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Concepts of Adding in Base 2


Date: 08/18/98 at 10:52:34
From: Andre Ross
Subject: Base 2

Dr. Math, 
         
I don't understand the whole concept of base 2 or how to add in base 
2. Please help.

Thank you,
                                                                                     
Andre Ross


Date: 08/19/98 at 12:52:56
From: Doctor Peterson
Subject: Re: Base 2

Hi, Andre. Let's see if we can relate base two to something you can 
picture easily. You've probably seen an odometer in a car, or a tape 
counter in a cassette player, or things like that. They have a set of 
wheels, each of which has the ten digits 0, 1, ... 9 on it. The one on 
the right is turned by whatever is being counted, and each time it 
finishes a turn, going from 9 back to 0, it pulls the wheel to its 
left one space, say from 3 to 4:

    +-+-+-+-+
    |0|0|3|9|
    +-+-+-+-+
          /|
    +-+-+-+-+
    |0|0|4|0|
    +-+-+-+-+

When that digit reaches 9, then rolls around to 0, it pulls the next 
digit forward one place with it:

    +-+-+-+-+
    |0|0|9|9|
    +-+-+-+-+
        / /|
    +-+-+-+-+
    |0|1|0|0|
    +-+-+-+-+

This way, the rightmost wheel counts "ones," the second counts "tens," 
and so on.

A binary counter would work the same way, but would only have two 
digits on each wheel: 0 and 1. So when one digit rolled around from 1 
to 0, it would pull the next digit forward:

    +-+-+-+-+
    |0|0|0|1| = 1
    +-+-+-+-+
          /|
    +-+-+-+-+
    |0|0|1|0| = 2
    +-+-+-+-+
           |
    +-+-+-+-+
    |0|0|1|1| = 3
    +-+-+-+-+
        / /|
    +-+-+-+-+
    |0|1|0|0| = 4
    +-+-+-+-+

This is the binary system: representing numbers with nothing but 0 and 
1. We can interpret a binary number by noticing that the rightmost 
digit counts every time (counting "ones"); the next digit counts every 
second time (counting "twos"); the next counts "fours," and so on. So 
for instance, the binary number 0011 means 1 one and 1 two, which is 
1 + 2 = 3.

Now how do you add? Again, it's the same as for ordinary decimal 
numbers, except that you go by twos. Picture one of those little 
plastic money counters you may have seen, that look like the odometer 
I've just been talking about but have a button above every digit that 
moves that wheel once for every push. To add 23 to the total, I push 
the rightmost button three times (adding 3 "ones") and the second 
button twice (adding 2 "tens"). If either wheel is pushed past 9, it 
moves the wheel next to it, just as when it's counting. (We call that 
"carrying".)

    +-+-+-+-+
    |0|0|3|9| + 3
    +-+-+-+-+
          /|
    +-+-+-+-+
    |0|0|4|2| + 20
    +-+-+-+-+
         | 
    +-+-+-+-+
    |0|0|6|2|        39 + 23 = 62
    +-+-+-+-+

A binary counter would work the same, but we'd never have to push any 
button more than once, since each digit we have to add would be a 0 
or a 1!

    +-+-+-+-+
    |0|0|1|0| + 1
    +-+-+-+-+
           |
    +-+-+-+-+
    |0|0|1|1| + 10
    +-+-+-+-+
        /| 
    +-+-+-+-+
    |0|1|0|1|        10 + 11 = 101
    +-+-+-+-+       ( 2 + 3  = 5  )

When we write this down, it looks just like adding decimal numbers, 
except that we only write ones and zeroes, and when we add 1 + 1 we 
get 10, so we write down 0 and carry 1:

     1   <-- carry
      10
    + 11
    ----
     101

So working with binary is actually much easier than decimal because 
you hardly have to remember any addition facts, but everything else is 
the same.

If you want more, here's a nice explanation of how to add, subtract, 
multiply, and divide in binary, from our Dr. Math archives:

   http://mathforum.org/dr.math/problems/matt4.7.97.html   

- Doctor Peterson, The Math Forum
Check out our web site! http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory
Middle School Number Sense/About Numbers

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