Introduction to Bases in MathDate: 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/ |
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