Question

Let a be a base-10 number with both integer and fractional parts such that a=1.09. How many bits do we need to represent the value of a in base-2? Let A be a set such that A={1.09,1.10,1.11}. How many bits do we need to encode the elements in set A? You should aim to find the minimum number of bits. Hint: You can come up with different representation/encoding schemes.

Answer 1

To represent the value of a=1.09 in base-2, we need 1 bit for the integer part and a repeating binary fraction. To encode the elements in set A, each element requires 2 bits.

Step-by-step explanation:

To represent the value of a=1.09 in base-2, we need to convert the integer and fractional parts separately. The integer part is 1, which can be represented by 1 bit. The fractional part, 0.09, can be converted using repeated multiplication by 2.

Multiplying 0.09 by 2 gives us 0.18, so the first bit after the decimal point is 0. Multiplying 0.18 by 2 gives us 0.36, so the next bit is 0. Continuing this process, we get the binary representation of the fractional part as 0.0001100110011...

Adding the integer and fractional parts, we get the binary representation of a=1.09 as 1.0001100110011...

To encode the elements in set A ({1.09, 1.10, 1.11}) in base-2, we follow the same process as above for each element. We find that we need 2 bits to represent each element in set A, as the binary representations are 1.0001100110011..., 1.1, and 1.11 respectively. Therefore, we need a minimum of 2 bits to encode the elements in set A.