 Binary Binary  Binary Number System Binary Number System  Binary to Decimal Binary to Decimal  Decimal to Binary Decimal to Binary  Octal and Hexadecimal Octal and Hexadecimal  Binary to Hexadecimal Binary to Hexadecimal  Hexadecimal to Binary Hexadecimal to Binary  Hexadecimal to Decimal Hexadecimal to Decimal  Any Number Base to Decimal Any Number Base to Decimal  Decimal to Any Number Base Decimal to Any Number Base  Binary Coded Decimal (BCD) Binary Coded Decimal (BCD)

 A computer is a “bistable” device  A bistable device: ◦ Easy to design and build ◦ Has 2 states: 0 and 1  One Binary digit (bit) represents 2 possible states (0, 1)

 With 2 bits, 4 states are possible (2 2 = 4) With n bits, 2 n states are possible With 3 bits, 8 states are possible (2 3 = 8) Bit 1 Bit 0 State Bit 2 Bit 1 Bit 0 State Go back

 From left to right, the position of the digit indicates its magnitude (in decreasing order) ◦ E.g. in decimal, 123 is less than 321 ◦ In binary, 011 is less than 100  A subscript indicates the number’s base ◦ E.g. is 100 decimal or binary? We don’t know! ◦ But = is clear Go back

 Binary is base 2  Example: convert (binary) to decimal = 1x x x x x2 0 = 1x16 + 0x8 + 1x4 + 1x2 + 0x1 = =22  So = Go back

 Binary is base 2  Example: convert 35 (decimal) to binary QuotientRemainder 35 / 2 = / 2 = 81 8 / 2 = 40 4 / 2 = 20 2 / 2 = 10 1 / 2 = 01  So = Go back

 It is difficult for a human to work with long strings of 0’s and 1’s  Octal and Hexadecimal are ways to group bits together  Octal: base 8  Hexadecimal: base 16

 With 4 bits, there are 16 possibilities  Use 0, 1, 2, 3, …9 for the first 10 symbols  Use a, b, c, d, e, and f for the last 6 Bit 3 Bit 2 Bit 1 Bit 0 Symbol a 101 1b 110 0c 110 1d 111 0e 111 1f Go back

 = ? in hex  Group into 4 bits, from the right:  0101, 0110, 1011,  Now translate each (see previous table): => 5, => 6, => b, => 3 So this is 56b3 16  What if there are not enough bits? ◦ Pad with 0’s on the left Go back

f0e5 16 = ? in binary Translate each into a group of 4 bits: f 16 => , 0 16 => , e 16 => , 5 16 => So this is Go back

 Hexadecimal is base 16  Example: convert 16 (hex) to decimal = 1x x16 0 = 1x16 + 6x1 = =22  So =  Not surprising, since = 0001, ◦ If one of the hex digits had been > 9, say c, then we would have used 12 in its place. Go back

 From right to left, multiply the digit of the number-to-convert by its base position  Sum all results Go back

 Take the decimal number, and divide by the new number base  Keep track of the quotient and remainder  Repeat until quotient = 0  Read number from the bottom to the top Go back

 Why not use 4 bits to represent decimal?  Let 0000 represent 0  Let 0001 represent 1  Let 0010 represent 2  Let 0011 represent 3, etc. ◦ This is called BCD ◦ Only uses 10 of the 16 possibilities Go back