Conversion From Base 2 to Base 10 and From Base 10 to Base 2

Number Base Conversion is the process of converting a number from one base to another.

In this post, we will be converting from base 2 to base 10, and vice versa.

Base 2 has only two digits, which are 0 and 1.

On the other hand, Base 10 has 10 digits, namely; 0,1,2,3,4,5,6,7,8,9.

Conversion From Base 2 to Base 10

To convert from base 2 to base 10, you must follow the following steps:

1. Write the number you want to convert from base 2 to base 10.
2. Starting from the rightmost digit and starting from zero, assign digits and raise 2 to the power of the digits.
3. Multiply each digit by its corresponding value.
4. Add up the product together.

To put this into practice, we take some examples.

Example 1

Convert \(10111_2\) to decimal.

Solution:

10111 has five digits. So the righter most digit (fifth digit) will be multiplied by 2 to the power of 0. The second right to the right (fourth digit) will be multiplied by 2 to the power of 1, and so on.

\(1×2^4+0×2^3+1×2^2+1×2^1+1×2^0\)

Let’s now add the result

\(16+0+4+2+1=23\)

Example 2

Convert \(1000011_2\) to base 10.

Solution:

1000011 has 7 digits. The seventh digit will be multiplied by 2 raised to the power of 0, the sixth digit will be multiplied by 2 raised to the power of 1, the fifth digit will be multiplied by 2 raised to the power of 2, and so on.

\(\small{1×2^6+0×2^5+0×2^4+0×2^3+0×2^2+1×2^1+1×2^0}\)

We can then add the result

\(64+0+0+0+0+2+1=67\)

Conversion from Base 10 to Base 2

To convert a digit from base 10 to base 2, you must follow the following steps:

1. Write the digit down
2. Divide the digit by 2
3. Write the remainder after the division. The remainder will be either 0 or 1.
4. Divide the result of the division by 2.
5. Repeat steps 3 and 4 until the division results in 0.
6. The sequence of remainders written down in reverse order represents the digit in base 2.

Example 3

Convert \(58_{10}\) to base 2.

Solution:

We will continue to divide 58 by 2 until we get to 2.

So, reading the remainders from the whole from upward, \(58_{10}=111010_2\)

Example 4

Convert 27 from decimal to binary.

Solution:

Now, let’s divide the digit by 2 until we get a quotient that is zero.

Conversion of Floating-point Numbers From Base 2 to Base 10

Some base 2 numbers usually have a decimal number. These are called floating-point numbers.

Examples of floating point numbers are 101.0111.

To convert a floating-point base 2 digits to base 10, you should follow these steps:

1. Divide the digit into the integer part and the fractional part.
2. You can convert the integer part into decimals using the usual way of converting base 2 to base 10.
3. Now, To convert the numbers after the decimal point to decimal, you should follow these steps:
   • Starting from the leftmost digit and starting from -1, assign digits of -1, -2,-3 and so on to the digits after the decimal point.
   • Multiply each digit of the fractional part by the corresponding power of 2.
   • Add the results.
4. Add the result of 2 and 3 together. The final result is the digit in base 10.

Example 5

Convert \(111.101_2\) to base 10.

Solution:

First, we need to separate the digits into integer and fractional parts. 111 is the integer part while 0.101 is the fractional part.

Let’s now convert using the normal way of converting:

\(1 \times 2^2+ 1 \times 2^1 + 1 \times 2^0=7\)

Now, we assign power to raise to the power of 2.

\(1 \times 2^{-1} +0 \times 2^{-2} + 1 \times 2^{-3}\)

We can then add the numbers

\(0.5+0+0.125=0.625\)

Adding the integral part and fractional part

\(7+0.625=7.625\)

Conversion of Floating-point numbers from Base 10 to Base 2

The following steps can be used to convert floating numbers from decimal to binary.

1. Separate the numbers into integer and fractional parts.
2. Converts the integer as you would normally convert decimal numbers to binary.
3. Now, for the decimal part, follow the steps:
   • Multiply the fractional part by 2 and keep track of the integer part of the result.
   • Continue the above step until you get a fractional part of 0 or after 5 significant figures.

Example 6

Convert \(10.625_{10}\) to a number in base 2

Solution:

10 is the integer part while 0.625 is the fractional part. Now, let’s convert the way we usually cover base 10 to base 2.

So, \(10_{10}=1010_2\)

Now, let’s convert the fractional part 0.625 to base 2

Reading the integer part, \(0.625_{10}=0.101_2\)

Now, \(10.625_{10}\) in base 2 will be the sum of 10 and 0.625 in base 2

\(10.625_{10}=1010_2+0.101_2=1010.101_2\)