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Digital Computers and Binary


Date: 07/02/2000 at 15:23:19
From: Marcia
Subject: Digital computers and the binary number system

How do digital computers use the binary number system?


Date: 07/10/2000 at 16:32:55
From: Doctor TWE
Subject: Re: Digital computers and the binary number system

Hi Marcia - thanks for writing to Dr. Math.

Wow! There's a lot to that question, but I'll try to cut to the chase.

Computers are electronic devices that use small components called 
transistors to record and process information. As used in digital 
computers, each transistor is like an on/off switch. (They can also be 
used as amplifiers, where the output is a fixed multiple of the input, 
but that's not how they're used in computers.) When used to store 
numbers, a transistor that's "on" (or saturated) represents the digit 
1 and a transistor that's "off" (or cut off) represents the digit 0.

With a single transistor, therefore, I can only count 0 and 1. But as 
in the decimal system, I can use more digits to represent a number. 
How do we represent numbers larger than 9 in decimal? We add more 
digits. The next place value in decimal is 10 (because we had to stop 
at 9 with 1 digit), and we can proceed 10, 11, 12, etc. These 
represent:

     10   1  <- Place values
     --   -
      1   0  =  1*10 + 0*1  =  10 + 0  =  10
      1   1  =  1*10 + 1*1  =  10 + 1  =  11
      1   2  =  1*10 + 2*1  =  10 + 2  =  12
      :   :      :      :         :        :
      1   9  =  1*10 + 9*1  =  10 + 9  =  19
      2   0  =  2*10 + 0*1  =  20 + 0  =  20
      2   1  =  2*10 + 1*1  =  20 + 1  =  21
      2   2  =  2*10 + 2*1  =  20 + 2  =  22
      :   :      :      :         :        :
      9   9  =  9*10 + 9*1  =  90 + 9  =  99

Once I reach 99, I've exhausted all combinations of two digits, so I 
have to add a third digit to represent 100. Then I can continue to 999 
before requiring a fourth digit, and so on.

In binary, I can't represent a 2 as a single digit (I only have 0 and 
1 as digits), so I'll need a new place value for 2's. Thus the second 
place value in binary will be 2's and

     2   1  <- Place values
     -   -
     1   0  =  1*2 + 0*1  =  2 + 0  =  2 (decimal)
     1   1  =  1*2 + 1*1  =  2 + 1  =  3 (decimal)

Now I've exhausted all combinations of 2 digits, so I have to go to 3 
digits. My next place value will be 4 (since 3 was the largest number 
I could count to using 2 digits.) So I'll have:

     4   2   1  <- Place values
     -   -   -
     1   0   0  =  1*4 + 0*2 + 0*1  =  4 + 0 + 0  =  4 (decimal)
     1   0   1  =  1*4 + 0*2 + 1*1  =  4 + 0 + 1  =  5 (decimal)
     1   1   0  =  1*4 + 1*2 + 0*1  =  4 + 2 + 0  =  6 (decimal)
     1   1   1  =  1*4 + 1*2 + 1*1  =  4 + 2 + 1  =  7 (decimal)

My next place value will need to be 8, and I could continue.

Let's look at the place values for a moment. In decimal, our place 
values are:

     1,000    100     10      1

which is the same as:

      10^3   10^2   10^1   10^0

They are all powers of 10. Why? Because we're in base 10 and we have 
10 digits (0 through 9) to represent numbers. In binary, our place 
values are:

       8     4     2     1

which is the same as:

     2^3   2^2   2^1   2^0

They are all powers of 2. Why? Because we're in base 2 and we only 
have 2 digits (0 and 1) to represent numbers.

Using this system, we can develop rules for adding, subtracting, 
multiplying, dividing, representing fractions, etc. that parallel 
what we do in decimal.

For more information on binary and binary operations visit our FAQ on 
Number Bases at:

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

Computers use millions of transistors to represent numbers in binary 
and to do operations such as addition and subtraction. These 
transistors are usually arranged in groups of 8, 16, 32 or 64 (and 
sometimes 40 or 80) for convenience.

You might ask, "why didn't they just use decimal instead?" Because 
on/off is easy. It's easy to make a circuit that is either on or off, 
and to make circuits that detect whether another circuit is on or off. 
It is much more difficult to make circuits that can be in one of ten 
different states, and to make circuits that can detect which of those 
states another circuit is in. Think of an on/off light switch as 
opposed to those three-way switches. Designing a switch with ten 
different positions is even more complicated. Now imagine trying to 
detect which of the ten "brightnesses" the light bulb is at. So even 
though binary requires more digits (called "bits"), it is easier to 
work with on a circuit design level.

I hope this helps. If you have any more questions, or want more 
specific information, write back.

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

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