The representation of the decimal number (27.625)₁₀ in base-2 number system is
- A11011.110
- B11101.101
- C11011.101
- D10111.110
Solution & Step-by-step Explanation
Decimal to Binary Conversion Explained
Converting a decimal number with both an integer and a fractional part into its binary (base-2) equivalent involves two main steps: converting the integer part and converting the fractional part separately. The given decimal number is . We will convert 27 and 0.625 independently.
Integer Part Conversion: Converting 27 to Binary
To convert the integer part, 27, from decimal to binary, we use successive division by 2. We record the remainders at each step and then read them from bottom to top to get the binary equivalent.
Reading the remainders from bottom to top, the binary representation of 27 is .
Fractional Part Conversion: Converting 0.625 to Binary
To convert the fractional part, 0.625, from decimal to binary, we use successive multiplication by 2. We take the integer part of the result at each step, and these integer parts, read from top to bottom, form the binary fraction. We continue this process until the fractional part becomes 0 or until we reach the desired precision.
Reading the integer parts from top to bottom, the binary representation of 0.625 is .
Combining the Parts for the Final Binary Representation
Now, we combine the binary equivalents of the integer and fractional parts.
- Integer part =
- Fractional part =
Therefore, the representation of the decimal number in base-2 number system is .
Converting a decimal number with both an integer and a fractional part into its binary (base-2) equivalent involves two main steps: converting the integer part and converting the fractional part separately. The given decimal number is . We will convert 27 and 0.625 independently.
Integer Part Conversion: Converting 27 to Binary
To convert the integer part, 27, from decimal to binary, we use successive division by 2. We record the remainders at each step and then read them from bottom to top to get the binary equivalent.
| Operation | Quotient | Remainder |
|---|---|---|
| 13 | 1 | |
| 6 | 1 | |
| 3 | 0 | |
| 1 | 1 | |
| 0 | 1 |
Fractional Part Conversion: Converting 0.625 to Binary
To convert the fractional part, 0.625, from decimal to binary, we use successive multiplication by 2. We take the integer part of the result at each step, and these integer parts, read from top to bottom, form the binary fraction. We continue this process until the fractional part becomes 0 or until we reach the desired precision.
| Operation | Result | Integer Part | New Fractional Part |
|---|---|---|---|
| 1.25 | 1 | 0.25 | |
| 0.50 | 0 | 0.50 | |
| 1.00 | 1 | 0.00 |
Combining the Parts for the Final Binary Representation
Now, we combine the binary equivalents of the integer and fractional parts.
- Integer part =
- Fractional part =
Therefore, the representation of the decimal number in base-2 number system is .