Binary Number System (from http://www.mathsisfun.com/binarynumbersystem.html)
by techtiptom
Computers use binary digits. And some puzzles can be solved using binary numbers.
A Binary Number is made up of only 0s and 1s.
110100 
Example of a Binary Number 
There is no 2,3,4,5,6,7,8 or 9 in Binary!
How do we Count using Binary?
Binary 

0 
We start at 0  
1 
Then 1  
??? 
But then there is no symbol for 2 … what do we do? 
Decimal 

Well how do we count in Decimal? 
0 
Start at 0  
… 
Count 1,2,3,4,5,6,7,8, and then…  
9 
This is the last digit in Decimal  
10 
So we start back at 0 again, but add 1 on the left 
The same thing is done in binary …
Binary 

0 
Start at 0  
• 
1 
Then 1  
•• 
10 
Now start back at 0 again, but add 1 on the left  
••• 
11 
1 more  
•••• 
??? 
But NOW what … ? 
Decimal 

What happens in Decimal … ? 
99 
When we run out of digits, we …  
100 
… start back at 0 again, but add 1 on the left 
And that is what we do in binary …
Binary 

0 
Start at 0  
• 
1 
Then 1  
•• 
10 
Start back at 0 again, but add 1 on the left  
••• 
11 

•••• 
100 
start back at 0 again, and add one to the number on the left… … but that number is already at 1 so it also goes back to 0 … … and 1 is added to the next position on the left 

••••• 
101 

•••••• 
110 

••••••• 
111 

•••••••• 
1000 
Start back at 0 again (for all 3 digits), add 1 on the left 

••••••••• 
1001 
And so on! 
Decimal vs Binary
Here are some equivalent values:
Decimal: 
0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
Binary: 
0 
1 
10 
11 
100 
101 
110 
111 
1000 
1001 
1010 
1011 
1100 
1101 
1110 
1111 
Here are some larger equivalent values:
Decimal: 
20 
25 
30 
40 
50 
100 
200 
500 
Binary: 
10100 
11001 
11110 
101000 
110010 
1100100 
11001000 
111110100 
“Binary is as easy as 1, 10, 11.”
Position
In the Decimal System there are the Units, Tens, Hundreds, etc
In Binary, there are Units, Twos, Fours, etc, like this:
This is 1×8 + 1×4 + 0×2 + 1 + 1×(1/2) + 0×(1/4) + 1×(1/8) 
Numbers can be placed to the left or right of the point, to indicate values greater than one or less than one.
10.1  
The number to the left of the point is a whole number (10 for example)  
As we move further left, every number place gets 2 times bigger. 

The first digit on the right means halves (1/2).  
As we move further right, every number place 
Example: 10.1
 The “10” means 2 in decimal,
 The “.1” means half,
 So “10.1” in binary is 2.5 in decimal
Words
The word binary comes from “Bi” meaning two. We see “bi” in words such as “bicycle” (two wheels) or “binocular” (two eyes).
When you say a binary number, pronounce each digit (example, the binary number “101” is spoken as “one zero one”, or sometimes “oneohone”). This way people don’t get confused with the decimal number. 
A single binary digit (like “0” or “1”) is called a “bit”. For example 11010 is five bits long.
The word bit is made up from the words “binary digit”
How to Show that a Number is Binary
To show that a number is a binary number, follow it with a little 2 like this: 101_{2}
This way people won’t think it is the decimal number “101” (one hundred and one).
Examples
Example: What is 1111_{2} in Decimal?
 The “1” on the left is in the “2×2×2” position, so that means 1×2×2×2 (=8)
 The next “1” is in the “2×2” position, so that means 1×2×2 (=4)
 The next “1” is in the “2” position, so that means 1×2 (=2)
 The last “1” is in the units position, so that means 1
 Answer: 1111 = 8+4+2+1 = 15 in Decimal
Example: What is 1001_{2} in Decimal?
 The “1” on the left is in the “2×2×2” position, so that means 1×2×2×2 (=8)
 The “0” is in the “2×2” position, so that means 0×2×2 (=0)
 The next “0” is in the “2” position, so that means 0×2 (=0)
 The last “1” is in the units position, so that means 1
 Answer: 1001 = 8+0+0+1 = 9 in Decimal
Example: What is 1.1_{2} in Decimal?
 The “1” on the left side is in the units position, so that means 1.
 The 1 on the right side is in the “halves” position, so that means 1×(1/2)
 So, 1.1 is “1 and 1 half” = 1.5 in Decimal
Example: What is 10.11_{2} in Decimal?
 The “1” is in the “2” position, so that means 1×2 (=2)
 The “0” is in the units position, so that means 0
 The “1” on the right of the point is in the “halves” position, so that means 1×(1/2)
 The last “1” on the right side is in the “quarters” position, so that means 1×(1/4)
 So, 10.11 is 2+0+1/2+1/4 = 2.75 in Decimal
“There are 10 kinds of people in the world,
those who understand binary numbers, and those who don’t.”