Before starting to convert hexadecimal numbers in decimal number, go through the following articles:
- Numbers Representation Systems – Decimal, Binary, Octal and Hexadecimal: this articles explains the different types of number representation systems: decimal, binary, hexadecimal and octal
- Decimal to Hexadecimal Conversion: this article explains how to convert a decimal number into a hexadecimal number
Let’s start by writing down the first four powers of 16 (we’ll use them in our conversion example):
\[\begin{split}16^0 &= 1 \\
16^1 &= 16 \\
16^2 &= 256 \\
16^3 &= 4096
\end{split}\]
For example, let’s convert the hexadecimal number 0xFACE into a decimal number.
Method 1
This method is straightforward and it’s using the powers of 16.
Step 1. Convert each of the hexadecimal symbols into decimal symbols, as follows:
| Hexadecimal | F | A | C | E |
| Decimal | 15 | 10 | 12 | 14 |
Step 2. Multiply each of the decimal numbers with the corresponding power of 16, as follows:
| Decimal | 15 | 10 | 12 | 14 |
| Powers of 16 | 163 | 162 | 161 | 160 |
| Multiplication | 15·163 | 10·162 | 12·161 | 14·160 |
Step 3. Add the products of all multiplication operations. The result is going to be the decimal number.
\[\bbox[#FFFF9D]{15 \cdot 4096 + 10 \cdot 256 + 12 \cdot 16 + 14 \cdot 1 = 64206}\]The hexadecimal number 0xFACE converted to decimal is 64206.
Method 2
This method is not as direct as the first one, but it’s easier to execute because it does not involve heavy multiplications. It can also be performed without a calculator or software. The principle of this method is to use a hexadecimal to binary transformation first and a binary to decimal conversion second.
A hexadecimal number can be represented on 4 bits. For example, the highest symbol in hexadecimal notation is F, which in binary is 0b1111.
Step 1. Convert the hexadecimal numbers into binary, as follows:
| Hexadecimal | F | A | C | E |
| Decimal | 15 | 10 | 12 | 14 |
| Binary | 1111 | 1010 | 1100 | 1110 |
Step 2. Concatenate the four binary numbers into one, as follows:
1111101011001110
Step 3. Convert the binary number into decimal, using the powers of 2:
\[ \begin{split}1 \cdot 2^{15}+1 \cdot 2^{14}+1 \cdot 2^{13}+1 \cdot 2^{12}+1 \cdot 2^{11}+0 \cdot 2^{10}+1 \cdot 2^9+ …\\
0 \cdot 2^8+1 \cdot 2^7+1 \cdot 2^6+0 \cdot 2^5+0 \cdot 2^4+1 \cdot 2^3+1 \cdot 2^2+1 \cdot 2^1+0 \cdot 2^0
\end{split} \]
Step 4. Perform the sum of the multiplications to get the decimal number
\[32768+16384+8192+4096+2048+512+128+64+8+4+2=64206\]As you can see, we obtained the same result with both methods. The advantage of the second one is that it deals with simpler multiplications and can be performed without a handheld calculator or software.
Of course, we can also use the Scilab function hex2dec() to convert from hexadecimal to decimal.
--> hex2dec('FACE')
ans =
64206.
-->
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