Question

1. Write down the binary expansion of 50.5.

2. Consider the gaps between representable floating-point numbers. Which of the following has the larger gap between it and the next larger representable floating-point number?

• 2

• 201

3. Suppose that you approximate a number y by a number x. We will say that x approximates y to k decimal places if the decimal expansions of roundk(x) and roundk(y) match, where roundk(·), for the purposes of this problem, indicates rounding to the nearest number with at most k nonzero digits after the decimal point. E.g., round10(π) = 3.141592654, while round5(π) = 3.14159. So, for example, 3.14159222222 approximates π to 5 decimal places. Find a positive number z such that the following implication holds: if |x − y| 6 z, then x approximates y to 4 decimal places.

Answer 1

1.Binary Expansion of the given number 50.5 is

the number 50.5₁₀ in decimal number .This number has to translated in to binary number System.To convert the given number in to binary

the whole number part should be divided on the basis of new

after dividing the number 50 with 2 we get result as 50₁₀=110010₂

the fractional part of number is found by multiplying on the basis new

0.5₁₀=0.1₂

the add both whole and fractional part then we get

110010₂+0.1₂=110010.1₂

result of 50.5₁₀=110010.1₂