COSC2410: LAB 2 BINARY ARITHMETIC SIGNED NUMBERS FLOATING POINT REPRESENTATION BOOLEAN ALGEBRA 1

Review Convert from base 10 to 2, 8, Base 2: Base 8: 51 Base 16: 29 2

BINARY ARITHMETIC ADDITION, SUBTRACTION, MULTIPLICATION, DIVISION 3

Binary Arithmetic: Addition and Subtraction Addition = = = = 0 (carry: 1) Subtraction 0 – 0 = 0 0 – 1 = 1 (borrow: 1) 1 – 0 = 1 1 – 1 = 0 4

Binary Addition: Example Note: Incase of adding 3 1’s. Instead of the sum being 0 with a carry of 1, the sum is 1 with a carry of 1. 5

Binary Subtraction: Examples Note: After borrowing from the next digit, 0 becomes 1 0 and not becomes 1 when borrowed from. 6

Binary Addition and Binary Subtraction Practice 1. Add and Subtract 1011 from Add and Subtract 010 from

Binary Arithmetic Multiplication Multiplication  0 x 0 = 0  0 x 1 = 0  1 x 0 = 0  1 x 1 = 1 Note: When multiplying, add two of the partial products at a time to prevent mistakes that my occur from adding many numbers 8

Binary Multiplication Practice 1. Multiply 1001 and Multiply 1101 and Multiply 0001 and Multiply 1010 and

SIGNED NUMBERS 3 WAYS TO REPRESENT 10

Method 1: Sign And Magnitude  Obvious solution: sign and magnitude  define leftmost bit (most significant bit) to be sign!  0 => +(positive), 1 => - (negative)  Rest of bits can be numerical value of number  MIPS uses 32-bit integers. +1then would be:  And - 1 in sign and magnitude would be:

Method 2: 1’s Complement  Complement the bits  Example: 7 10 = =  What is ?

Method 3: 2’s Complement  Method 1: Subtract from 2 n. For Example 3: 0011, n = 4 -3: (2 n ) – 0011 = 1101  Method 2: Complement and add 1 3: : = 1101  The leftmost bit of a signed binary numeral indicates the sign  0 (+), all the bits are the numerical value of number  1 (-), all the bits are inverted then 1 is added to the result ► Remember for positive numbers, just make sure a 0 is the MSB. You do not need to convert it using the formula above! 13

Signed Number Representation Table 14

Signed Number Representation Practice 1. Represent each negative number as an 8 bit binary number using a) sign and magnitude, b) 1’s complement, and c) 2’s complement

Addition/Subtraction using 2’s Complement  Subtraction = Addition with opposite sign. 4 – 6 = 4 + (-6) = -2 4: :1010(2’s complement) Sign bit =-2

Overflow Error: Sum Is Too Large  4 bit binary with 2’s complement representation has a number range between -8 and 7  What happens if 6 + 4? (-7) + (-2)? 6:0110-7:1001 4:0100-2: Sign bit = 1 (negative) Sign bit = 0 (positive)

Overflow Error: Cont’d…  Only 2 cases where overflow can occur:  Sum of 2 positive numbers yields a negative result  Sum of 2 negative numbers yields a positive result  Note: We always ignore the carry to ensure number of bits does not change. 18

2’s Complement Addition/Subtraction Practice Convert the following decimal numbers into 6 bit binary and then solve using 2’s complement. Indicate if Overflow Occurs

IEEE FLOATING POINT REPRESENTATION 20

IEEEFloating Point Representation  Most standard floating point representation use: 1 bit for the sign (positive or negative) 8 bits for the range (exponent field) 23 bits for the precision (fraction field) 21 Note: Biased Notation. The exponent is obtained by adding 127 to the value. That is: 5 -> 132, 1-> 128, 0 -> 127, >0

Floating Point Exceptions Overflow & Underflow  Overflow occurs when the exponent is too large to fit in the exponent field.  Underflow occurs when the exponent is too small to fit in the exponent field. 22 Negative Overflow Negative numbersPositive Numbers Negative UnderflowPositive UnderflowPositive Overflow 0

Floating Point Example  How is the decimal number represented in floating point? Show Steps of your work. 23 Thus the exponent is given by:

Floating Point Example Solution

Floating Point Example 2  Convert the following binary floating point number into decimal FPexponent = = 125

Floating Point Representation Practice  How is the decimal number represented in floating point? Show all steps of your work.  How is the number (17.4) 16 represented in binary floating point? Show all steps of your work.  Represent ( ) 2 as floating point. Show all steps of your work. 26

Floating Point Practice Cont’d… 27  Convert the following binary floating point number into an un-normalized binary number  Convert the following binary floating point number into decimal