Binary System

Date: 5/22/96 at 11:33:35
From: Anonymous
Subject: Binary

Can you please explain to me how to use binary?  Can I use it on my 
calculator?  How about on the microwave?  

Thanks.  Justin Rae

Date: 7/10/96 at 9:38:7
From: Doctor Patrick
Subject: Re: Binary

Hi!  That's a really good question. 

Binary is a system of counting that uses only 0 and 1  to make up all 
of the numbers, just like we use 0-9 to make up all of the numbers in 
our number system.

In our system we use the symbols 0 1 2 3 4 5 6 7 8 and 9 to make up 
all of the numbers.  Whenever we want to go above a 9 we have to add a 
new place to the number that is ten times greater then the one before 
it.  

For example, if our number is 99 (9 tens and 9 ones) and we want to 
add to it, we will need to add a hundreds place (since a hundred is 
ten times greater then ten).  The number 256 is equal to 2 hundreds, 
5 tens, and 6 ones.  

Have you learned about exponents yet?  If you have, then you know that 
that same number could also be equal to 2*10^2 + 5*10^1 + 6*10^0.  Any 
place beyond the hundreds place can also be written as increasing 
powers of 10. For example, the thousands place is 10^3 and the ten 
thousands place is 10^4.

If you don't know what exponents are, here's a quick lesson.  
Basically an exponent tells you to multiply a number by itself as many 
times as the number that the exponent is equal to.  10^2 means 10*10 
and 10^3 means 10*10*10.  

Likewise 2^2 = 2*2 and 2^3 = 2*2*2.  Does that make sense?  What 
would 3^2 and 3^3 be equal to?

Binary numbers work just like ours do, only they only use the symbols 
0 and 1, so whenever you want to go above a 1 you need to add a new 
place that is TWO times greater then the one before it.  For example, 
101 in binary is 1 four plus 0 twos plus 1 one. What number would that 
be equal to in our system?  

Using exponents, this is 1*2^2 + 1*2^1 + 1*2^0.  

If you added 1 to 101 in binary it would become 110, since to go 
beyond the number one you need to move on to the next place.  Now 
there would be 1 four and 1 two and 0 ones.  How do you think the 
number will change as we keep adding 1 to it?

Let's see, 110 + 1=111 (1 four, 1 two, one 1), right?  That part is 
easy.

Now, what happens if we add another 1?  The 1 in the ones place 
becomes equal to 2, so it has to move over into the twos place.  But 
since that makes 2 twos (or 4) we have to make that a 0 also and add 
to the fours place. With one four already there, we now have 2 fours, 
or 8, and must add a new place for eights (remember that each place is 
twice the one before it, so the eights place comes after the fours).  
This makes our new number 1000 since we have 1 eight (1*2^3) + 0 fours 
(0*2^2) + 0 twos (0*2^1) + 0 ones (0*2^0).  

In our system what would this equal?

Both your calculater and your microwave use binary, since that is the 
system that computers (and other electronic equipment) use to count.
Even though we see all ten numbers when we use them, it is as a result
of the computer (or other equipment) interpreting them for us.  All 
that they use is the binary system.  You will probably never see the 
binary side of most computers or electronic equipment like your 
microwave, but many calculaters do have a feature for using binary.

Hope this helps to answer some of your questions,

-Doctor Patrick,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/