https://file.aiquickdraw.com/mathbot/file/1726097151212cpg436pj.jpg

The image shows a binary floating-point number with the format:

- **Sign bit**: `0` (positive)
- **Exponent**: `10000010`
- **Mantissa**: `1.11000000000000000000000`

This follows the IEEE 754 single-precision floating-point format.

### Steps to decode:
1. **Sign bit**: `0` means the number is positive.
   
2. **Exponent**: `10000010` in binary translates to `130` in decimal. The exponent in IEEE 754 is stored with a bias of 127, so the actual exponent is:
   \[
   130 - 127 = 3
   \]

3. **Mantissa**: The mantissa is given as `1.11000000000000000000000`. In IEEE 754, the leading `1` is implicit, so we interpret the mantissa as:
   \[
   1 + 0.5 + 0.25 = 1.75
   \]

4. **Putting it all together**:
   \[
   \text{Value} = (-1)^{\text{sign}} \times 1.75 \times 2^{3}
   \]
   \[
   \text{Value} = 1.75 \times 8 = 14
   \]

So, the decimal equivalent of this binary floating-point number is **14**.

Would you like further details or explanations?

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Here are 5 related questions:
1. How does IEEE 754 handle denormalized numbers?
2. What is the role of bias in floating-point representations?
3. How are floating-point numbers compared for equality in binary systems?
4. What is the smallest and largest number that can be represented using single-precision IEEE 754 format?
5. How does rounding affect floating-point calculations in practical applications?

**Tip:** Understanding how floating-point numbers are stored and interpreted can help avoid precision errors in calculations, especially in programming.

The binary floating-point number 0 10000010 1.11000000000000000000000 has decimal equivalent.

Convert the binary floating-point number 0 10000010 1.11000000000000000000000 to its decimal equivalent using the IEEE 754 standard. Learn about sign bits, exponent bias, and mantissa handling in this college-level problem.

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