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In mathematics we use decimal numbers, that is, 0 to 9. But in digital system we use binary number system which has only 0 and 1. There are other number systems such as octal and hexadecimal which are used in digital systems. A decimal number has a base of 10, a binary number has a base of 2, an octal number has a base of 8 and a hexadecimal number has a base of 16.

Hexadecimal number is way to represent the number, which base is 16. We are going to convert a number in base 16 to an equivalent number in base 10. It is generally use in digital electronic system. A hexadecimal number can be converted to a decimal number and vice versa.

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Numbers to the base 16 is called as hexadecimal numbers. It uses digits from 0 to 9 and alphabets from A to F. That is, it will be a combination of numbers and alphabets. As in the decimal system we have the the numbers 0 to 9, whereas in hexadecimal we have numbers 0 to 9, as well as numbers 10 to 15 are represented by alphabets A to F respectively.
10 - A, 11 - B, 12 - C, 13 - D, 14 - E, 15 - F
A hexadecimal number A67 = $10 \times 16^2 + 6\times 16^1 + 7\times 16^0$.

Examples: 1) 6FDA 2) 2A9B 3) EB2

Decimal System

Numbers with base 10 is called as decimal numbers. Decimal numbers include digits from 0 to 9. Decimal number system is the basic number system and is widely used in everyday mathematics. The decimal numbers work in power of 10. If a number is given as 567, it means $5\times10^2 + 6\times 10^1 + 7\times 10^0 = 500 + 60 + 7$. The decimal system is used extensively in mathematics but when it comes to digital systems the binary, octal and hexadecimal systems are used. So, it becomes important to know conversion from decimal system to these systems and vice versa.

Examples: 1) 25 2) 984 4) 21

How to Convert Hexadecimal to Decimal

See the below steps -

Step 1: First we find the number of hexadecimal digits in the number. Let there be n numbers.

Step 2: Then we multiply each hexadecimal digit with $16^{n-1}$, when n is equal to number of position from right side.

Step 3: Then we add each number after multiplication.

Step 4: The resultant is equivalent hexadecimal number of the given decimal number.

If the given number contain decimal then

Step 1: Take digits from the right of decimal point. Let the value of the position of digit to the right of decimal be m. Take a hexadecimal number 2.51.

Step 2: Multiply each digit after decimal with $\frac{1}{16^m}$, where m is the number of position of digit from the decimal point. 2 = $2\times 16^0$
5 = $5\times \frac{1}{16}$
1 = $1\times \frac{1}{16^2}$

Step 3: Add all the obtained numbers.Obtained number is 2 + 0.3125 + 0.00390625 = 2.31640625.

Step 4: The resultant is equivalent hexadecimal number of the given decimal number.

Below are the examples on converting Hexadecimal to Decimal -

Solved Examples

Question 1: Convert 7B$_{16}$ into decimal number.
Solution:

Given hexadecimal number is 7B$_{16}$.

7B$_{16}$ = 16$^1$ $\times$ 7 + 16$^0$ $\times$ B

= 16 $\times$ 7 + 1 $\times$ B

= 112 + 1 $\times$ 11

= 112 + 11

= 123

Question 2: Convert 7E$_{16}$ into decimal number.
Solution:

Given hexadecimal number is 7E$_{16}$.

7E$_{16}$ = 16$^1$ $\times$ 7 + 16$^0$ $\times$ E

= 16 $\times$ 7 + 1 $\times$ E

= 112 + 1 $\times$ 14

= 112 + 14

= 126

Question 3: Convert CA$_{16}$ into decimal number.
Solution:

Given hexadecimal number is CA$_{16}$.

CA$_{16}$ = 16$^1$ $\times$ C + 16$^0$ $\times$ A

= 16 $\times$ C + 1 $\times$ A

= 16 $\times$ C + A

= 16 $\times$ 12 +9

= 192 + 9

= 202

Question 4: Convert 20$_{16}$ into decimal number.
Solution:

Given hexadecimal number is 20$_{16}$.

20$_{16}$ = 2 $\times$ 16$^1$ + 0 $\times$ 16$^0$

= 2 $\times$ 16 + 0 $\times$ 1

= 2 $\times$ 16 + 0

= 2 $\times$ 16

= 32

Question 5: Convert 24$_{16}$ into decimal number.
Solution:

Given hexadecimal number is 24$_{16}$.

24$_{16}$ = 2 $\times$ 16$^1$ + 4 $\times$ 16$^0$

= 2 $\times$ 16 + 4 $\times$ 1

= 2 $\times$ 16 + 4

= 32+ 4

= 36

Practice Problems

Question 1: Convert  AB$_{16}$ into decimal number.
Question 2: Convert D3$_{16}$ into decimal number.
Question 3: Convert 52$_{16}$ into decimal number.