Answered Essay: The following 5 questions pertain to the normalized floating-point number system F with B = 4, p = 2, m

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The following 5 questions pertain to the normalized floating-point number system F with B = 4, p = 2, m = -1, and M = 2: 1. How many positive numbers are there in F? 2. List all of these positive numbers, in some reasonable order (in base 4, of course). 3. What is the smallest positive number? 4. What is the largest number? 5. What are the spacings between numbers in each interval bounded by the various table powers of 4? A Floating-point number system is the set of numbers defined by the following integral parameters: A base, B A precision, p A minimum exponent, m (usually negative) A maximum exponent, M

Expert Answer

Total numbers in the system is given by 2(B-1)(B^p-1)(M^-m+1)+1
1. This gives the total numbers be 97(which includes zero); Therefore total positive numbers is (97-1)/2 which gives 48.
2. Positive number in the system is 0.01, 0.02, 0.03, 0.10, 0.11, 0.12, 0.13, 0.20, 0.21, 0.22, 0.23, 0.30, 0.31, 0.32, 0.33, 1.0, 1.01, 1.02, 1.03, 1.10, 1.11, 1.12, 1.13, 1.20, 1.21, 1.22, 1.23, 1.30, 1.31, 1.32, 1.33, 2.0, 2.01, 2.02, 2.03, 2.10, 2.11, 2.12, 2.13, 2.20, 2.21, 2.22, 2.23, 2.30, 2.31, 2.32, 2.33, 3.0, 3.10, 3.11, 3.12, 3.13, 3.20, 3.21, 3.22, 3.23, 3.30, 3.31, 3.32, 3.33, 10
3. Smallest number is B^m which is 4^-1 which is 1/4= (0.25)10 = (0.1000)4 ;
4. Largest number is (1-B^-p)(B^U+1) which is (1- 4^-2) (0.0625)*(4³)= (4)₁₀ = (10)₄;
5. Spacings between numbers in each interval bounded by the power of 4 is (1/4)₁₀ which is (0.01)₄

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