 3*1 = 3*82 = 240 + 6 = 246, and (45*6)9 in base 10 is (4*9 + 5*1)6*1 = 41*6 = 246 . Since We're used to base 10, I didn't write out the above in all its gory detail as I did with base 9. :biggrin:
     
    Last edited: Apr 9, 2005
  20. Apr 9, 2005 #19
    How did you get from (5*6*90) + (4*6*91) to
    (3*91 + 3*90) + [(2*91 + 6*90) and then the rest?

    This makes sense.

    This involves converting, which is fine.

    I will go through it again in my mind and then see.

    Cheers.

    The Bob (2004 ©)
     
  21. Apr 9, 2005 #20
    In base 9, 5*6 = 33. It's just the base 9 multiplication table. We're so used to our base ten multiplication table that we seldom remember that we pretty much memorized most of it as opposed to deriving every multiplication from first principles. You can avoid learning the base 9 table by doing the conversion from 30 in base ten to 33 in base 9 in your head.
    Where did I convert (besides pretending to know the base 9 multiplication table :wink: ) ?
     
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