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."