part2

Author: RaihanSharif

number-systems/Part-2.md

@@ -9,14 +9,55 @@ The answers to these questions will require a bit of explanation, not just a sim
 Q16: How can you test if a binary number is a power of two (e.g. 1, 2, 4, 8, 16, ...)?
 Answer:
 
+A power of two has exactly one bit set (e.g. 1000 or 0100 — leading zeros
+don't count). E.g. 0100 is 2^2 but 0110 is 2^2 + 2^1.
+
+One method to figure this out is to repeatedly divide the number by 2:
+
+Since 0 has no bits set and the power of 2 is always positive these are not checked. So, for any n > 0 , repeatedly right-shift n, which is equivalent to dividing by 2.
+The bit dropped off by each shift is the remainder: if it's ever 1, n is not a
+power of two. If you shift all the way down to a single remaining bit — the
+original number's leading 1 — without ever dropping a 1, then n is a power of two.
+
+Another, more efficient method is to check the bitwise AND of n with (n-1).
+
+For example:
+
+```
+n = 1000
+n-1 = 0111
+
+n & (n - 1):
+1000
+0111
+----
+0000
+```
+
+If the result is 0000 then it is a power of two only if the result is 0.
+
 Q17: If reading the byte 0x21 as an ASCII character, what character would it mean?
-Answer:
+Answer: !
 
-Q18: If reading the byte 0x21 as a greyscale colour, as described in "Approaches for Representing Colors and Images", what colour would it mean?
-Answer:
+Q18: If reading the byte 0x21 as a greyscale colour, as described in "Approaches for Representing Colors and Images", what colour would it mean? Answer:
+Figure 2.2 illustrates 4 shades of gray, 0x00 = black, 0x55 = dark gray, 0xAA = light gray, and 0xFF = white. As 0x21 falls between black and dark gray, it would be classified as dark gray or very dark gray.
 
 Q19: If reading the bytes 0xAA00FF as a sequence of three one-byte decimal numbers, what decimal numbers would they be?
 Answer:
 
+```
+AA = 10 * 16^1 + 10 * 16^0 = 170
+00 = 0
+FF = 0b11111111 = 2^8-1 = 255
+```
+
 Q20: If reading the bytes 0xAA00FF as an RGB colour, as described in "Approaches for Representing Colors and Images", what colour would it mean?
 Answer:
+
+Red: 170
+
+Green: 0
+
+Blue: 255
+
+A lot of blue and red = purple