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About Base Five

Date: 10/22/2001 at 23:09:05
From: David Prescott
Subject: Base ten and base five

I don't understand any of it. I've read time and time again and still 
I don't understand it. I can't do anything at all. I'm home schooled 
and my Mom is trying to get as much on it as she can. If you can tell 
me where I can get all the help I can, please!

Thanks, David

Date: 10/23/2001 at 09:11:44
From: Doctor Peterson
Subject: Re: Base ten and base five

Hi, David.

We have a number of explanations of this concept in the Dr. Math  FAQ:

   Number Bases   

There's a lot there, so you will have to look for any discussion that 
seems to be at your level; some are very introductory and others deal 
with more advanced questions. You might want to start here:

   Base Number   

This, dealing with base 2 (binary) may also help:

   Concepts of Adding in Base 2   

Here's a quick summary of base 5:

In our usual base 10 system, you can think of a number as a collection 
of coins: pennies (worth 1 cent), dimes (worth 10 cents), and dollars 
(worth 100 cents each). Each digit represents the number of one kind 
of coin:

   123 = 100 + 20 + 3
       = 1 x 100 + 2 x 10 + 3 x 1
       = 1 dollar, 2 dimes, 3 pennies

Each kind of coin is worth ten times as much as the next one; and you 
are only allowed up to 9 of any one kind. If you had ten pennies, you 
would change them to one dime.

So base ten means that the value of each place is a power of ten, and 
ten different digits (0 through 9) are needed.

Base 5 is just the same, but with "five" everywhere I used "ten." Our 
"coins" are therefore worth 1, 5, and 25 units (I can't call them 
cents, because the word "cent" means "hundredth"). We can have 0, 1, 
2, 3, or 4 in each digit. For example,

    123 (base 5) = 1 x 25 + 2 x 5 + 3 x 1
                 = 25 + 10 + 3
                 = 38

One way to get used to this idea is to make a physical representation 
of a base 5 number by using cups to represent each digit, and putting 
up to 4 counters in each cup. See what happens when you add two such 
numbers together. Or, use different colored chips (or paper 1, 5, and 
25 dollar bills) to represent the "coins"; one blue one might be worth 
five white ones, so when you add them together you have to use that 
rule to "make change."

This is just the basic concept. You probably have specific problems 
you've been unable to do; I can help you best if you show me those 
problems, tell me what you were able to do and where you got stuck, 
and then let me see your mistakes so I can correct them. Working 
together, we should be able to get you back on track pretty quickly.

- Doctor Peterson, The Math Forum   
Associated Topics:
Elementary Number Sense/About Numbers
Middle School Number Sense/About Numbers

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