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
[tex]= (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[/tex]
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:
[tex]2A = (2 * 16^{1} ) + (A * 16^{0})\\2A = (2 * 16^{1} ) + (10 * 16^{0})\\2A = (2 * 16 ) + (10 * 1)\\2A = 32 + 10\\2A = 42[/tex]
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
