Question

1. Many operating systems now operate on 64 bits of memory. What is the highest base 10 number it could store?

2. How many bits would you need to store the number 8,000,000?

3. How many bits is 5 Gigabytes?

4. In order to get back the original signal without distortion I must sample at _____________ the highest frequency in the signal.

a. at least 2 times

b. exactly

c. exactly 2 times

d. at least half

Answer 1

Ans 1. For unsigned numbers

For 3 bits, the highest number which could be stored is 7 i.e. 111 in binary

For 4 bits, it is 15 i.e. 1111 in binary

Similarly for 64 bits, 2⁶⁴ - 1 (= 18,446,744,073,709,551,615) is the highest base 10 number it could store

Ans 2. 23 bits

Using the same logic from before, we take floor(log₂(n)) + 1 as the number of bits

Suppose, we have to calculate for 7

floor(log2(7)) = 2

2 + 1 = 3 bits

Now suppose we have to calculate for 8

log2(8) = 3

floor(log2(8)) = 3

3 + 1 = 4 bits

Ans 3.

5 Gigabytes = 5 * 2³⁰ * 8 bits or 42949672960

Explanation: 1 byte = 8 bits

1 Kilobyte = 1024 bytes or 2^10 bytes

1 Megabyte = 1024 Kilobytes or 2^20 bytes

1 Gigabyte = 1024 Megabytes or 2^30 bytes

Note: Some manufacturers use 1 KB = 1000 bytes, 1 MB = 1000 KBs and so on

Ans 4. a) At least 2 times

Refer to Nyquist-Shannon Sampling Theorem. If the frequency is lower than that(The Nyquist frequency), aliasing will occur which is like folding or overlapping of the upper frequency region resulting in distortion.