In my experience of teaching networking many students struggle with IP addresses because they lack an basic understanding of binary numbers.
An understanding of binary numbers,the binary system, and how to convert between binary and decimal is essential for anyone involved in computers, coding, and networking.
Binary 101 – What You will Learn
- The Basics of Number Bases -Base 10, Base 2 and Base 16
- How to convert binary to decimal and vice versa
- How to convert binary numbers to Hexadecimal and vice versa,
- How to convert Hexadecimal to decimal and vice versa,
Overview of Base 10 System or Decimal Numbers
Before we learn about the binary number system we will look in more detail at our normal decimal numbering system.
The principals are the same for all numbering systems, and they are easier to learn using a system that you are more familiar with.
Firstly our decimal system uses 10 as a base and the numbers range from 0 to 9
Let’s look at a few example numbers
We start with the three digit number 129 (one hundred and twenty nine).
This made up of 100 +20 +9 =129
If we look at the diagram below we see that as we move from right to left the columns increase by a factor of 10.
The 2 in the second column isn’t 2 but is 2*10=20 and the 1 in the third column isn’t 1 but 1*10*10=100.
means 10 raised to the power of 0. This is equal to 1 and represents our units column.
The short table below shows a few more entries using the power notation.
When writing decimal numbers we rarely write column values above the numbers as we already know what they are so we simply write:
129 and not
I have introduced the power notation because it is fundamental to understanding binary numbers.
The minimum number possible with three digits is 000 and the maximum is 999. For numbers larger than 999 we need a 4th column which with be the 1000s column.
The Binary Number System
Binary numbers are base 2 numbers, and have only two values – 0 and 1.
If we look at a binary number like 101, then we can again assign column values as we did with our decimal number, but this time we use 2, and not 10 as the base.
So binary 101 binary has 1 in the units column,0 in the 2s column and 1 in the 4s column.
Again if we work our way from right to left then:
The 1 is a 1 as it is in the units column but the next 1 is not 1 but 1*4=4
Binary numbers use base 2 and so the columns are
Binary To Decimal Conversion
Let’s look at a few binary numbers and convert them to decimal
We start with the three digit binary number 101 (see image above
The number can be converted to decimal by multiplying out as follows:
1*1 + 0*2 + 1*4 = 5
The maximum value we can have with three binary digits is 111 = decimal 7 calculated as follows-
1*1 + 1*2 + 1*4
1011 binary = 1*1+1*2+0*4+1*8=11
1111 binary = 1*1+1*2+1*4+1*8=15
Try It Yourself
1001 binary = ?
1100 binary = ?
Converting from Decimal to Binary
How do you convert a decimal number to a binary number.
Example what is decimal 10 in binary.
The way I do it is by using the following list of 2 multiples.
Here’s an handy chart
Note: error in diagram above should be 27=128
The procedure is to subtract the number with largest power of two from the decimal number
The largest power of two number that we can subtract is 8 which is 23.
So 10-8 =2
we now do the same with the remainder so the largest number we can subtract is 2 which is =21
so we have 1 eight , No fours, 1 two, No units = 1010 = 23+ 21.
Example 2: Decimal 13 to binary code
1 eight , 1 four, 0 two, 1 units = 1101.
Example 3: Decimal 7 to binary code
0 eight , 1 four, 1 two, 1 units = 0111.
Answers to try it yourself questions
1001 binary = 9
1100 binary = 12
Bytes and Octets and Hexadecimal Numbers
In computers, coding and networking 8 bit numbers are common.
An 8 bit number is known as an octet, and also more commonly it is called a byte. See Wiki for details.
Binary to Decimal and Decimal to Binary Conversion 8 Bit Numbers
An 8 bit binary number can represent a maximum of decimal 255= binary 11111111.
Calculated as follows:
1*128 +1*64+1*32+1*16+1*8+1*4+1*2+1+1 = decimal 255
Here is another 8 bit binary number –01101011.
To convert it to decimal we write the number with the column numbers above, as follows:
if we convert our columns to decimal equivalents using the following chart.
then the binary number 01101011 = 1*1 +1*2+0*4=1*8+0*16=1*32+1*64+0*128
Notice it consists purely of 1’s and 0’s.
To convert this number into decimal we need to understand what each 1 represents.
If we write the column values above the numbers then it becomes easy to convert the binary number to decimal.
Decimal to Binary Conversion Example
A final larger example convert decimal 200 to binary code
200=128+64+8=27+ 26+ 23 =11001000
Once you are happy with the process then you can use a binary to decimal calculator like the one on windows.
This converts binary numbers to decimal
and this converts decimal numbers to binary
Understanding Hexadecimal Numbers
A hexadecimal number (base 16) requires 4 bits and and has a maximum value of 15. It uses the symbols 0-9,A,B,C,D,E,F.
They are represented in binary form as follows:
A byte (8 bits) can be represented as two hexadecimal numbers.
FF=binary 11111111 and decimal 255
F0=11110000 binary and decimal 240
Binary Numbers Basics Quiz
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Basic test of students understanding of Binary Numbers.
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Question 1 of 10
Is This a valid binary number
Question 2 of 10
Is this a valid decimal number
Question 3 of 10
Convert decimal 8 to binaryCorrect
Question 4 of 10
Convert decimal 3 to binaryCorrect
Question 5 of 10
Convert binary 0100 to decimalCorrect
Question 6 of 10
What is 01001011 in decimal?Correct
Question 7 of 10
What is 200 in binaryCorrect
Question 8 of 10
What isFE in binaryCorrect
Question 9 of 10
What is 01001011 in hexadecimalCorrect
Question 10 of 10
Is AG a valid Hexadecimal number?Correct
I produced a video that covers the above if you prefer videos- Understanding Binary numbers video
Resources and Related Articles:
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- IPv6 Explained for Beginners
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- JSON Basics For Beginners-With Examples and Exercises