Binary to Decimal Conversion Explained

Converting a binary number, which uses base-2, to its decimal equivalent, which uses base-10, involves multiplying each binary digit by the corresponding power of 2, based on its position. For binary numbers with a fractional part (digits after the binary point), the powers of 2 become negative.

Calculating Decimal Value of 1001.0010

We need to find the decimal value for the binary number 1001.0010. We will process the integer part (1001) and the fractional part (0010) separately.

Integer Part Conversion: 1001

The integer part is 1001. We assign positional values starting from $2^0$ for the rightmost digit and increasing towards the left.

• 1 is in the $2^3$ position ( $2 \times 2 \times 2 = 8$ )
• 0 is in the $2^2$ position ( $2 \times 2 = 4$ )
• 0 is in the $2^1$ position ( $2$ )
• 1 is in the $2^0$ position ( $1$ )

To get the decimal value, we multiply each digit by its corresponding power of 2 and sum the results:

$ (1 \times 2^3) + (0 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) $

$ = (1 \times 8) + (0 \times 4) + (0 \times 2) + (1 \times 1) $

$ = 8 + 0 + 0 + 1 $

$ = 9 $

So, the integer part 1001 in binary is 9 in decimal.

Fractional Part Conversion: 0010

The fractional part is 0010. We assign positional values starting from $2^{-1}$ for the leftmost digit after the binary point and decreasing towards the right.

• 0 is in the $2^{-1}$ position ( $\frac{1}{2}$ )
• 0 is in the $2^{-2}$ position ( $\frac{1}{4}$ )
• 1 is in the $2^{-3}$ position ( $\frac{1}{8}$ )
• 0 is in the $2^{-4}$ position ( $\frac{1}{16}$ )

Now, we multiply each digit by its corresponding negative power of 2 and sum the results:

$ (0 \times 2^{-1}) + (0 \times 2^{-2}) + (1 \times 2^{-3}) + (0 \times 2^{-4}) $

$ = (0 \times \frac{1}{2}) + (0 \times \frac{1}{4}) + (1 \times \frac{1}{8}) + (0 \times \frac{1}{16}) $

$ = 0 + 0 + \frac{1}{8} + 0 $

$ = \frac{1}{8} $

$ = 0.125 $

So, the fractional part 0010 in binary is 0.125 in decimal.

Combining Results

Finally, we add the decimal values of the integer and fractional parts to get the overall decimal value of the binary number:

$ \text{Decimal Value} = (\text{Integer Part Value}) + (\text{Fractional Part Value}) $

$ = 9 + 0.125 $

$ = 9.125 $

Therefore, the binary number 1001.0010 is equal to 9.125 in decimal.