Binary Number System
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! |
See how it is done in this little demonstration (press play):
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) = 13.625 in Decimal |
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
gets 2 times smaller (half as big). |
|
Example: 10.1
- The "10" means 2 in decimal,
- The ".1" means half,
- So "10.1" in binary is 2.5 in decimal
You can do conversions at Binary to Decimal to Hexadecimal Converter.
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 "one-oh-one"). 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: 1012
This way people won't think it is the decimal number "101" (one hundred and one).
Examples
Example: What is 11112 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 10012 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 1.12 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 10.112 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."