Question

The number 0.60 is rational and can be represented in finitely many digits in decimal.

a. Can it be represented by finitely many bits in binary? Explain why or why not.
b. If it can be done in finitely many bits, give its exact binary representation. If not, give the first ten bits of the approximate representation.
c. Give an upper bound on absolute error in this representation.

Answer 1

The number 0.60 cannot be represented by finitely many bits in binary because it is a repeating decimal, requiring an infinite number of bits. The first ten bits of the approximate binary representation are 0.1001100110.

Step-by-step explanation:

The number 0.60 can be represented by finitely many digits in decimal because it is rational. However, it cannot be represented by finitely many bits in binary. This is because in binary, the number 0.60 is a repeating decimal, which means it requires an infinite number of bits to represent it accurately.

In binary, 0.60 is approximately equal to 0.1001100110011... (repeating). Since we cannot represent an infinite number of bits, we can only provide an approximation. The first ten bits of the approximate binary representation of 0.60 would be 0.1001100110.

The upper bound on the absolute error in this representation would be the difference between the exact value of 0.60 and the approximate binary representation (0.1001100110). However, calculating this difference would require comparing an infinite number of bits, so we cannot provide an exact value for the absolute error.