Understanding Number Representation and Precision in Computing for Engineers, Study notes of Computer Science

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CS1371 Introduction to Computing for Engineers Numbers and Arrays

How are Numbers Represented? Numeric representation is a key part of defining dataused in a computer program. We will consider severaltopics: • Limitations in representation when using finite sizednumbers (precision & accuracy) • Binary representation • Integers versus real numbers 4

(aka: fixed vs floating point numbers) • IEEE Floating Point • How Matlab handles this

Numbers: Precision and Accuracy Good Accuracy^ Good PrecisionGood Precision^ Poor Accuracy 5

Good Accuracy^ Poor Precision

Poor Accuracy Poor Precision

-^ Low precision:

π^ = 3.

-^ High precision:

π^ = 3.

-^ Low accuracy:

π^ = 3.

-^ High accuracy:

π^ = 3.

-^ High accuracy & precision:

π^ = 3. In most cases, computers must deal with finite precision(some software provides unlimited precision).

Inside the Bytes

-^ A byte is the smallest memory allocation available. •^ A byte contains 8 bits so that:^ – Smallest: 0 0 0 0 0 0 0 0 = 0 7

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• Largest:^ 1 1 1 1 1 1 1 1 =^7 6 5 _12+12+12_^ _+1_

_4 32 1 +12+12+12+1_**

0 8 = 255(or 2^ -1)^10

-^ Result: a single byte can be used to store an integer numberranging from 0 to 255

(256 different numbers)

-^ If negative numbers are included, one bit must be dedicated to thesign, leaving only 7 bits for the number^ – Smallest: 0^ – Largest: +

or -128^10

Signed-Magnitude

(OPTIONAL MATERIAL)

•^ Example of signed magnitude:^ „^^00000101 8

•^ Notice also that signed-magnitude gives plus and minuszero!^ –^^00000000

Two’s Complement

(OPTIONAL MATERIAL)

-^ Two’s Complement has no negative zero; for a onebyte integer:^ –^ 01111111 = 127,^ –^ 00000001 = 1,^ –^ 00000000 = 0,^ –^ 11111111 = - 1,^ –^ 10000000 = -128 •^ Range: [ -128, 127 ] 10

Inside the Bytes: Integers 8 • 2 -1 (255) is not a very big number, so computers generally usemultiple bytes to represent numbers: • Integers are defined in various sizes: 11

Other languagesuse differentnames! Size^ Precision

Matlab^

Java (^8) 1 byte 2 = 256 uint8, int*^

byte 2 byte^ 65,

uint16, int*

short 4 byte^ 4,294,967,

uint32, int*^

int (^64) 8 byte 2 --^ long

• Matlab uint’s are “unsigned integers” (no sign bit; positive values only) Note: A word is the basic size of the CPU registers and for Pentium chips it is 4 bytes or 32 bits; it is 8 bytes for the new Itanium and some unix chipsets. It was 2 bytes for early Intel chips; some new game consoles use 16 byte words.

Inside the Bytes: Real Numbers • Matlab’s real is called double and is an 8 byte IEEE 754 format. • Because only a finite number of bits are used for each part of thenumber, not all possible real numbers can be represented in acomputer using IEEE 754 13

(^0) Positive numbers smaller than 2

• ( positive underflow, Matlab “realmin”

) Positive numbers greater-52^1023 than (2-2) x 2^ ( positive overflow, Matlab“realmax” ) Negative numbers less than -(2-

-52^1023 ) x 2^ Negative numbers-1022greater than -2^ ( negative underflow ( negative overflow ) )

Zero^ (actually is a specialcombination of bits) RESULT : about 16 digits of precision in the range ±

±

Inside the Bytes

Others sources of error in computation: • Errors in the input data - measurement errors, errors^ introduced by the conversion of decimal data to binary,^ roundoff errors. • Roundoff errors during computation (as discussed) • Truncation errors - using approximate calculation is inevitable^ when computing quantities involving limits and other infinite^ processes on a computer^ – Never try to compare two floating point numbers for equality because all^ 16+ digits would have to match perfectly…^ – Use Matlab routi 14

nes whenever possible (they are optimized to avoid truncation and roundoff^ problems)

• Representing Numbers Learning OutlineObjectivesPrecision and accuracy Understand howNumbers on a computernumbers areExercisesstored in aSummarycomputer andwhat theimplications are Not in our textbook … pay attention … create your own knowledge base!
• Memory Allocation in MATLAB

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What it Means…

-^ In the previous slide, we see: •^ What is the size of the variable “

i”

-^ What does “class

” represent? Name^ • How many bytes are used to store the value? Size^

Bytes^ Class i^ 1x^

8 double^ array s^ 1x^

86 char^ array t^ 1x

1600

double^ array

SUMMARY

-^ Describe memory. List different kinds of memory. •^ What is IEEE 754? Describe how MATLAB representsnumbers. •^ Draw a number line and identify ranges where computerswill generate errors. •^ Describe three potential sources of errors incomputation. •^ Describe precision. Describe accuracy. •^ Describe how we can protect ourselves fromcomputation error. 19

Summary

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Action Items • Review the lecture • How will you use this informationin your current work? … futurework at GT? • Start Exploring Matlab! • Increment your record ofcommands learned • Come prepared to ask questionsabout MATLAB Lecture Outline • Numbers in the Computer • Precision and Accuracy