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 signed-magnitude 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