What is a Binary Number?
We are used to a decimal number system where the numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0 are used to represent quantities. In order for the system to work, we use place values - for example counting from 0 to 15 requires that we place a 1 in the 10s column to represent the number 10 and higher because there are no single numbers available to represent those quantities.
If we continued counting to 150 we find that we need to place a 1 in the hundreds column after 99 in order to represent 100 units. Once again, the system requires that once all possible combination of numbers are taken, we simply add 1 to the next place value.
A decimal number system is known as base 10.
This system is not unique to a decimal system. It can be applied just as successfully to binary which uses just two numbers 0 and 1.
A binary number system is known as base 2.
The base 2 number system allows only two numbers to be used before starting the count again at the next place value. The two usable numbers in binary are 0 and 1. Place values are used in the same way as our decimal system.
Related Information
The following table shows how the place values increment when counting in binary.
DECIMAL
BINARY
BINARY VALUE VALUE
VALUE Place Values Place Values
Place Values 10 1 16 8 4 2 1
0 0
1 1
2 1 0
3 1 1
4 1 0 0
5 1 0 1
6 1 1 0
7 1 1 1
8 1 0 0 0
9 1 0 0 1
1 0 1 0 1 0
1 1 1 0 1 1
1 2 1 1 0 0
1 3 1 1 0 1
1 4 1 1 1 0
1 5 1 1 1 1
1 6 1 0 0 0 0
1 7 1 0 0 0 1
1 8 1 0 0 1 0
All binary values can be calculated using place values of 2^n
In the example above, the place values for the first 5 digits are calculated as such:
Place Value2^42^32^22^12^0
2^42^32^22^12^0Decimal Value16 8 4 2 1
16 8 4 2 1Example 1. Using the aforementioned place values you can calculate any decimal equivalent. For example the decimal number 23 is represented as 10111 in binary.
Place Value2^42^32^22^12^0
2^42^32^22^12^0Decimal Value16 8 4 2 1
16 8 4 2 1Binary Bit 1 0 1 1 1 = 10111=16+4+2+1 = 23
1 0 1 1 1 = 10111=16+4+2+1 = 23The number 23 (10111 binary) was calculated by adding the columns that have a 1 in their place value. In this case 23 was calculated in binary by adding the 16 + 4 + 2 + 1 place values.
Example 2. The number 19 can be represented as:
Place Value2^42^32^22^12^0
2^42^32^22^12^0Decimal Value16 8 4 2 1
16 8 4 2 1Binary Bit 1 0 0 1 1 = 10011=16+2+1 = 19
1 0 0 1 1 = 10011=16+2+1 = 19The number 19 (10011 binary) was calculated by adding the columns that have a 1 in their place value. In this case 19 was calculated in binary by adding the 16 + 2 + 1 place values.
Practical Use for Binary
Binary numbers are used in digital circuits because the numbers 0 and 1 can be logically represented by varying voltage levels over a period of time. Digital systems often use a +2V to +5V supply to represent a logical 1 and a voltage input below 0.8V to represent a logical 0. This way, digital circuits can transmit and perform calculations on binary numbers by simply varying voltage levels.
Binary numbers are also used extensively in IP addressing. An IP address is a decimal representation of an 8 bit binary number, this is why the highest possible IP address is 255 (2^7 = 256). So each IP address octet (0 to 255) can be derived from its 8 binary bits.
