3. (9 points) Convert the following numbers as indicated. Include a sentence of

Question

3. (9 points) Convert the following numbers as indicated. Include a sentence of explanation about how you’re doing the conversion. For full credit, the explanation must be a grammatically correct, complete sentence. For example, in support of a calculation for converting (101)2 to decimal, you might write: “(101)2 = 1 · 2 2 + 0 · 2 + 1 = (5)10. I calculated the value contributed by each column in the binary representation, using the definition of base expansion, and then added up these values together.” a. (145)10 (decimal) to base 2 (binary). b. (AA)16 (hexadecimal) to base 8 (octal). c. (10110000001)2 (binary) to base 16 (hexadecimal).

Explanation / Answer

a.) In order to convert a decimal no., we need to recursively divide the decimal no. by 2 and keep track of the remainder and final result will be the respecive remainders in reverse order.

Now, 145/2 = 72 and 145 %2 = 1,

72/2 = 36 and 72 % 2 = 0,

36/2 = 18 and 36 % 2 = 0,

18/2 = 9 and 18 % 2 = 0,

9/2 = 4 and 9 %2 =1,

4 / 2 = 2 and 4 % 2 =0,

2/ 2 = 1 and 2 % 2 =0,

1/ 2 = 0 and 1 % 2 = 1.

Now the result will be (10010001) base 2.

b.) In order to convert a hexadecimal no. into a octal, first we need to convert it in binary no. and then in to octal representation.

(AA)16 (hexadecimal) will be (1010 1010) in binaryandin order to convert binary into octal we need to group them in pair of three numbers from LSB i.e., (010 101 010) base 2. Now, octal representation will be (252)8 (octal).

c.) In order to convert a binary number into hexadecimal no., we need to group binary no. in form of 4 bits. i.e.,

(10110000001)2(binary) -> ( 0101 1000 0001)2 (binary).

Now, we can easily represent this no. into hexadecimal format by simply converting each 4-bit pair into corresponding hexadecimal format. i.e.,

( 0101 1000 0001)2 (binary) -> (581)16 (hexadecimal)

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