Hexadecimal number systems help computers to process the large quantity of data they are supposed to handle. Hexadecimal number systems is a numerical system of base 16. Different combinations of the 16 basic digits are used after 9 in the decimal system. Understanding the conversion between hexadecimal and binary numbers is vital in the field of digital electronics just like understanding the conversion of rupees to dollar or dollar to euro is in the physical world. Before understanding these conversions let us try and familiarize binary and hexadecimal number systems.
The binary number system is a system of numbers with base 2. This numerical system has only two numbers- 0 and 1. We use some combinations of these two digits to generate the entire binary number system. So (0)10( decimal number)is represented as (00)2in binary and (1)10 is 01. The next numbers are 10,11 etc.
We can easily find the corresponding binary representation to any decimal digit by continuously dividing the digit by 2 till the quotient is 0. For example, 24 in binary representation can be found by :
24/2 = 12; reminder = 0
12/2 = 6; reminder = 0
6/2 = 3; reminder = 0
3/2 = 1; reminder = 1
1/2 = 0; reminder = 1
Hence , (24)10 = (11000)2
We can even represent decimal point numbers like 0.205 and 1.234 in the binary system. In this case, instead of dividing with 2, we multiply the numbers after decimal point with 2 till the numbers after decimal point become 0. For example, to represent 0.25 in the binary system,
Take 0.25 out of .25 and multiply with 2.
0.25*2=0.50; The number before decimal point = 0;
0.50*2 = 1.00; The number before decimal point = 1;
Hence, (0.25)10 = (0.01)2
Decimal Digit | Binary Digit |
0 | 0000 |
1 | 0001 |
2 | 0010 |
3 | 0011 |
4 | 0100 |
5 | 0101 |
6 | 0110 |
7 | 0111 |
8 | 1000 |
9 | 1001 |
10 | 1010 |
11 | 1011 |
12 | 1100 |
13 | 1101 |
14 | 1110 |
15 | 1111 |
Hexadecimal number systems is a numerical system of base 16. Here just as we use 10,11,12 etc after 9 in the decimal system, we use different combinations of the 16 basic digits of the hexadecimal number system. Let us see the representation of decimal numbers in the hexadecimal number system.
Decimal Digit | Hexadecimal Digit |
0 | 0 |
1 | 1 |
2 | 2 |
3 | 3 |
4 | 4 |
5 | 5 |
6 | 6 |
7 | 7 |
8 | 8 |
9 | 9 |
10 | A |
11 | B |
12 | C |
13 | D |
14 | E |
15 | F |
These are the basic hexadecimal numbers. Any decimal number can be converted to a hexadecimal number by dividing the number by 16 and then following a similar algorithm as seen above.
First, let us see how to convert a whole number from hexadecimal to binary.
Suppose we have a number E9A; To find the corresponding binary number we write each hex digit with the corresponding binary number. From the above table, we know,
E is 14 in decimal representation and 14 is 1110 in binary representation.
9 is 9 in decimal representation and 1000 in binary.
A is A in decimal and 111 in binary.
E | 9 | A |
1110 | 1001 | 1010 |
Hence (E9A)16 = 111010011010
Now let us see how to convert a decimal point hexadecimal number to binary. Convert 0.A39 to binary.
Here the rules are the same as above except we avoid the rightmost zeros in this case. These are called trailing zeros.
A | 3 | 8 |
1010 | 0011 | 1000 |
That is (0.A38)16 = (0.101000111)2
It will be very helpful to have a ‘hexadecimal to binary’ table. This can be achieved by combining the above tables.
Decimal Digit | Hexadecimal Digit | Binary Digit |
0 | 0 | 0000 |
1 | 1 | 0001 |
2 | 2 | 0010 |
3 | 3 | 0011 |
4 | 4 | 0100 |
5 | 5 | 0101 |
6 | 6 | 0110 |
7 | 7 | 0111 |
8 | 8 | 1000 |
9 | 9 | 1001 |
10 | A | 1010 |
11 | B | 1011 |
12 | C | 1100 |
13 | D | 1101 |
14 | E | 1110 |
15 | F | 1111 |
We have covered the basic idea of binary and hexadecimal number systems and the conversion of the ‘ hexadecimal to binary’ system. Let us try to summarize the rules:
Write the hexadecimal number neatly.
Underneath each digit write the corresponding decimal digit.
Underneath each decimal term write the corresponding binary digit.
String together all the digits, without any space in between.
Rewrite the answer after leaving out leading and trailing zeros. We have obtained our final answer.
1. What is the Binary Number System? What is the Hexadecimal Number System?
Answer: The binary number system is a system of numbers with base 2. This numerical system has only two numbers- 0 and 1. Combinations of these two digits form the entire system. Hexadecimal number system has the base 16 and also consists of 16 symbols. It contains numeric values 0-9 and alphabets A-F. A table is shown above. We can even represent decimal point numbers like 0.205 and 1.234 in the binary system.
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