Completed checkpoint 2

questions.md

@@ -52,9 +52,13 @@ talk with others.
 9. If `a` represents a positive integer, and `one = Bits(1, length=len(a))`, 
    give an expression equivalent to `-a`, but which does not use negation.
 
+~a + 1
+
 10. It is extremely easy to double a binary number: just shift all the bits 
     to the left. (`a << 1` is twice `a`.) Explain why this trick works. 
 
+Shifting to the left moves all the bits one over and a 0 on the right. Multyplying by 2 is the same as adding itself. So in the addtion, If you have a 1 in a postion, it will result in 0 with a 1 carried to the next postion. This is exactly what doubling looks like in binary adddition. For example, 1 = 0001, 2 = 0010, 4 = 0100, 8 = 1000. Here the one is moved to the next postion for each double. The same is true for: 3 = 0011, 6 = 0110, 12 = 1100. 
+
 11. Consider the following: 
    ```
    >>> hundred = Bits(100, 8)
@@ -67,17 +71,27 @@ talk with others.
    ```
    Apparently 100 + 100 = -56. What's going on here?
 
+   When expressing data that can be positive or negaitve, the first bit represents the sign. 0 is positive and 1 is negative. In this example, we have a length of 8 bits to work with. If the first bit is denotes the sign, we have now 7 bits left to express an integer. Since each bit has 2 possiblities (1 or 0), we have 2^7 possible positve integers, or 128 possible positive integers (and their negative counterparts). With 100 + 100, we would get 200. Since one of the integers we must account for is 0, that leaves a maximum sum of 127. Now, the first bit does not simply switch the sign. It actually adds -128 to the rest of the data. Now, if we take the result of hundred + hundred, 1100100, we would have -128 + 64 + 8  = -56. 
+
 12. What is the bit representation of negative zero? Explain your answer.
 
+An integer + its negative should be 0. Since 0 is already 0, nothing needs to change. So, -0 should be exactly the same as 0. Also if we use the algorithm to make a postive integer a negative, we flip all the bits and add 1. This gives -0 = 11111111 + 1 = 00000000 = 0. So our logic follows. 
+
 13. What's the largest integer that can be represented in a single byte? 
     Explain your reasoning.
 
+    A signle byte has a length 8. Since the first bit contribute to sign, the max amount of integers we can store (with negative conterparts) is 2^7 = 128. However, one of these integers is 0, so the maximum integer we can represent is 127 which is 01111111
+
 14. What's the smallest integer that can be represented in a single byte? 
     Explain your reasoning.
 
+    The first bit of a byte adds -128 to the data. The rest of the bits add positive integers. So the smalles we can have is -128 which is 10000000. 
+
 15. What's the largest integer that can be represented in `n` bits? 
     Explain your reasoning.
 
+    Using our method from before, we cannot include the first bit in determining the max. So we are left with (n-1) bits that contribute to the max. Since every bit has two possiblities, we take 2^(n-1) to find the amount of positive integers (and 0) we can store. Then we subtract 1 from that value to account for 0 (since 0 does not contribute to the sum), so we get 2^(n-1) -1 as the maximum integer for 'n' bits. 
+
 ## Text questions
 
 16. Look at the bits for a few different characters using the `utf8` encoding.