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Introduction to Bases in Math


Date: 02/27/2002 at 12:23:28
From: Billy Menhart
Subject: Bases in Math

I'm a home-schooler and I'm having trouble with the bases system. I 
don't understand it.

My question is:  Rewrite the base 10 numeral in base 5:  13

I don't understand. Please help. Thank you.


Date: 02/27/2002 at 13:02:40
From: Doctor Peterson
Subject: Re: Bases in Math

Hi, Billy.

Have you seen our FAQ on Number Bases?

   http://mathforum.org/dr.math/faq/faq.bases.html   

There are several basic explanations of bases there. I'll give a brief 
introduction.

You are given the number 13 in base ten; this means 1 ten and 3 ones. 
You want to write it in base five, as some number of fives and some 
number of ones. (If the number were big enough, you might also need 
25's, 125's, and so on, just as in base ten you might need hundreds or 
thousands.)

So here is 13:

     ___10___  1 1 1
    /        \ 
    oooooooooo o o o
    \________/ \___/
         1       3
        ten     ones

Let's group it by 5's rather than 10's:

     _5_   _5_  1 1 1
    /   \ /   \
    ooooo ooooo o o o
    \_________/ \___/
         2        3
       fives     ones

So we have 2 fives and 3 ones; in base five, we write this as 
23 (base 5).

Let's try a bigger number: 68 (base ten). We can group it in fives by 
just dividing by 5, rather than drawing pictures:

    68 / 5 = 13 r 3

So we have 13 fives and 3 ones.

But we're not done, because in base five we can't have any digits 
greater than 4 (just as in base ten we have no digits greater than 9). 
So now we have to group our 13 fives into groups of five fives. 
Divide again:

    13 / 5 = 2 r 3

This tells us that 13 is 2 fives and 3 ones; so 13 FIVES is 
2 twenty-fives and 3 fives. Our number is then

    68 (base ten) = 2 (25's) + 3 (5's) + 3 (1's)

which we write as 233.

     _25   _25   _5_   _5_   _5_  1 1 1
    /   \ /   \ /   \ /   \ /   \
    ooooo ooooo ooooo ooooo ooooo o o o
    ooooo ooooo
    ooooo ooooo
    ooooo ooooo
    ooooo ooooo
    \_________/ \_______________/ \___/
         2              3           3
    twenty-fives      fives        ones

Check it out:

    2x25 + 3x5 + 3x1 = 50 + 15 + 3 = 68

If you aren't sure what I mean by bases in the first place, try 
reading this explanation referred to in our FAQ, then come back here 
and see if you can follow what I said:

   Counting: Base 6, 12, 16
   http://mathforum.org/dr.math/problems/asplund5.31.96.html   

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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