Chapter 1Data Storage(3)

Yonsei University

1st Semester, 2015 Sanghyun Park

Outline Bits and their storage (prev. file) Main memory (prev. file) Mass storage (prev. file) Representing information as bit patterns (prev. file) Binary system (prev. file) Storing integers Storing fractions

Representing Integers Unsigned integers can be represented in base ___

______ integers =numbers that can be positive or negative

Sign and magnitude notation Two’s complement notation Excess notation

Sign and Magnitude Notation Left hand bit is used to determine the ___ of the number

Left hand bit 0 = positive number 0010 = +2

Left hand bit 1 = negative number1010 = -2

Using 4 bits with sign and magnitude,largest positive number represented is __(-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7)

___ values overall can be represented

+2 added to -2 using regular binary addition does ___ _____ to 0

Two’s Complement Notation (1/2) For a positive number, the two’s complement

representation is _____

For a negative number, ___________ positive valueand then ____ 1

3 in two’s complement is 011

-3 in two’s complement is 100 + 1 = 101

Two’s Complement Notation (2/2) What is the decimal value of 1010 in two’s complement?

It is a negative number since left hand bit is 1 Complement it and add 1: 0101 + 1 = 0110 (+6) Therefore the decimal value is -6

Two’s Complement Addition

Overflow Problem Try adding 5 + 4 in 4-bit two’s complement notation

Result is negative value (-7)

Such an error is called ________

When using two’s complement notation, this might occur when adding two ________ values orwhen adding two ________ values

Normally integers are represented in 32-bit patterns, allowing for positive values as large as 2,147,483,647

Excess Notation (1/3) Excess four notation

All positive numbers begin with 1 All negative numbers begin with 0 0 is represented as ___ Smallest negative number is ___ Largest positive number is ___ Why 4?

(2#of bits –1) = 23-1 = 4

Excess Notation (2/3)

Excess Notation (3/3) What is the decimal value of 101 in excess four

notation? 101 if interpreted unsigned is 5 101 in excess four notation is (5-4) = 1

What is the excess four notation for decimal value 3? Add four: 3+4 = 7 Represent it in 3-bit binary pattern = 111

Excess Eight Notation

Storing Fractions (1/6) Need to represent ______ and ______ of radix point Use ___________ notation

1 bit: Sign bit (0 positive, 1 negative)

3 bits: Exponent (encodes position of radix point in excess four )

4 bits: Mantissa (encodes number)

Storing Fractions (2/6) What is the decimal value of 01101011 in floating-point

notation? First bit is 0, then positive Exponent is 110 and mantissa is 1011 Extract mantissa and place a radix point on its left side 0.1011 Extract exponent and interpret as excess four notation 110 in excess four is +2 (make sure?) +2 exponent means move radix point to the right by two bits

(a negative exponent means move radix to left) 0.1011 becomes 10.11

Storing Fractions (3/6) What is the decimal value of 10111100 in floating-point

notation? First bit is 1, then negative Exponent is 011and mantissa is 1100 Extract mantissa and place a radix point on its left side 0.1100 Extract exponent and interpret as excess four notation 011 in excess four is -1 (make sure?) -1 exponent means move radix point to the left by 1 bit 0.1100 becomes 0.01100

Storing Fractions (4/6) What is the floating point notation of the number ?

Express the number in binary to obtain 1.001 (make sure?) Copy bit pattern into mantissa field from left to right

starting with the leftmost 1 in binary representation Mantissa is 1001 Compute exponent to get 1.001 from .1001

(imagine mantissa with radix point at its left) Need to move radix point to right one bit Exponent is +1 Express exponent in excess four notation: 1+4 = 5 (101) Therefore, 01011001

811

Storing Fractions (5/6) What is the floating point notation of the number ?

Express number in binary to obtain 0.01 Copy bit pattern into mantissa field from left to right

starting with the leftmost 1 in binary representation Mantissa is 1000 (you append zeros to fill the 4-bit mantissa) Compute exponent to get 0.01 from 0.1000 Need to move radix point to left one bit Exponent is -1 Express exponent in excess four notation: -1+4 = 3 (011) Therefore, 10111000

41

Storing Fractions (6/6) What is the floating point notation of the number ?

We end up with the bit pattern 01101010,which represents 2 1/2 instead of 2 5/8

What has occurred is called a truncation error or round-off error

852