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Section 4.8 Real Numbers in Binary
Checkpoint 4.8.1. Representing Real Numbers.
Which of the following types of numbers can we represent precisely in binary given a fixed number of bits?
Rational Numbers
Incorrect.
Irrational Numbers
Incorrect
Both A and B
Incorrect
Neither A nor B
Correct!
Hint.
Think about the type of numbers we can represent entirely \(\textit{in binary} \text{.}\)
Checkpoint 4.8.2. Identifying Operand Types.
Convert the following fixed-point binary numbers to their equivalent decimal representations:
Hint.
Recall that the bit immediately to the left of the point has a weight of \(2^0\) while the one immediately to the right has a weight of \(2^{-1}\)
Checkpoint 4.8.3. Floating-Point Numbers.
True or False: Floating-point representation can represent both positive and negative numbers.
True.
Floating-point representation can represent both positive and negative numbers.
False.
Floating-point representation can represent both positive and negative numbers.
Checkpoint 4.8.4. SG7 IEEE 754 Floating Point Formula.
Given the following equation used to calculate the value of a number given in IEEE 754 floating-point format:
\(\textit{value} = (-1)^A * 2^B * C \)
Match A, B, and C with its appropriate value.
- A
- \(n\)
- B
- \(e^{-127}\)
- C
- \(1.s\)
Checkpoint 4.8.5.
Checkpoint 4.8.6. Floating-Point Conversion.
Given the following real number expressed in the IEEE 754 floating-point format:
1101 0110 1011 0110 1011 0000 0000 0000
Select which option depicts how the bits should be grouped to determine its value.
1 1010110 101101101011000000000000
Correct!
1 10101101 01101101011000000000000
- Incorrect
110101101011011010110000 0000000 0
Incorrect.
None of the Above
Incorrect.
Checkpoint 4.8.7. Floating Point Format.
Given the real number expressed in IEEE 754 floating-point format:
1101 0110 1011 0110 1011 0000 0000 0000
Answer the following questions:
(a)
(b)
This number is _____.
Positive
Incorrect
Negative
Correct!
(c)
The sign bit represents the sign of _________.
The value
Correct!
The exponent of the value
Incorrect.
(d)
(e)
(f)
Checkpoint 4.8.8. IEEE 754 Floating-Point Format (Part II).
Select the expression for converting the following IEEE 754 floating-point value to decimal.
1101 0110 1011 0100 0000 0000 0000 0000
\((2^{86 - 127}) * (2^{-1} + 2^{-3} + 2^{-4} + 2^{-6}) \)
Incorrect.
\(-(2^{86 - 127}) * (2^{-1} + 2^{-3} + 2^{-4} + 2^{-6})\)
Incorrect
\((2^{173 - 127}) * (2^{-2} + 2^{-3} + 2^{-5}) \)
Incorrect
\(-(2^{173 - 127}) * (2^{-2} + 2^{-3} + 2^{-5})\)
Correct!
None of the above
Incorrect
Checkpoint 4.8.9. Floating-Point Conversion.
Hint.
Use the formula to compute the final value: \((-1)^0 \cdot 2^{(132-127)} \cdot 1.375 \)
