number-systems-solutions

Author: Pezhman-Azizi

number-systems/README.md

@@ -7,59 +7,100 @@ The goal of these exercises is for you to gain an intuition for binary numbers.
 Convert the decimal number 14 to binary.
 Answer:
 
+1110
+
 Convert the binary number 101101 to decimal:
 Answer:
 
+32,8,4,1 = 45
+
 Which is larger: 1000 or 0111?
 Answer:
 
+1000 is larger since it has more leading ones ( literally digit 1) compared to 0111. By leading one, i mean the outer most on the left side.
+
 Which is larger: 00100 or 01011?
 Answer:
 
+01011 is larger as it is got again the more leading ones ( digit 1) on the left side.
+
 What is 10101 + 01010?
 Answer:
+It is 11111
 
 What is 10001 + 10001?
 Answer:
+It is 100010
+
 
 What's the largest number you can store with 4 bits, if you want to be able to represent the number 0?
 Answer:
 
+Answer1: If representing 0 means having 0000 AS A POSSIBLE NUMBER in the range of 0 to largest number produced by four digits, then the largest possible number is when all the digits are one which makes 1111 the largest number.
+
+Answer2: If it means having 0 in one of the digits as a possible occurrence, then the largest possible number produced when zero goes to the last digit to have the least effect which  1110 the largest number.
+
 How many bits would you need in order to store the numbers between 0 and 255 inclusive?
 Answer:
 
+since 256 is 2^8 the biggest number of digits with 255 inclusive is made when we have 8 digits.
+
 How many bits would you need in order to store the numbers between 0 and 3 inclusive?
 Answer:
 
+2 bits.
+
 How many bits would you need in order to store the numbers between 0 and 1000 inclusive?
 Answer:
 
+10 bits. since 2^n - 1 = the biggest number with n bits.
+
 How can you test if a binary number is a power of two (e.g. 1, 2, 4, 8, 16, ...)?
 Answer:
 
+if a number is a power of 2, then it has only one bit as 1 and the rest are zeros.
+
 Convert the decimal number 14 to hex.
 Answer:
+It is E
 
 Convert the decimal number 386 to hex.
 Answer:
+ox182
 
 Convert the hex number 386 to decimal.
 Answer:
+6*(16^0)+8*(16^1)+3*(16^2) = 902
 
 Convert the hex number B to decimal.
 Answer:
+It is 11
 
 If reading the byte 0x21 as a number, what decimal number would it mean?
 Answer:
+2(16^1)+1(16^0) = 33
 
 If reading the byte 0x21 as an ASCII character, what character would it mean?
 Answer:
+!
 
-If reading the byte 0x21 as a greyscale colour, as described in "Approaches for Representing Colors and Images", what colour would it mean?
+If reading the byte ox21 as a greyscale colour, as described in "Approaches for Representing Colors and Images", what colour would it mean?
 Answer:
 
+Since Ox21 is 33 in decimal and we have a range of 0(black) to 255(white) in decimal, then 33 is relatively close to zero. So it the gray scale it a dark grey
+
 If reading the bytes 0xAA00FF as an RGB colour, as described in "Approaches for Representing Colors and Images", what colour would it mean?
 Answer:
 
+Each set of  2 hex digits stands for each of primary colors in the following order:
+RRGGBB: comparing it to AA00FF means that we don't have any greens. so we would have a mixture of blue and red. So eventually we have shade of purpule
+
+
 If reading the bytes 0xAA00FF as a sequence of three one-byte decimal numbers, what decimal numbers would they be?
 Answer:
+
+So A in hex represents 10 in decimal
+And F in hex represent 15, so:
+AA => (16^0)*10 + (16^1)*10 = 10+160 = 170;
+00 => 0
+FF => (16^0)*15 + (16^1)*15 = 15+240 = 255