ECE 331 – Digital System Design Number Systems and Conversion, Binary Arithmetic, and Representation of Negative Numbers (Lecture #10) The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6th Edition, by Roth and Kinney, and were used with permission from Cengage Learning.

ECE 331 - Digital System Design 52 What does this number represent? Consider the “context” in which it is used. Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design 1011001.101 What is the decimal value of this number? Consider the base (or radix) of this number. Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Number Systems Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Number Systems R is the radix (or base) of the number system. Must be a positive number R digits in the number system: [0 .. R-1] Important number systems for digital systems: Base 2 (binary) [0, 1] Base 8 (octal) [0 .. 7] Base 16 (hexadecimal) [0 .. 9, A .. F] Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Number Systems Positional Notation [a4a3a2a1a0.a-1a-2a-3]R ai = ith position in the number R = radix or base of the number radix point Spring 2011 ECE 331 - Digital System Design

+ a-1 x R-1 + a-2 x R-2 + … a-m x R-m Number Systems Power Series Expansion D = an x R4 + an-1 x R3 + … + a0 x R0 + a-1 x R-1 + a-2 x R-2 + … a-m x R-m D = decimal value ai = ith position in the number R = radix or base of the number Spring 2011 ECE 331 - Digital System Design

Number Systems: Example Decimal 927.4510 = 9 x 102 + 2 x 101 + 7 x 100 + 4 x 10-1 + 5 x 10-2 Spring 2011 ECE 331 - Digital System Design

Number Systems: Example Binary 1101.1012 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20 + 1 x 2-1 + 0 x 2-2 + 1 x 2-3 Spring 2011 ECE 331 - Digital System Design

Number Systems: Example Octal 326.478 = 3 x 82 + 2 x 81 + 6 x 80 + 4 x 8-1 + 7 x 8-2 Spring 2011 ECE 331 - Digital System Design

Number Systems: Example Hexadecimal E5A.2B16 = 14 x 162 + 5 x 161 + 10 x 160 + 2 x 16-1 + 11 x 16-2 Spring 2011 ECE 331 - Digital System Design

Conversion between Number Systems Spring 2011 ECE 331 - Digital System Design

Conversion of a Decimal Integer Use repeated division to convert a decimal integer to any other base. Spring 2011 ECE 331 - Digital System Design

Conversion of a Decimal Integer Example: Convert the decimal number 57 to binary and to octal: 57 / 2 = 28: rem = 1 = a0 28 / 2 = 14: rem = 0 = a1 14 / 2 = 7: rem = 0 = a2 7 / 2 = 3: rem = 1 = a3 3 / 2 = 1: rem = 1 = a4 1 / 2 = 0: rem = 1 = a5 5710 = 1110012 57 / 8 = 7: rem = 1 = a0 7 / 8 = 0: rem = 7 = a1 5710 = 718 Spring 2011 ECE 331 - Digital System Design

Conversion of a Decimal Fraction Use repeated multiplication to convert a decimal fraction to any other base. Spring 2011 ECE 331 - Digital System Design

Conversion of a Decimal Fraction Example: Convert the decimal number 0.625 to binary and to octal. 0.625 * 2 = 1.250: a-1 = 1 0.250 * 2 = 0.500: a-2 = 0 0.500 * 2 = 1.000: a-3 = 1 0.62510 = 0.1012 0.625 * 8 = 5.000: a0 = 5 0.62510 = 0.58 Spring 2011 ECE 331 - Digital System Design

Conversion of a Decimal Fraction Example: Convert the decimal number 0.7 to binary. 0.7 * 2 = 1.4: a-1 = 1 0.4 * 2 = 0.8: a-2 = 0 0.8 * 2 = 1.6: a-3 = 1 0.6 * 2 = 1.2: a-4 = 1 0.2 * 2 = 0.4: a-5 = 0 0.4 * 2 = 0.8: a-6 = 0 0.710 = 0.1 0110 0110 0110 ...2 In some cases, conversion results in a repeating fraction. process begins repeating here! Spring 2011 ECE 331 - Digital System Design

Conversion of a Mixed Decimal Number Convert the integer part of the decimal number using repeated division. Convert the fractional part of the decimal number using repeated multiplication. Combine the integer and fractional parts in the new base. Spring 2011 ECE 331 - Digital System Design

Conversion of a Mixed Decimal Number Example: Convert 48.562510 to binary. Confirm the results using the Power Series Expansion. Spring 2011 ECE 331 - Digital System Design

Conversion between Bases Conversion between any two bases can be carried out directly using repeated division and repeated multiplication. Base A → Base B However, it is, generally, easier to convert Base A to its decimal equivalent and then convert the decimal value to Base B. Base A → decimal value → Base B Power Series Expansion Repeated Division, Repeated Multiplication Spring 2011 ECE 331 - Digital System Design

Conversion between Bases Conversion between binary and octal can be carried out by inspection. Each octal digit corresponds to 3 bits 101 110 010 . 011 0012 = 5 6 2 . 3 18 010 011 100 . 101 0012 = 2 3 4 . 5 18 7 4 5 . 3 28 = 111 100 101 . 011 0102 3 0 6 . 0 58 = 011 000 110 . 000 1012 Is the number 392.248 a valid octal number? Spring 2011 ECE 331 - Digital System Design

Conversion between Bases Conversion between binary and hexadecimal can be carried out by inspection. Each hexadecimal digit corresponds to 4 bits 1001 1010 0110 . 1011 01012 = 9 A 6 . B 516 1100 1011 1000 . 1110 01112 = C B 8 . E 716 E 9 4 . D 216 = 1110 1001 0100 . 1101 00102 1 C 7 . 8 F16 = 0001 1100 0111 . 1000 11112 Note that the hexadecimal number system requires additional characters to represent its 16 values. Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Number Systems Base: 10 2 8 16 What is the value of 12? Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Binary Arithmetic Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Binary Addition 0 0 1 1 + 0 + 1 + 0 + 1 0 1 1 10 Sum Carry Spring 2011 ECE 331 - Digital System Design

Binary Addition: Examples 01011011 + 01110010 00111100 + 10101010 10110101 + 01101100 Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Binary Subtraction 0 10 1 1 - 0 - 1 - 0 - 1 0 1 1 0 Difference Borrow Spring 2011 ECE 331 - Digital System Design

Binary Subtraction: Examples 01110101 - 00110010 00111100 - 10101100 10110001 - 01101100 Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Binary Arithmetic Single-bit Addition Single-bit Subtraction What logic function is this? A B Carry Sum 1 A B Difference 1 Spring 2011 ECE 331 - Digital System Design

Binary Multiplication 0 0 1 1 x 0 x 1 x 0 x 1 0 0 0 1 Product Spring 2011 ECE 331 - Digital System Design

Binary Multiplication: Examples 0110 x 1010 1011 x 0110 1001 x 1101 Spring 2011 ECE 331 - Digital System Design

Representation of Negative Numbers Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design 10011010 What is the decimal value of this number? Is it positive or negative? If negative, what representation are we using? Spring 2011 ECE 331 - Digital System Design

Unsigned and Signed Binary Numbers 1 – Magnitude MSB Unsigned number Sign Signed number 2 0 denotes 1 denotes + Spring 2011 ECE 331 - Digital System Design

Unsigned Binary Numbers For an n-bit unsigned binary number, all n bits are used to represent the magnitude of the number. ** Cannot represent negative numbers. Spring 2011 ECE 331 - Digital System Design

Unsigned Binary Numbers For an n-bit binary number 0 <= D <= 2n – 1 where D = decimal equivalent value For an 8-bit binary number: 0 <= D <= 28 – 1 28 = 256 For a 16-bit binary number: 0 <= D <= 216 – 1 216 = 65536 Spring 2011 ECE 331 - Digital System Design

Signed Binary Numbers For an n-bit signed binary number, n-1 bits are used to represent the magnitude of the number; the leftmost bit is, generally, used to indicate the sign of the number. 0 = positive number 1 = negative number Spring 2011 ECE 331 - Digital System Design

Signed Binary Numbers Representations for signed binary numbers: 1. Sign and Magnitude 2. 1's Complement 3. 2's Complement Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Sign and Magnitude For an n-bit signed binary number, The leftmost bit is the sign bit. The remaining n-1 bits represent the magnitude. Includes a representation for +0 and -0 - (2n-1 – 1) <= N <= + (2n-1 – 1) Spring 2011 ECE 331 - Digital System Design

Sign and Magnitude: Example What is the Sign and Magnitude representation for the following decimal values, using 8 bits? + 97 - 68 - 97 + 68 Spring 2011 ECE 331 - Digital System Design

Sign and Magnitude: Example Can the following decimal numbers be represented using 8-bit Sign and Magnitude representation? - 212 - 127 +128 +255 Spring 2011 ECE 331 - Digital System Design

ECE 331 - Digital System Design Questions? Spring 2011 ECE 331 - Digital System Design