1. Copyright © 2003-2012 Curt Hill Numerics Representation and Errors

2. Copyright © 2003-2012 Curt Hill Integers Typically stored as a series of bits Two’s complement binary Each bit represents a zero or one –Similar to a light switch –Only two states Binary is base 2

3. Copyright © 2003-2012 Curt Hill Decimal In base 10 the 10 occurs twice The number of digits The power to which a place is raised In decimal each digit may be one of 10 possibilities, 0 - 9 Each digit is multiplied by 10 raised to some power

4. Copyright © 2003-2012 Curt Hill A Decimal Number 4,809 9  10 0 = 9 0  10 1 = 0 8  10 2 = 800 4  10 3 = 4000

5. Copyright © 2003-2012 Curt Hill Binary is just the same In base 2 the 2 occurs twice The number of digits The power to which a place is raised In binary each digit may be one of two possibilities, 0 or 1 Each digit is multiplied by two raised to some power

6. Copyright © 2003-2012 Curt Hill A Binary Number 101101 1  2 3 = 8 0  2 4 = 0 1  2 0 = 1 0  2 1 = 0 1  2 2 = 4 1  2 5 = 32 45

7. Copyright © 2003-2012 Curt Hill Binary and other bases We use it because it is the only base easy to implement in hardware We almost never display it –Binary is quite bulky We often use other bases for certain displays –Octal, base 8 –Hexadecimal, base 16 –Converting these to and from binary is quite simple –Conversion to decimal is much harder

8. Copyright © 2003-2012 Curt Hill Integers Come in several sizes: 16 bit –-32768 to 32767 Most common is 32 bit –Largest integer is (2 31 )-1 –-2,147,483,648 to 2,147,483,647 –There is always one more negative than positive –Zero is taken from positives Large integers may be 64 bit –Largest is 9,223,372,036,854,780,000

9. Copyright © 2003-2012 Curt Hill Integer Errors Integers are only subject to one error Overflow Add 1 to largest positive integer and result is smallest negative integer Subtract 1 from largest (in absolute value) negative and result is largest positive Similar to clock arithmetic, except there are positives and negatives

10. Another Look at Overflow Copyright © 2003-2012 Curt Hill

11. Real Numbers Real numbers (float, double) are actually two numbers Similar to scientific notation: 2.543  10 8 The 2.543 is called the mantissa The 8 is called the exponent The 10 is the base –Not represented, since always a 2 or some other constant

12. Copyright © 2003-2012 Curt Hill Real Number Errors Real numbers are subject to a variety of errors –Some quite subtle In order to demonstrate these we will model a computation using three digit arithmetic –Two digit of mantissa –One digit of exponent –Base of 10

13. Copyright © 2003-2012 Curt Hill 3 Digit Arithmetic Game A perfect circle is found in nature We want to know its area The formula is a = πr 2 The radius (r) is: 11.4962790000000000 Unfortunately, we measured this radius to be: 11.495 Which becomes 11E0 on our machine

14. Copyright © 2003-2012 Curt Hill Computing… Take the 11 E 0 and square 11 E 0 121 E 0 The121E0 has too many digits, adjust: 12 E 1 Multiply times π 12 E 1 31 E -1 372 E 0 Adjust again 37 E 1

15. Copyright © 2003-2012 Curt Hill What happened? This exercise demonstrates several important sources of error It is never the case that if its close its accurate –Which is what most believe The three demonstrated errors are –Observational –Representational –Computational

16. Copyright © 2003-2012 Curt Hill The Results The real value should be 415.2060805 We ended up with 370 About 12% error One half of a significant digit

17. Copyright © 2003-2012 Curt Hill Observational Error We can never measure anything completely accurately –To infinite precision The introduced error in this case is 0.02% This is not a problem for integers The Olympics have a problem with this, in measuring to hundredths of a second

18. Olympic Observational Error LA 1984 Women’s 100 M. Freestyle –Carrie Steinseifer and Nancy Hogshead won in 55.92 –Both received gold – no silver awarded Sydney 2000 Men’s 50 M. Freestyle –Anthony Ervin and Gary Hall Jr. 21.98 London 2012 two events, four silvers –Men’s 200 M. Freestyle Taehwan Park and Yang Sun –Men’s 100 M. Butterfly Chad le Clos and Evgeny Korotyshkin Copyright © 2003-2012 Curt Hill

19. Representational Error The measurement cannot be represented accurately on this machine 11.495 to 11 causes an error amounting to 8.4% in addition to previous Truncating π will cause an additional 1.3% This will be minimized but not eliminated with many more than 2 digits of precision This is the only integer arithmetic error –Overflow There is more of this than you might think

20. Copyright © 2003-2012 Curt Hill Computational Error A digit was lost twice –121 to 120 –372 to 370 We may round or truncate this but we still lose information This will account for another 1.3% error in addition to the previous Another problem that integer arithmetic does not have

21. Copyright © 2003-2012 Curt Hill Representational Error Revisited Recall from grade school days there are three types of decimal numbers: –Rational Terminating Repeating –Irrational π could not be represented since it has infinite digits For computational purposes, irrationals are no worse than repeating decimals

22. Copyright © 2003-2012 Curt Hill Repeating decimals Many fractions convert to repeating decimals 1/3 repeats with one digit 0.33333… 1/11 has two digits 0.09090909… 1/7 repeats with 6:.142857 142857… As does 1/13:.076923 076923…

23. Copyright © 2003-2012 Curt Hill Terminating decimals Terminating decimals are what we want They end in an infinite number of zeros 1/2 is 0.50 1/4 is 0.25 1/5 is 0.2 1/8 is 0.125 1/10 is 0.1 No representational errors with these Or is there?

24. Copyright © 2003-2012 Curt Hill Terminating and Repeating Why does 1/5 terminate and 1/3 repeat? It all has to do with the prime factors of the denominator –If the prime factors only contain 2s and 5s the number terminates, otherwise it repeats –These are the prime factors of 10

25. Copyright © 2003-2012 Curt Hill Terminator A number terminates if its denominator has only the prime factors of the base 1/7 terminates in base 7 and base 14 but not in base 10 or base 2 What base does a computer use? Binary – base 2 We have representational error for any fraction whose denominator has something other than two for a prime factor

26. Copyright © 2003-2012 Curt Hill Example: Money Cents are always fractions, with a denominator of 100 –100 has prime factors of 2 and 5 Only the cent amounts of.00,.25,.50 and.75 reduce to having a denominator of 2 or 4 Thus 96% of all monetary amounts have representational error

27. Copyright © 2003-2012 Curt Hill Computational Error Revisited Certain operations produce more or less error The subtraction of two values that are close in value has a deleterious effect The most significant and most accurate digits cancel each other This leaves only the most error prone digits left

28. Copyright © 2003-2012 Curt Hill Multiplication and Division Generally multiplication and division are better at preserving significant digits than addition and subtraction –The result has the precision of the least precise value The result of multiplication has the sum of digits –Most of these are thrown away Errors are cumulative Once we have error digits we cannot get rid of them

29. Copyright © 2003-2012 Curt Hill Examples 0.20348 -0.20321 0.00027 Most accurate digits are lost 0.95  0.83 0.0285 + 0.760 0.7885 The most accurate digits contribute most