To convert the binary number **10111.112** to a mixed number, we need to understand both the whole number and fractional parts of a binary number.
The binary number **10111.112** consists of two parts: **10111** and **.11**.
Converting the Whole Number Part **10111**
The process of converting a binary number to decimal involves multiplying each bit by the power of 2 corresponding to its position, starting from the right (0th position). Let's do this step by step:
- The rightmost digit (least significant bit) is 1: \(1 \times 2^0 = 1\)
- The next digit to the left is 1: \(1 \times 2^1 = 2\)
- The next digit is 1: \(1 \times 2^2 = 4\)
- The next digit is 0: \(0 \times 2^3 = 0\)
- The leftmost digit (most significant bit) is 1: \(1 \times 2^4 = 16\)
Add their decimal values together:
**16 + 0 + 4 + 2 + 1 = 23**
Converting the Fractional Part **.11**
For the fractional part, we work with negative powers of 2, starting from \(-1\) for the digit immediately after the binary point:
- The first digit is 1: \(1 \times 2^{-1} = 0.5\)
- The second digit is 1: \(1 \times 2^{-2} = 0.25\)
Add these values together:
**0.5 + 0.25 = 0.75**
Forming the Mixed Number
Now, we need to combine both parts (whole number and fractional parts):
**23 + 0.75 = 23.75**
To express this in a mixed number form, recognize that **23.75** is equivalent to the mixed number **23 3/4**.
Therefore, the complete conversion of the binary number **10111.112** is **23 3/4**.