How to Convert a Decimal Number to Binary with an Optimistic Outlook

What is the binary floating number for the real decimal number −76.0625?

a. The binary floating number is −1001100.0001.
b. The binary floating number is −1001100.1001.
c. The binary floating number is −1001100.0111.
d. The binary floating number is −1001100.0101.

The binary floating number for the decimal number -76.0625 is -1001100.0101. (Option d)

Converting a decimal number to binary may seem daunting at first, but with a positive mindset and a clear understanding of the process, it can be an exciting challenge to tackle. Let's break down how to convert the real decimal number −76.0625 to its binary floating number representation.

Steps to convert a decimal number to binary:

1. Convert the absolute value of the decimal number to binary:

- For the integer part: Divide the number by 2 and keep track of the remainders until the quotient becomes 0. The remainders in reverse order give the binary representation.

- For the fractional part: Multiply the number by 2 and take the integer part as the binary digit. Repeat this process until the fractional part becomes 0 or until you reach the desired level of precision.

2. Assign the sign bit:

- If the decimal number is positive, the sign bit is 0.

- If the decimal number is negative, the sign bit is 1.

3. Combine the sign bit with the binary representation obtained in step 1.

In our case, let's apply these steps to convert −76.0625 to binary:

1. Converting the absolute value of 76.0625 to binary:

- For the integer part (76), the binary representation is 1001100.

- For the fractional part (0.0625), multiplying by 2 gives 0.125. Taking the integer part, we get 0. Multiplying 0.125 by 2 again gives 0.25, with an integer part of 0. Repeating this process until we reach the desired level of precision, we obtain 0101.

2. The sign bit for a negative number is 1.

3. Combining the sign bit with the binary representation from step 1:

-1001100.0101

Therefore, option d, -1001100.0101, is the correct binary floating number for the real decimal number −76.0625.