Question

For the array given in FindDuplicateCount.java, which index is the last check we need to make to count all the duplicates? 5. Interpret the number 101010 in bases 2, 10, 16. For each, convert to the other two bases. You need to show all 6 conversions: a) When interpreting 101010 as a base 2 number, convert to bases 10 and 16; b) When interpreting 101010 as a base 10 number, convert to bases 2 and 16; c) When interpreting 101010 as a base 16 number, convert to bases 2 and 10;

Answer 1

Answer:

a) 101010 as a base 2 number is the same as 42 in base 10 and 2A in base 16

b) 42 as a base 10 number is same as 101010 in bases 2 and 2A in base 16

c) 2A as a base 16 number is same as 101010 in base 2 and 42 in base 10

Step-by-step explanation:

a) When interpreting 101010 as a base 2 number, convert to bases 10 and 16;

First convert 101010 from base 2 to base 10

We number the digit from the rightmost digit starting from 0 and we have the following index:

1 = 5, 0 = 4, 1 = 3, 0 = 2, 1 = 1, 0 = 0

The we expand in the power of the index to base 2

$= (1*2^{5}) + (0*2^{4}) + (1*2^{3}) + (0*2^{2}) + (1*2^{1}) + (0*2^{0})\\= (1 * 32) + (0 * 16) + (1 * 8) + (0 * 4) + (1 * 2) + (0 * 1)\\= 32 + 0 + 8 + 0 + 2 + 0\\= 42$

Then convert to base 16: divide base 10 by 16

42 ÷ 16 = 2 remainder 10(A)

2 ÷ 16 = 0 remainder 2

Reading the remainder bottom-up; we have 2A. 10 is represented as A in Base 16.

b) When interpreting 101010 as a base 10 number, convert to bases 2 and 16;

101010 in base 10 = 42

Converting to base 2

42 ÷ 2 = 21 remainder 0

21 ÷ 2 = 10 remainder 1

10 ÷ 2 = 5 remainder 0

5 ÷ 2 = 2 remainder 1

2 ÷ 2 = 1 remainder 0

1 ÷ 2 = 0 remainder 1

Reading the remainder bottom-up; we have 101010 in base 2

Then convert to base 16: divide base 10 by 16

42 ÷ 16 = 2 remainder 10(A)

2 ÷ 16 = 0 remainder 2

Reading the remainder bottom-up; we have 2A. 10 is represented as A in Base 16.

c) When interpreting 101010 as a base 16 number, convert to bases 2 and 10

Converting from base 16 to base 10; we expand in the power of the index:

$2A = (2 * 16^{1} ) + (A * 16^{0})\\2A = (2 * 16^{1} ) + (10 * 16^{0})\\2A = (2 * 16 ) + (10 * 1)\\2A = 32 + 10\\2A = 42$

Converting to base 2

42 ÷ 2 = 21 remainder 0

21 ÷ 2 = 10 remainder 1

10 ÷ 2 = 5 remainder 0

5 ÷ 2 = 2 remainder 1

2 ÷ 2 = 1 remainder 0

1 ÷ 2 = 0 remainder 1

Reading the remainder bottom-up; we have 101010 in base 2