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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 (base2), 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 28-1 = 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 0th 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 0th 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 20 * 1 = 1 21 * 1 = 2 22 * 0 = 0 23 * 0 = 0 24 * 1 = 16 25 * 1 = 32 26 * 1 = 64 27 * 0 = 0 The sum is 115. Convert 10101010 to decimal. Convert 00011111 to decimal. Convert 11110110 to decimal. Convert 234 base-8 to decimal. 80 * 4 = 4 81 * 3 = 24 82 * 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: 160 * 6 = 6 161 * 1 = 16 162 * 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 Base-16 0 1 2 3 4 5 6 7 Base-2 1000 1001 1010 1011 1100 1101 1110 1111 Base-16 8 9 A B C D E F 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 Answer = 7FC8D C 8 D Convert 73254 base-16 to base-2 Groups: Base-16: 7 Base-2: 0111 3 0011 2 0010 Answer: 01110011001001010100 5 0101 4 0100 Convert 1a2b3c to base-2 Groups: Base-16: 1 a Base-2: 0001 1010 2 0010 b 1011 Answer: 000110100010101100111100 3 0011 c 1100 Convert 101101011010101101 to hexadecimal. Convert 111111111111111111101010 to hexadecimal. Convert 3f5a86 to binary. Convert aa4fc to binary.