Introduction to Hexadecimal and Binary system
Hexadecimal number systems help computers process the large quantity of data they are supposed to handle. Hexadecimal number systems are the 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.
Binary Number System
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)2 in 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 :
\[\frac {24}{2}\] = 12; reminder = 0
\[\frac {12}{2}\] = 6; reminder = 0
\[\frac {6}{2}\] = 3; reminder = 0
\[\frac {3}{2}\] = 1; reminder = 1
\[\frac {1}{2}\] = 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 the decimal point with 2 till the numbers after the 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
Some Decimal Numbers in Binary Representation
Hexadecimal Numbers
The hexadecimal number system 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.
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.
Conversion of Hexadecimal to Binary
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.
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.
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.
Hexadecimal to Binary Table
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.
Examples of Conversion of Hexadecimal (Hex) numbers to Binary system
Example 1: Convert a hexadecimal number A2B16 to its equivalent Binary number.
Steps: First convert the hexadecimal number to its equivalent decimal number as shown below:
A2B16 = (A*162) + (2*161) + (B*160)
= (A*256)+(2*16)+(B*1)
= (10*256)+32+11
= 2560+43
= 2603 (Equivalent decimal number)
We will now convert this decimal number into its binary form:
Taking the base value is 2, we divide 2603 up until we receive a quotient equal to zero.
We then write the obtained set of quotients from right to left as 1010001010112
Example 2: Convert the hexadecimal number E16 to its equivalent binary number.
Steps: First we will convert the given hexadecimal number to its equivalent decimal number as shown below:
E16= E*160
=E*1
=E
=14 (Decimal number value obtained from the table above)
We shall now convert this decimal number to its binary equivalent
This can be done by dividing the number 14 by 2 until a quotient of zero is achieved.
The unique set of quotients is listed and written from right to left.
This shall be represented as 11102
Therefore the binary equivalent of the hexadecimal E16 is 11102
Example 3: To convert the hexadecimal number 3016 to its binary equivalent.
Steps: In order to obtain the binary value for the given hexadecimal number we follow the given steps in order:
First, we obtain the decimal value of the given hexadecimal number as
3016 = (3*161)+ (0*160)
=48+0
=48 (Decimal value of 3016)
Now in the following step, we can convert the decimal to binary, by dividing the value by 2 such that the ultimate quotient obtained in the end is zero. The binary number obtained is 1100002.
Therefore the binary equivalent of the hexadecimal 3016 is 1100002
Uses of Binary system
Following are the most important uses of Binary system in the modern world:
We all know the importance of computers and computer systems in our work and societal structure. It will be interesting to know that all computer systems are purely dependent on the binary numeral system for data storage, retrieval, processing, and essentially all machine-level operations.
In communication systems, signals are “digitized” by means of a binary system.
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FAQs on Convert Hexadecimal to Binary
1. What is the Binary Number System? What is the Hexadecimal Number System?
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. The 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.
2. How do you convert hexadecimal to binary?
Using the conversion system, a number can be converted from hexadecimal to binary from the following steps:
First, find the number of digits in the given hexadecimal number.
Say if there are ‘n’ number of digits, multiply each of the digits with 16n-1, where the digit is in its nth position.
Add each of the terms post multiplication.
The result is the decimal number.
In the next step, we will convert this decimal number into a binary number.
We do this by dividing the decimal by 2. Note the remainder and continue to divide by 2 until the quotient is zero.
Finally, we write the remainders in reverse order. The resultant number is our binary number.
3. How do you convert binary to hexadecimal?
In order to convert a binary number to a hexadecimal number, follow the steps below:
Assume that we have a base value of 2.
Now segregate and create groups of 4 digits each starting from right to left and look up all the values for each of these groups in the table above. In case the last group is in a deficit of 4 digits, simply add zero to the left to make it a group of 4 digits.
Then we get the desired result in base 16.
We now have the required hexadecimal value of the binary digit.
4. What are the various number systems in mathematics?
There are some major number systems used since ancient times in major world civilisations. The most commonly used number system has been the decimal number system. Other commonly used systems are:
The Integer system (developed from the Hindu- Arabic numerical system)
The Urinary system (in which every natural number is denoted by a corresponding number of symbols such as if the selected symbol is ‘/’, then the number seven will be denoted by ‘///////’ Tally marks use this way of numeral representation.
The Chinese numeral system
The Positional system (where the place-value notation is used), and lastly
The Binary system
5. What are the characteristics of a number system?
Following are the major characteristics of a number system:
Firstly, they represent a useful set of numbers such as integers, rational numbers, etc
Second, they give a unique standard representation for every number in the number system
Lastly, they reflect the algebraic and arithmetic structure of numbers.