The hexadecimal number is one of the number systems which has a value is 16 and it has only 16 symbols − 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and A, B, C, D, E, F. Where A, B, C, D, E, and F are single-bit representations of decimal values 10, 11, 12, 13, 14, and 15 respectively. Whereas, The binary is a simple kind of number system that uses two digits of 0 and 1 (value of base 2). Since digital electronics have only these two states (0 and 1), the binary number is most preferred by modern computer engineers, networking and communication specialists, and other professionals. The Hexadecimal is useful because large numbers that represented using fewer digits. For example, color values and MAC addresses are often represented in hexadecimal. Additionally, hexadecimal is easy for humans to understand than binary. Programmers often use hexadecimal to represent binary values as they are simpler to write and check than when using binary.
Converting a number from Hexadecimal to Binary Number System
There are two ways to convert hexadecimal to binary, which are explained below.
1- The direct method(grouping method) of hexadecimal to decimal then to binary conversion using table.
2- Convert hexadecimal to decimal and then decimal to binary
1- Direct Method
Before we directly move to the steps have a look at the conversion table that we are going to use while converting.
Hex to the binary table:
To convert hexadecimal numbers to binary follow the steps:-
◆ Step 1 −Split the hex number into individual values.
◆ Step 2 −Convert each hex value into its decimal equivalent.
◆ Step 3 −Next, convert each decimal digit into binary, making sure to write four digits for each value.
◆ Step 4 −Combine all four digits to make one binary number.
Examples :
Example 1 - Convert the A25 into binary.
We can obtain the decimal equivalent of A25 using the conversion table as,
A=(10)₁₀, 2=(2)₁₀, and 5=(5)₁₀
Again using the conversion table we can obtain the binary equivalent of the decimal equivalent we obtained in the last step as,
(10)₁₀=(1010)₂,(2)₁₀=(0010)₂,and(5)₁₀=(0101)₂
Now, combine all the binary numbers in the same order as they are in the original number to obtain the final binary equivalent number
(A25)₁₆=(101000100101)₂
Example 2 - Convert the (0.CA5)₁₆ into binary.
For converting hexadecimal to binary with a decimal point, we use a similar method to method 2, which is using the conversion table. Remember that the steps will be the same, but do not take the zero placed on the leftmost side (trailing zeros) into consideration.
We can obtain the decimal equivalent of (0.CA5)₁₆ using the conversion table as,
C=(12)₁₀,A=(10)₁₀,and5=(5)₁₀
Again using the conversion table we can obtain the binary equivalent of the decimal equivalent we obtained in the last step as,
(12)₁₀=(1100)₂,(10)₁₀=(1010)₂,and(5)₁₀=(0101)₂
Now, combine all the binary numbers in the same order they are in the original number to obtain the final binary equivalent number
(0.CA5)₁₆=(110010100101)2
2- Convert hexadecimal to decimal and then decimal to binary
This method requires both multiplication and division of numbers using the respective base numbers. The hexadecimal base number is 16, the base number of a decimal number is 10, and the base of a binary number is 2. Let us look at the steps:
◆ Step 1 − Write the hexadecimal number and find its equivalent decimal number.
◆ Step 2 − To find the decimal equivalent, we multiply each digit with 16n-1, where the digit is in its nth position.
◆ Step 3 − After multiplying the numbers, add the product of those numbers to obtain the decimal number.
◆ Step 4 − To convert decimal to binary, we divide the decimal number by 2 by keeping the remainder aside and quotient dividing the by 2 until we arrive at zero.
◆ Step 5 − Once the quotient is zero, we arrange the remainder from bottom to top. Reverse order to obtain the binary number.
Examples :
Example 1 - Convert a hexadecimal number (A2B)₁₆ to its equivalent Binary number.
Steps: First convert the hexadecimal number to its equivalent decimal number as shown below:
(A2B)₁₆ = (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 as 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 (E)₁₆ to its equivalent binary number.
Steps: First we will convert the given hexadecimal number to its equivalent decimal number as shown below:
(E)₁₆= 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 (1110)₂
Therefore the binary equivalent of the hexadecimal E16 is (1110)₂
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