Oct: 50 8 = 6 (r = 2) 6 8 = 0 (r = 6) Writing the remainders in reverse order we get: (50) 10 = (62) 8


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1 ECE Department Summer LECTURE #5: Number Systems EEL : Digital Logic and Computer Systems Based on lecture notes by Dr. Eric M. Schwartz Decimal Number System: Our standard number system is base, also called radix This means we have possible digits: {,,,, 4, 5, 6,,, 9} We use a positional notation to show value Example: 9.46 = Number Systems in General: In positional notation for a rational number system of radix R >, a number N can be represented as the sum of its digits d i : N = m i= k d i R i, where k = # of digits right of the decimal m = # of digits left of the decimal  Any radix greater than can be used, but the most common are: Binary => Base, Octal => Base, Hexadecimal => Base 6 IncrementingCounting: Decimal Binary Octal Hexadecimal A B 4 C 5 D 4 6 E 5 F 6 Converting Other Number Systems to Decimal: Use the equation: N = Examples: (.) = (.4) (.4) 6 = m i= k = 6 d i R i = (.5) = (5.5) = (9.5) Page of
2 ECE Department Summer Converting Decimal to Other Number Systems: Divide the decimal number by the radix repeatedly Save the remainder after each division Continue dividing the new quotient until it is equal to Write the remainders in reverse order (the last remainder is first) Example: Convert (5) to Binary, Octal, and Hexadecimal Bin: 5 = 5 (r = ) 6 = (r = ) 5 = (r = ) = (r = ) = 6 (r = ) = (r = ) Writing the remainders in reverse order we get: (5) = () Oct: 5 = 6 (r = ) 6 = (r = 6) Writing the remainders in reverse order we get: (5) = (6) Hex: 5 6 = (r = ) 6 = (r = ) Writing the remainders in reverse order we get: (5) = () 6 Converting Decimal Fractions Multiply repeatedly by the desired radix. Remove the integer and write them in order after the decimal. Example: Convert (.5) to Binary, Octal, and Hex Bin:.5 =.65 (int = ).65 =.5 (int = ).5 =.5 (int = ).5 = (int = ) Writing the remainders in order we get: (.5) = (.) Oct:.5 = 6.5 (int = 6).5 = 4. (int = 4) Writing the remainders in order we get: (.5) = (.64) Hex:.5 6 =. (int = D) Writing the remainders in order we get: (.5) = (.D) 6 Page of
3 ECE Department Summer How many bits will it take? Compare the decimal number to n Find the smallest n that yields a result ( n ) greater than the decimal number Example: How many bits will () take? 6 = 64 = Therefore, () requires bits. Converting Between Binary, Octal, and Hexadecimal: There is a direct correlation between Binary bits and Octal or Hex numbers Binary => Base => Base Octal => Base => Base => A group of bits Hex => Base 6 => Base 4 => A group of 4 bits (Also, see IncrementingCounting table above.) Converting Octal or Hex to Binary: Write the or 4 bits corresponding to Octal or Hex numbers Example: Convert (4.6) to Binary Octal: 4. 6 {}{}{}.{}{} Therefore, (4.) = (.) Example: Convert (C.9A) 6 to Binary Hex: C. 9 A {}{}.{}{} Therefore, (C.9A) 6 = (.) Converting from Binary to Octal or Hex: Group the binary number by or 4 bits Write the corresponding Octal or Hex numbers Note: For numbers to the left of the decimal, start grouping with the LSb. For numbers to the right of the decimal, start grouping with the MSb. Note: If the bits do not group evenly, assume zeroes where they do not affect the original number s value. (Additional zeroes are bolded below.) Example: Convert (.) to Octal and Hex Octal: Groups of => {}{}{}{}.{}{} Therefore, (.) = (56.6) Hex: Groups of 5 => {}{}{}.{}{} 5 6. D Therefore, () = (56.D) 6 Page of
4 ECE Department Summer Page 4 of Converting from Octal to Hex or Hex to Octal It is easiest to convert to binary as an intermediate step. Note: It is also easier to convert Dec => Hex => Bin than Dec => Bin directly. Binary Arithmetic: Addition Like decimal, but carry over at rather than. Example: Add to. Subtraction Like decimal, but borrowing gives rather than. Example: Subtract from. Multiplication Like decimal, but easier since you only multiply by. Example: Multiply by. Tip: To double (multiply by ) a binary number, shift the bits to the left. Division Like decimal, but easy to get confused. Example: Divide by.. (or R = ) Tip: To halve (divide by ) a binary number, shift the bits to the right.
5 ECE Department Summer Arithmetic with Octal and Hexadecimal: Addition and Subtraction in Oct and Hex can be done in a similar manner Examples: Hex Oct Add: D C A 9 B.5. A. F Subtract: C D C 9 F A B 5.5. A. B For multiplication or division, it is best to convert to decimal Negative Numbers in Binary: So far, we have only looked at positive, or unsigned, binary numbers There are several ways to represent negative numbers Signed Magnitude The MSb is denotes the sign ( for positive, for negative) of the number The remaining bits give the magnitude of the number Example: () = (4) () = (4) What is the range of an bit unsigned binary number? => to 55 What is the range of a signedmagnitude binary number? =>  to It is cumbersome to try to do arithmetic with signed magnitude. s Complement Positive Number (N) MSb is a Remaining bits give the magnitude of the number Negative Number (N^) N^ = ( n ) N, where n is the number of bits This is the same as to flipping all s to s and s to s Example: What is the s complement of? Answer: Note: () = (5), () = (5) What is the range of an bit s complement number? =>  to Note: s complement has numbers that represent. Page 5 of
6 ECE Department Summer s Complement Positive Number (N) MSb is a Remaining bits give the magnitude of the number Negative Number (N*) N* = n N This is the same as flipping s to s and s to s, then adding Example: What is the s complement of? Answer: = Note: () = (5), () = (5) What is the range of an bit s complement number? =>  to Note: Negative numbers start with and positive numbers with. s Complement Arithmetic:  s complement is the most widely used representation of negative numbers Why? Addition, Subtraction, and Multiplication can all be done in s complement Example: Add (5) and () as 4 bit numbers in s Complement => () Dealing with 4 bit numbers so drop the 5 th bit. Example: Subtract () from (5) as 4 bit numbers in s Complement => () Borrow from the left even if there is no number there. Overflow occurs when: Adding two positive numbers results in a negative number Example: = Wrong! Adding two negative numbers results in a positive number Example: = => Wrong! Subtracting a negative from a positive results in a negative number Example: = Wrong! Subtracting a positive from a negative results in a positive number Example: = Wrong! Page 6 of
7 ECE Department Summer Binary Coded Decimal (BCD): Binary coded decimal is a way of representing decimal numbers in binary Each decimal number uses 4 bits (similar to hexadecimal) Decimal BCD Example: Convert (49) to BCD 4 9 {} {} {} {} Answer: (49) = ( ) BCD Example: Convert ( ) BCD to Decimal {} {} {} {} 6 Answer: ( ) BCD = (6) ASCII: American Standard Code for Information Interchange Uses bit binary to represent numbers, letters, and symbols See page in Roth book. Example: Write Joel in ASCII. J o e l ASCII: Page of
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