Conversion From any Base to Base 10 and From Base 10 to any Base

Number base conversion is the process of converting from one base to another.

We can convert a digit from any base to base 10 and from base 10 to any base. The process is very simple. We will divide this post into two headings.

Conversion from One base to Base 10

To convert a digit from any base to base 10, the steps are as follows:

1. Write out the number you want to convert to base 10.
2. Identify the base of the number you want to convert to base 10. For example, the number might be in base 2, or base 8.
3. Multiply each digit by the base raised to a power equal to its position in the number, starting from 0 for the rightmost digit and starting with the rightmost digit. For example, the rightmost digit will be raised to power 0, while the second rightmost digit will be raised to power 1.
4. Add up the product derived from step 3. The sum will be the decimal equivalent of the original digit.

Example 1

Convert \(13_{4}\) to a number in base 10.

Solution:

The number we want to convert is in base 4, as a result, the rightmost digit will be multiplied by 4 raised to the power of 0, and the second rightmost digit will be multiplied by 4 raised to the power of 1.

So, 7 is the decimal equivalent of the base 4 number 13.

Example 2

Convert 6762 from base 8 to base 10

Solution:

The number we want to convert is in base 8, as a result, the rightmost digit will be multiplied by 8 raised to a power of 0, the second rightmost digit will be multiplied by 8 raised to a power of 1 and so on.

Therefore, The decimal equivalent of the octal number 6762 is 3570.

Example 3

Convert \(1E5_{16}\) to a number in base 10.

Solution:

E represent 14 in base 16. So, the rightmost digit will be multiplied by 16 and raised to a power of 0, the second rightmost digit will be multiplied by 16 and raised to a power of 1 and so on.

Therefore, \(1ES_{16}=485_{10}\).

Conversion From Base 10 to Any Base

You can convert a digit from base 10 to another base using the following steps:

1. Write down the base 10 digits you want to convert to another base.
2. Divide the digit by the base number you want to convert and write the remainder. For example, if you are trying to convert to base 4, then you will divide by 4 and write the remainder.
3. Divide the quotient from step 2 and write the remainder.
4. Repeat steps 3 and 4 until the quotient is 0.
5. The remainder’s sequence in reverse order represents the original number in the desired base.

Example 4

Convert \(567_{10}\) to a number in base 16.

Solution:

Since we want to convert to Base 16, We will divide by 16 until the quotient is 0.

Reading the remainder, 237 is the hexadecimal equivalent of decimal 567.

Example 5

Convert 1256 from decimal to a number base 6.

Solution:

Now, we will divide through by 6.

Therefore \(1256_{10}=5452_6\)

Example 6

Convert \(420_{10}\) to an octal number.

Solution:

The octal number is base 8, we will divide by 8.

Hence, 644 is the octal number of the decimal number 420.