The Wonders of Conversion

 A number system is a system in which a number is represented. There are potential infinite number systems that can exist (there are infinite numbers, after all), but you are only responsible for a very small subset.

 For the AP Exam, you will need to know binary (base 2), octal (base-8), decimal (base-10) and hexadecimal (base-16).

 Notice that all of those bases are powers of 2!

 The binary number system is the one your computer explicitly understands.  All numbers are represented by bits, which is either a 0 or a 1.

 A byte is a collection of 8 bits, and can represent numbers from -256 to 255. (The max value of a collection of bits is always 2^ numbits -1  2 8 -1 = 255)  For example, 10110110 is 182 in decimal.

 Octal is base-8. Only digits 0-7 are used.

 Using 182 again, it is 266 in octal.

 (That is not a typo – the number appears to be bigger!)

 Decimal is good old base-10. You’ve been using this all of your lives!

 182 is 182 in decimal!

 Hexadecimal is base-16. It uses the digits 0-9 and the letters A-F to represent 10-15, respectfully.

 For example, 15 base 10 is F in hexadecimal.

 16 is 10.

 890 is 37A.

 The following is called the expansion method and only works on converting a number TO BASE-10 !!!

 You need to understand how these numbers are written. You have to analyze the number starting on the right. This number represents the base number raised to the 0 th power.

 The second number from the right represents the base number raised to the first power.

 …and so on

     Consider the following binary number: 01110011 What is this number in decimal format?

Start looking at the rightmost digit. This represents the base number raised to the 0 th power. Multiply this number by the digit present (which is a 1). Save this number.

Look at the second rightmost digit. This represents the base number raised to the first power. Multiply this number by the digit present (which is a 1). Save this number.

…do this for all numbers present and add all of products together to get your base-10 number.

 01110011 to decimal         2 0 * 1 = 1 2 1 * 1 = 2 2 2 * 0 = 0 2 3 * 0 = 0 2 4 * 1 = 16 2 5 * 1 = 32 2 6 * 1 = 64 2 7 * 0 = 0  The sum is 115 .

 Convert 10101010 to decimal.

 Convert 00011111 to decimal.

 Convert 11110110 to decimal.

 Convert 234 base-8 to decimal.

   8 0 * 4 = 4 8 1 * 3 = 24 8 2 * 2 = 128  The sum is 156 .

 Convert 716 base-8 to decimal.

 Convert 45 base-8 to decimal.

 Convert 10 base-8 to decimal.

 REMEMBER:  A = 10      B = 11 C = 12 D = 13 E = 14 F = 15  Convert F16 to decimal.

 Work:    16 0 * 6 = 6 16 1 * 1 = 16 16 2 * 15 = 3840  The sum is 3862 .

 Convert C10 to decimal.

 Convert FF to decimal.

 Convert 16 to decimal.

 One method of converting any base number to base-10 is by continuously dividing the original decimal number by the desired base until you get a quotient of 0, and then read the remainders backwards. Note: if you are converting to hexadecimal, remember that 10..15 are represented by A..F respectively!)

   Convert 201 to binary.

Work:         201 / 2 = 100 remainder 1 100 / 2 = 50 remainder 0 50 / 2 = 25 remainder 0 25 / 2 = 12 remainder 1 12 / 2 = 6 remainder 0 6 / 2 = 3 remainder 0 3 / 2 = 1 remainder 1 1 / 2 = 0 remainder 1.

201 in binary is 11001001 .

 Convert 1076 to binary.

 Convert 200 to binary.

 Convert 450 to binary.

 Convert 173 to octal.

 Work:    173 / 8 = 21 remainder 5 21 / 8 = 2 remainder 5 2 / 8 = 0 remainder 2  173 base-10 is 255 base-8.

 Convert 1076 to octal.

 Convert 200 to octal.

 Convert 450 to octal.

 Convert 506 to hexadecimal.

 Work:    506 / 16 = 31 remainder 10 31 / 16 = 1 remainder 15 1 / 16 = 0 remainder 1  BUT 10 is A and 15 is F so…  506 base-10 is 1FA base-16.

 Convert 1076 to hexadecimal.

 Convert 200 to hexadecimal.

 Convert 450 to hexadecimal.

 There is a neat trick that allows one to convert from binary to hexadecimal, without converting the binary to base-10 first.

 Every base-16 digit (including letters) can be represented by four bits:

Base -2 0000 0001 0010 0011 0100 0101 0110 0111 3 4 5 6 7 Base-16 0 1 2 Base-2 1000 1001 1010 1011 1100 1101 1110 1111 B C D E F Base-16 8 9 A

 Convert 1001001010111 base-2 to base-16.

 Starting from the right, break up the binary number into groups of 4 bits. If the last group doesn’t have four bytes, pad it on the left with zeros.

 Base-2 groups: 0001 0010 0101 0111  Base-16: 1 2 5 7  Answer = 1257

 1111111110010001101 to base-16  Groups:   Base-2: 0111 1111 1100 1000 1101 Base-16: 7 F C 8 D  Answer = 7FC8D

 Convert 73254 base-16 to base-2  Groups:   Base-16: 7 3 2 5 4 Base-2: 0111 0011 0010 0101 0100  Answer: 01110011001001010100

 Convert 1a2b3c to base-2  Groups:   Base-16: 1 a 2 b 3 c Base-2: 0001 1010 0010 1011 0011 1100  Answer: 000110100010101100111100

 Convert 101101011010101101 to hexadecimal.

 Convert 111111111111111111101010 to hexadecimal.

 Convert 3f5a86 to binary.

 Convert aa4fc to binary.