Simple Data Types and Numeric Representations in C

1. Chapter 7 Simple Data Types

2. Objectives • No programming language can predefine all the data types that a programmer may need. • C allows a programmer to create new data types. • Simple Data Type: a data type used to store a single value.

3. Representation and Conversion of Numeric Types • int vs. double: why having more than one numeric type is necessary? • Can the data type double be used for all numbers? • Operations involving integers are faster than those involving numbers of type double. • Operations with integers are always precise, whereas some loss of accuracy or round-off error may occur when dealing with double numbers.

4. Internals Formats • All data are represented in memory as binary strings. • Integers are represented by standard binary numbers. • For example, 13 = 01101 • Doubles are represented by two sections: mantissa and exponent. • real number = mantissa * 2exponent e.g. 4.0 = 0010 * 20001 = 2 * 21

5. Implementation-Specific Ranges for Positive Numeric Data • Run demo (Fig. 7.2)

6. Integer Types in C

7. Floating-Point Types in C

8. Numerical Inaccuracies • Certain fractions cannot be represented exactly in the decimal number system • e.g., 1/3= 0.33333…… • The representational error (round-off error) will depend on the number of binary numbers in the mantissa.

9. Example: Representational Error • for (trial = 0.0; trial != 10.0; trial = trial + 0.1) { …. } • Adding 0.1 one hundred times in not exactly 10.0. The above loop may fail to terminate on some computers. • trail < 10.0: the loop may execute 100 times on one computer and 101 times on another. • It is best to use integer variable for loop control whenever you can predict the exact number of times a loop body should be repeated.

10. Manipulating Very Large and Very Small Real Numbers • Cancellation Error: an error resulting from applying an arithmetic operation to operands of vastly different magnitudes; effect of smaller operand is lost. • e.g., 1000.0 + 0.0000001234 is equal to 1000.0 • Arithmetic Underflow: an error in which a very small computational result is represented as zero. • e.g., 0.00000001 * 10-1000000 is equal to 0 • Arithmetic Overflow: a computational result is too large. • e.g., 999999999 * 109999999 may become a negative value on some machines.

11. Automatic Conversion of Data Types • The data of one numeric type may be automatically converted to another numeric types. int k = 5, m = 4; n; double x = 1.5, y = 2.1, z;

12. Explicit Conversion of Data Types • C also provides an explicit type conversion operation called a cast. e.g., int n = 2, d = 4; double frac; frac = n / d;//frac = 0.0 frac = (double) n / (double) d; //frac = 0.5 frac = (double) (n / d); //frac = 0.0

13. Representation and Conversion of char • Character values can be compared by the equality operators == and !=, or by the relational operators <, <=, >, and >=.e.g., letter = ‘A’;if (letter < ‘Z’) … • Character values may also be compared, scanned, printed, and converted to type int.

14. ASCII (American Standard Code for Information Interchange) • Each character has its own unique numeric code. • A widely used standard is called American Standard Code for Information Interchange (ASCII). (See Appendix A in the textbook) • The printable characters have codes from 32 to 126, and others are the control characters. • For example, the digit characters from ‘0’ to ‘9’ have code values from 48 to 57 in ASCII. • The comparison of characters (e.g., ‘a’<‘c’) depends on the corresponding code values in ASCII.

15. Print Part of the Collating Sequence (Fig. 7.3)

16. Enumerated Types • Good solutions to many programming problems require new data types. • Enumerated type: a data type whose list of values is specified by the programmer in a type declaration.

17. Accumulating Weekday Hours Worked (Fig 7.5)

18. Iterative Approximations • Numerical Analysis: to develop algorithms for solving computational problems. • Finding solutions to sets of equations, • Performing operations on matrices, • Finding roots of equations, and • Performing mathematical integration. • Many real-world problems can be solved by finding roots of equations.

19. Six Roots for the Equation f(x) = 0 • Case Study: Bisection Method for Finding Roots

20. Function Parameters • The bisection routine would be far more useful if we could call it to find a root of any function. • Declaring a function parameter is accomplished by simply including a prototype of the function in the parameter list.

21. Case Study: Bisection Method for Finding Roots • First, tabulate function values to find an appropriate interval in which to search for a root.

22. Bisect this interval • Three possibilities that wrise when the Iinterval [xleft, xright] is Bisected

23. Epsilon • A fourth possibility is that the length of the interval is less than Epsilon. • Epsilon is a very small constant. • In this case, any point in the interval is an acceptable root approximation.

24. Finding a Function Root Using the Bisection MethodRun demo

25. Figure 7.11 Sample Run of Bisection Program with Trace Code Included

26. Homework #8 • Due: 2006/12/9 • 複數運算 • 以長度 2 的一維陣列 ( float [2] ) ，來表示複數，並實作出加減乘除、次方 ( 根號 ) 的運算，為強化乘除的計算，本題的乘除、次方 ( 根號 ) 運算需使用極座標系統 ( 複數的 乘法 、 除法 以及 指數 以及開方運算，在極坐標中會比在直角坐標中容易得多，請見reference 的複數部份說明 ) 。 • 作業要求 : 1. 使用者輸入二對 X,Y 代表二複數 a = (X 1 +iY 1 ), b = (X 2 +iY 2 ) 2. 計算出 a+b 3. 計算出 a/b 4. 使用者輸入欲計算 a 次方大小 (exp) 5. 計算出 a 的 exp 次方

27. Summary • Representation and Conversion of Numeric Types • Representation and Conversion of Type Char • Enumerated Types • Iterative Approximations