Module 10 Homework Problems

Easier homework this week! Lots of bonuses to help your overall homework grade! You MUST show all work and all computations to get credit.

Convert the following numbers from base 2 to decimal.

1. \(10101_2\)
2. \(11111_2\)
3. \(1101010110_2\)

Convert the following base 10 numbers to base 2

4. \(212_{10}\)
5. \(256_{10}\)
6. \(541_{10}\)
7. \(5121_{10}\)

(Bonus +5 each) Convert the following as indicated

8. Convert \(241_{10}\) to base 3
9. Convert \(132_{10}\) to base 5
10. Convert \(15_{10}\) to base 1 (Hint: if ternary (base 3) represents all numbers with only three different digits, and binary represents all numbers with only two digits, how many digits can we use to represent a number in base 1?)

Perform the indicated operations, keeping your answer in base 2.

11. \(11011_2+110001_2\)
12. \(10101_2+1111_2\)
13. \(1011_2+10001_2+101111_2\)
14. \(1001_2-111_2\)
15. \(101101_2-111011_2\)
16. (Bonus +5) \(10111_2-1101101_2\) (yes, your answer will be negative)
17. \(1010_2 \cdot 1101_2\)
18. \(11101_2\cdot 101_2\).
19. (Bonus +5) \(132_4+123_4\)
20. (Bonus +5) \(5123_9-3172_9\)

Here’s a crazy fact about bit strings and binary numbers that might blow your mind: every computer program is nothing more than a string of bits, like 1011010101010100100101010101111001010100100010100010100. Every bit string is a binary number that can be converted to a decimal number. Therefore, every program that has been or ever will be written is just a number! And since every decimal number can be converted to a bit string, every number (technically) represents a computer program. So, one could legitimately ask “I wonder what computer program 1,387,469,213 does…”

(Computability-Theoretic disclaimer on the above fact: context/encoding is important to determine what the 1,387,469,213th program does. For instance, different operating systems (OS’s) might interpret the binary conversion of 1,387,469,213 differently, and so 1,387,469,213 might mean different things to different OS’s. So not only does each natural number represent a program, it actually represents many different objects, depending on the “context” in which the number is used.)