Final answer:
The smallest number greater than 1 that can be exactly represented in a 32-bit IEEE floating point is approximately 1.0000001192092896, and this is due to the increment of the least significant bit in the mantissa part of the binary representation of 1. The gap between this number and 1 is approximately 1.192092896e-7.
Step-by-step explanation:
The smallest number greater than 1 that can be exactly represented in a 32-bit IEEE floating point number is achieved by incrementing the least significant bit (LSB) in the mantissa part of the number that represents 1. In a 32-bit floating point, there are 23 bits for the mantissa, 8 bits for the exponent, and 1 sign bit.
The number immediately greater than 1 can be represented as 1.00000000000000000000001 in binary, which translates to 1 + 2-23 in decimal, as the LSB in mantissa represents 2-23. This leads to a value of 1 + 1/8388608, which is approximately 1.0000001192092896 in decimal form. Therefore, the gap between 1 and this number would also be 1/8388608, or roughly 1.192092896e-7.