Binary SystemDate: 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/ |
Search the Dr. Math Library: |
[Privacy Policy] [Terms of Use]
Ask Dr. Math^{TM}
© 1994- The Math Forum at NCTM. All rights reserved.
http://mathforum.org/dr.math/