Lecture 2 Binary Arithmetic Topics Terminology Fractions Base-r to decimal Unsigned Integers Signed magnitude Two’s complement August 26, 2003 CSCE 211 Digital Design

– 2 – CSCE 211H Fall 2003 Overview Last Time: Readings 1.1, Analog vs Digital Conversion Base-r to decimal Conversion decimal to Base-r New: Readings , 3.1 Conversion of Fractions base-r  decimal Unsigned Arithmetic Signed Magnitude Two’s Complement Excess-1023VHDL Introductory Examples

– 3 – CSCE 211H Fall 2003 Review Last Time: Base-r to decimal Converting base-r to decimal by definition d n d n-1 d n-2 …d 2 d 1 d 0 = d n r n + d n-1 r n-1 … d 2 r 2 +d 1 r 1 + d 0 r 0 Example 4F0C = 4* F* * C*16 0 4F0C = 4* F* * C*16 0 = 4* * *1 = =20236

– 4 – CSCE 211H Fall 2003 Review Last Time: decimal to Base-r Repeated division algorithm Justification: d n d n-1 d n-2 …d 2 d 1 d 0 = d n r n + d n-1 r n-1 … d 2 r 2 +d 1 r 1 + d 0 r 0 Dividing each side by r yields (d n d n-1 d n-2 …d 2 d 1 d 0 ) / r = d n r n-1 + d n-1 r n-2 … d 2 r 1 +d 1 r 0 + d 0 r -1 So d 0 is the remainder of the first division ((q 1 ) / r = d n r n-2 + d n-1 r n-3 … d 3 r 1 +d 2 r 0 + d 1 r -1 So d 1 is the remainder of the next division and d 2 is the remainder of the next division …

– 5 – CSCE 211H Fall 2003 Review Last Time: decimal to Base-r Repeated division algorithm Example Convert 4343 to hex 4343/16 = 271 remainder = 7 271/16 = 16 remainder = /16 = 16 remainder = 15 16/16 = 1 remainder = 0 16/16 = 1 remainder = 0 1/16 = 0 remainder = 1 1/16 = 0 remainder = 1 So = 10F7 16 To check the answer convert back to decimal 10F7 = 1* *16 + 7*1 = = 4343

– 6 – CSCE 211H Fall 2003 Review Last Time: Hex to Binary One hex digit = four binary digits Example 3FAC = (spaces just for readability) 3FAC = (spaces just for readability) Binary to hex four digits = one (from right!!!) Example = = = 3 D A

– 7 – CSCE 211H Fall 2003 Hex to Decimal Fractions.d -1 d -2 d -3 …d –(n-2) d –(n-1) d -n = d -1 r -1 + d -2 r -2 …d -(n-1) r -(n-1) +d 1 r -n Example.1EF 16 = 1* E* F*16-3.1EF 16 = 1* E* F*16-3 = 1* * *2.4414e-4 = … (probably close but not right) = … (probably close but not right)

– 8 – CSCE 211H Fall 2003 Decimal Fractions to hex.d -1 d -2 d -3 …d –(n-2) d –(n-1) d -n = d -1 r -1 + d -2 r -2 …d -(n-1) r -(n-1) +d 1 r -n Multiplication by r yields r *(.d -1 r -1 + d -2 r -2 …d -(n-1) r -(n-1) +d 1 r -n = d -1 r 0 + d -2 r -1 …d -(n-1) r -(n-2) +d 1 r -(n-1) Whole number part = d -1 r 0 Multiplying again by r yields d -2 r 0 as the whole number part … till fraction = 0

– 9 – CSCE 211H Fall 2003 Example: Hex Fractions to decimal Convert.3FA to decimal.3FA16 = 3* F* A*16-3.3FA16 = 3* F* A*16-3 = 3* * * (1/4096) =

– 10 – CSCE 211H Fall 2003 Unsigned integers What is the biggest integer representable using n-bits(n digits)? What is its value in decimal? Special cases 8 bits 16 bits 16 bits 32 bits 32 bits

– 11 – CSCE 211H Fall 2003 Arithmetic with Binary Numbers x x 101 Problems with 8 bit operations

– 12 – CSCE 211H Fall 2003 Signed integers How do we represent? Signed-magnitude Excess representations w bits  0 <= unsigned_value < 2 w In excess-B we subtract the bias (B) to get the value. In excess-B we subtract the bias (B) to get the value. example example

– 13 – CSCE 211H Fall 2003 Complement Representations of Signed integers One’s complement Two’s complement

– 14 – CSCE 211H Fall 2003 Two’s Complement Operation One’s complement + 1 or Find rightmost 1, complement all bits to the left of it. Examples

– 15 – CSCE 211H Fall 2003 Two’s Complement Representation Consider a two’s complement binary number d n d n-1 d n-2 …d 2 d 1 d 0 If d n, the sign bit = 0 the number is positive and its magnitude is given by the other bits. If d n, the sign bit = 1 the number is negative and take its two’s complement to get the magnitude. Weighted Sum Interpretation … n-1 2 n-2 n -2 n-1

– 16 – CSCE 211H Fall 2003 Two’s Complement Representation Consider a two’s complement binary number d n d n-1 d n-2 …d 2 d 1 d 0 If d n, the sign bit = 0 the number is positive and its magnitude is given by the other bits. If d n, the sign bit = 1 the number is negative and take its two’s complement to get the magnitude. Weighted Sum 0 1 Example = 1 2 … n-1 2 n-2 n -2 n-1

– 17 – CSCE 211H Fall 2003 Two’s Complement Representation What is the 2’s complement representation in 16 bits of –5? +7?-1?0-2

– 18 – CSCE 211H Fall 2003 Arithmetic with Signed Integers Signed Magnitude Addition if the signs are the same add the magnitude if the signs are different subtract the smaller from the larger and use the sign of the larger if the signs are different subtract the smaller from the larger and use the sign of the largerSubtraction? Two’s complement Just add signs take care of themselves

– 19 – CSCE 211H Fall 2003 Overflow in Two’s Complement

– 20 – CSCE 211H Fall 2003 Binary Code Decimal

– 21 – CSCE 211H Fall 2003 Representations of Characters

– 22 – CSCE 211H Fall 2003 Basic Gates ANDORNOT

– 23 – CSCE 211H Fall 2003 Basic Gates NANDNORXOR

– 24 – CSCE 211H Fall 2003 Half Adder Circuit

– 25 – CSCE 211H Fall 2003 Full Adder