Question

In: Computer Science

Prove that every real number with a terminating binary representation (finite number

Prove that every real number with a terminating binary representation (finite number of digits to the right of the binary point) also has a terminating decimal representation (finite number of digits to the right of the decimal point).

Expert Solution

Real number with terminating binary representation also has terminating decimal representation:

Consider a fraction p/q has value all 0’s after some places in binary representation.

Represent p/q with numerical value as follows.

Here, k has some finite value.

Consider the value of p and q.

To prove the condition, consider any integer with power of k.

Consider an integer 5k. Multiply and divide the term p/q with 5k.

Substitute 2k for q.

To get the fraction value of p/q, the value of (5k)p is always less than (10k).

Therefore, in general for any proper fraction with finite decimal expansion can be represented as q/10k, with consideration of q is divisible by 5k.

For example if q is taken as value 5k, then

Therefore, real number with terminating binary representation also has terminating decimal representation.

Junaid answered 2 years ago

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