BINARY NUMBER FUN
Let's begin with a binary number
Suppose it is:
10011101
(it wouldn't have to be 10011101,
I just chose it because I thought it looked neat)
The first digit (starting from left) of the number is 1.
(1)0011101.
This first digit is the same as saying (1*2^7)
**Note:(^7 is the same as saying "to the 7th power")**
If we look at binary number 10011101
--------
then the power used for each digit --> 76543210
(directly under the binary number)
You once again see that for the first
digit (1) you should use (1*2^7)
Got it so far?
So here comes the fun stuff
(1) 0 0 1 1 1 0 1 =(1*2^7)
1 (0) 0 1 1 1 0 1 =(0*2^6)
1 0 (0) 1 1 1 0 1 =(0*2^5)
1 0 0 (1) 1 1 0 1 =(1*2^4)
1 0 0 1 (1) 1 0 1 =(1*2^3)
1 0 0 1 1 (1) 0 1 =(1*2^2)
1 0 0 1 1 1 (0) 1 =(0*2^1)
1 0 0 1 1 1 0 (1) =(1*2^0)
Notice how I put the digit in the ( ) as the first number in the
equation ( *2^#) and that # begins with 7 on the first row and then one is
subtracted every time until the last row is to 0? Also notice that in the
last row you are at the last ( ) digit?
If you don't see what I am talking about try writing the above number chart
on paper and looking for patterns. It is all a game of patterns.
Well now let's take a look at the column on the right, and get out our
calculators.
(1*2^7)
(0*2^6)
(0*2^5)
(1*2^4)
(1*2^3)
(1*2^2)
(0*2^1)
(1*2^0)
If you type in (1*2^7) you should get the answer to be 128.
**Note: Remember that exponents (^#) has authority over multiplication so
you should be sure to put (2^#) in your calculator before you multiply it
by 1 or 0**
So do this for each row and you should end up with these answers:
(1*2^7) = 128
(0*2^6) = 0
(0*2^5) = 0
(1*2^4) = 16
(1*2^3) = 8
(1*2^2) = 4
(0*2^1) = 0
(1*2^0) = + 1
---
157
Now you add them all up and you should get the base-10 answer for
binary number 10011101.
The answer is 157. **YOU'RE FINISHED** Wasn't that fun?
____
I know you don't want to hear anymore, but let's say you know the base-10
number, and what you are looking for is the binary number. Can you figure
that one out? Let's work it together :)
We'll use the above example to make it easier.
Let's say the base-10 number is 157.
How do you find out what the binary number is?
Well, to do this let's remember that 11111111 =255.
If you can't remember that then work it out the long way.
(1) 1 1 1 1 1 1 1 =(1*2^7) = 128
1 (1) 1 1 1 1 1 1 =(1*2^6) = 64
1 1 (1) 1 1 1 1 1 =(1*2^5) = 32
1 1 1 (1) 1 1 1 1 =(1*2^4) = 16
1 1 1 1 (1) 1 1 1 =(1*2^3) = 8
1 1 1 1 1 (1) 1 1 =(1*2^2) = 4
1 1 1 1 1 1 (1) 1 =(1*2^1) = 2
1 1 1 1 1 1 1 (1) =(1*2^0) = + 1
---
255
Okay either way now you know 11111111 (binary)= 255(base-10)
We will use the column on the far right (rewritten below):
=(1*2^7) = 128
=(1*2^6) = 64
=(1*2^5) = 32
=(1*2^4) = 16
=(1*2^3) = 8
=(1*2^2) = 4
=(1*2^1) = 2
=(1*2^0) = + 1
---
255
Okay now we will look at our number, 157.
We will compare it to the first row in the 157
far right column (also re-written above). - 128 (1)
Think to yourself . o O ( can 128 go into 157? ) -----
-Yes- So.. write down a 1, then subtract 128 from 157. 29
(as shown on right) - 0 (0)
Write down your new answer 29. -----
Now think . o O ( can 64 go into 29? ) 29
-No- So.. write a 0, then subtract 0 from 29. - 0 (0)
Write down your new (same) answer 29. -----
Now think . o O ( can 32 go into 29? ) 29
-No- So.. write a 0, then subtract 0 from 29 again. - 16 (1)
Write down your new (same) answer 29. -----
Now think . o O ( can 16 go into 29? ) 13
-Yes- So.. write down a 1, then subtract 16 from 29. - 8 (1)
Write down your new answer 13. -----
Now think . o O ( can 8 go into 13? ) 5
-Yes- So.. write down a 1, then subtract 8 from 13. - 4 (1)
Write down your new answer 5. -----
Now think . o O ( can 4 go into 5? ) 1
-Yes- So.. write down a 1, then subtract 4 from 5. - 0 (0)
Write down your new answer 1. -----
Now think . o O ( can 2 go into 1? ) 1
-No- So.. write a 0, then subtract 0 from 1. - 1 (1)
Write down your new (same) answer 1. -----
Now think . o O ( can 1 go into 1? ) 0
-Yes- So.. write down a 1, then subtract 1 from 1.
You are left with 0. You are done. Now look along the side of your long
subtraction equation in the ( ). Write down those numbers in order and you
have your binary number. Therefore, in this equation we have the binary
number:
10011101.
Isn't that great? If at the end of your long subtraction equation you are
not left with a 0, you need to redo the long subtraction equation paying
close attention.
1. **Make sure you are subtracting in order and using the correct
numbers: 128,64,32,16,8,4,2,1** [these numbers are simply (1*2^7),
(1*2^6), etc.]
2. **Double check to make sure you subtracted correctly (ie. 29-16 does
not equal 15)**
3. **and finally make sure the base-10 number is not more than 255**
[If it is, you can not make an 8-digit binary number out of it; instead
try a higher digit binary number, and make sure to the formulas
correspond to that new digit]
That's all, I think. Hope this helped. Print it up, study it, and practice it. Or just look at it once and throw it away. I don't care :) --Shannon-- @}>--,--`---