Chapter 3: Number Systems, Scalar Types, I/O · There are only two possible values for each digit:...

Created by Harry H. Cheng, 2009 McGraw-Hill, Inc. All rights reserved.

C for Engineers and Scientists: An Interpretive Approach

Chapter 3: Number Systems,

Scalar Types, I/O

Positional number systems:

• Base

• Digits

• Exponents / Powers



Decimal Numbers

The digits' weight increases by powers of 10. The weighted values for

each position is determined as follows:

For example,

A decimal number 4261 can be thought of as follows.

4 * 1000 + 2 * 100 + 6 * 10 + 1 * 1

= 4000 + 200 + 60 + 1

= 4261 (decimal)

104 103 102 101 100

10000 1000 100 10 1



Are there any non-positional number systems?



Binary

The base is 2 instead of 10

There are only two possible values for each digit: 0 and 1.

For example,

binary number decimal number

0 0

1 1

10 2

11 3

100 4

101 5

110 6

1100 1010 202



Binary

Basic skill: counting in binary (not in text)

“Poor man‟s” conversion: count in decimal and binary, side-by-side



Binary

The digits' weight increases by powers of 2. The weighted values for each position is

determined as follows:

27 26 25 24 23 22 21 20

128 64 32 16 8 4 2 1

For example,

binary 10 is decimal 2.

the binary value 1100 1010 represents the decimal value 202.

1 * 128 + 1 * 64 + 0 * 32 + 0 * 16 + 1 * 8 + 0 * 4 + 1 * 2 + 0 * 1

= 128 + 64 + 0 + 0 + 8 + 0 + 2 + 0

= 202 (decimal)



Binary practice

Convert the following binary integers to decimal:

11 =

101 =

101010 =



In reverse!

Converting from decimal to binary: repeated division by 2

Example: 13 =



Practice repeated division algorithm

Convert from decimal to binary:

6 =

41 =

127 =



Octal

The octal system is based on the binary system with a 3-bit boundary. The octal

number system uses base 8 includes 0 through 7. The weighted values for each

position is as follows:

83 82 81 80

512 64 8 1

1. Binary to Octal Conversion• Break the binary number into 3-bit sections from the least significant

bit (LSB) to the most significant bit (MSB).

• Convert the 3-bit binary number to its octal equivalent.

For example, the binary value 1 010 000 111 101 110 100 011 equals to

octal value (12075643)8.



2. Octal to Binary Conversion

• Convert the octal number to its 3-bit binary equivalent.

• Combine all the 3-bit sections.

For example, the octal value 45761023 equals to binary value

100 101 111 110 001 000 010 011.

3. Octal to Decimal Conversion

To convert octal number to decimal number, multiply the value in each position

by its octal weight and add each value together. For example, the octal value

(167)8 represents decimal value 119.

1*64 + 6*8 + 7*1 = 119



Hexadecimal, a.k.a. hex

Similar to octal, the hexadecimal system is also based on the binary system but

using 4-bit boundary. The hexadecimal number system uses base 16 including

the digits 0 through 9 and the letters A, B, C, D, E, and F. The letters A through F

represent the decimal numbers 10 through 15. For the decimal values from 0 to 15,

the corresponding hexadecimal values are listed below.

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

F E D C B A 9 8 7 6 5 4 3 2 1 0

Decimal

Hexadecimal



The weighted values for each position is as follows:

163 162 161 160

4096 256 16 1

The conversion between binary value and hexadecimal value is similar to octal

number,but using four-bit sections.

The hexadecimal value 20A represents decimal value 522.

2*256 + 0*16 + 10*1 = 522



Following table provides all the information you need to convert from one type

number into any other type number for the decimal values from 0 to16.

Binary Octal Decimal Hex Binary Octal Decimal Hex

0000 00 00 00 1001 11 09 09

0001 01 01 01 1010 12 10 A

0010 02 02 02 1011 13 11 B

0011 03 03 03 1100 14 12 C

0100 04 04 04 1101 15 13 D

0101 05 05 05 1110 16 14 E

0110 06 06 06 1111 17 15 F

0111 07 07 07 10000 20 16 10

1000 10 08 08

EoL



How about negative numbers?

First idea: sign-magnitude (not in text)

Good for pencil-and-paper, but not very good

for computers!



Binary Two’s Complement

The left-most bit is the sign bit. If it is 1, then the number is negative.

Otherwise, it is positive.

A negative value, represent it in binary two‟s complement form as follows:

1. write the number in its absolute value

2. complement the binary number bit by bit

3. add 1

Example, represent –2 in binary two‟s complement on 16 bits:

• Binary value of 2: 0b 0000 0000 0000 0010

• Binary complement of 2: 0b 1111 1111 1111 1101

• Plus 1: +1

• Two‟s complement representation of -2: 0b 1111 1111 1111 1110

Some C compilers accept this syntax, but

it‟s not standard!



Give binary two‟s complement form of a negative number, find its absolute value (a.k.a. magnitude)

1. Complement the binary number

2. Add 1

Example: find the decimal value of 0b1111 1111 1111 1110 in binary

two‟s complement form with 16 bits.

• Binary two‟s complement 0b 1111 1111 1111 1110

• Binary complement 0b 0000 0000 0000 0001

• Plus 1 +1

• Absolute value: 0b 0000 0000 0000 0010

• Absolute value in decimal: 210

• Negative value in decimal: -2



Subtraction of a value in the computer can be implemented as addition of its negative (in two‟s complement!)

Example: 2 – 2 can be performed as 2 + (-2) as follows:

0b 0000 0000 0000 0010 binary representation of 2

0b 1111 1111 1111 1110 two‟s comp. representation of -2

0b 0000 0000 0000 0000 Result (is it correct?)

Practice: calculate

• 5 – 4 =

• 4 – 5 =



We‟ve covered section 3.1

In notebook for next time

(do not turn in for grading!)

• End-of-chapter questions 1 and 2 a, b, c, d

– For 2, use only 8 bits, not 32

• Read and take notes: sub-section 3.1.3

Binary Fractions



1-minute quiz

What is the decimal value of this two‟s

complement numbers:

• 0111 1111

• 1001 0101



Extra-credit question

Is two‟s complement a positional number

system?

Explain, either way!



The following sections have

been covered in the lab:



3.2 Character Set

The character set in C includes the following members:

– the 26 uppercase letters of the Latin alphabet

A B C D E F G H I J K L M

N O P Q R S T U V W X Y Z

– the 26 lowercase letters of the Latin alphabet

a b c d e f g h i j k l m

n o p q r s t u v w x y z

– the 10 decimal digits

0 1 2 3 4 5 6 7 8 9

– the following 31 graphic characters

! " # % & ' ( ) * + , - . / :

; < = > ? [ \ ] ^ _ { | } ~ $ `



3.3 Comments

Comments of a C program can be enclosed within a pair of delimiters /* and */.

The symbol // will comment out a subsequent text terminated at the end of

a line. For example,

/* This is a comment */

/* This is a comment

across multiple lines */

printf(”Hello, world\n”); // This is a comment terminated at the end of line



– An identifier for a variable shall consist of lowercase and uppercase letters,

underscore _, and digits. It shall not start with digits.

– A variable has to be declared before it can be used inside a program. The following format can be used to declare a variable of simple type.

dataType varName;

where dataType is one of valid data types and varName is an identifier.

– In C99, variables can be declared almost anywhere even after executable

statements.

int main() {

int i;

i = 90;

int j;

j = 20;

return 0;

}

3.4 Declaration



• Integer Data Types

Integer is a basic data type and can be represented by one of the

following data types.

char long long

signed char signed long long

unsigned char unsigned long long

short

signed short

unsigned short

int

signed int

unsigned int

long

signed long

unsigned long

Data Types and Constants



> int i

> sizeof(int)

4

> sizeof(i)

4

> sizeof(2*i)

4

> sizeof(long long)

8

The sizeof operator gives the size of types or expression in bytes.

One byte equals 8 bits.



• Int Data Representation

– An int data is a signed integer. An int number is a whole number that can be negative, positive, or zero.

– An int data uses 4 bytes for storage with 1 bit for the sign.

– The int ranges from INT_MIN to INT_MAX, which are

–2147483648(–231) and 2147483647(231–1), respectively.

– The unsigned int ranges from 0 to UINT_MAX, which is equal to 4294967295(232–1).

INT_MAX 0b01111111 11111111 11111111 11111111

INT_MIN 0b10000000 00000000 00000000 00000000

UINT_MAX 0b11111111 11111111 11111111 11111111



#include <stdio.h>

#include <limits.h>

Int main() {

printf(”INT_MAX = %d\n”, INT_MAX);

printf(”INT_MIN = %d\n”, INT_MIN);

printf(”UINT_MAX = %d\n”, UINT_MAX);

return 0;

}

Output:

INT_MAX = 2147483647

INT_MIN = -2147483648

UINT_MAX = 4294967295

Program: limits.c



• Short Data Representation

– A short data uses 2 bytes for storage.

– The macros for minimum and maximum values of signed short are SHRT_MIN and SHRT_MAX defined in header file limits.h. SHRT_MIN is equal to -32768(-215) and SHRT_MAX is equal to 32767(215-1).

– The macro USHRT_MAX, defined in the header file limits.h, specifies the maximum value of unsigned short. It is equal to 65535(216-1).



• Long Long Data Representation

– Data of long long integral type contains 8 bytes. They have the

similar representation as the data type of int.

– The long long int ranges from LLONG_MIN to LLONG_MAX,

which are -9223372036854775808LL (-263) and

9223372036854775807LL (263–1), respectively.

– The long long int ranges from 0 to ULLONG_MAX, which is 264–1.



• Integer Constants

– An integer can be specified in decimal, binary, octal, or hexadecimal.

– A leading 0(zero) on an integer constant indicates an octal integer.

– A leading 0x or 0X indicates a hexadecimal integer.

– A leading 0b or 0B indicates a binary integer (in Ch only). •

Example:

30 (decimal) = 036 (octal)

= 0X1e or 0x1E (hexadecimal)

= 0b11110 or 0B11110 (binary)



Example:

C:/Ch> int i

C:/Ch> i = 036

30

C:/Ch> i = 0x1e

30

C:/Ch> i = 0b11110

30

C:/Ch> printf(”i = %d\n”, i)

i = 30

C:/Ch> printf(”i = 0%o\n”, i)

i = 036

C:/Ch> printf(”i = 0x%x\n”, i)

i = 0x1e

C:/Ch> printf(”i = 0b%b\n”, i)

i = 0b11110



End of sections have covered in

the lab



3.7 Boolean Type (C99 only!)

The keyword bool can be used to declare variables with boolean data type.

For example, the following statement

bool b;

declares a boolean variable b.

A boolean variable only has only two possible values: 1 and 0. Value 1 stands for true and value 0 stands for false.

true and false are defined in header file stdbool.h, which need to be included!



3.8 Characters– The char data are used to store characters such as letters and punctuation.

An array of char can be used to store a string.

– A character is stored as an integer according to a certain numerical code

such as the ASCII code that ranges from 0 to 127, which only requires 7 bits to

represent it.

– Typically, a char constant or variable occupies 1-byte (8 bits) of unit

memory.

Memory Address Binary value Character

20000 0 1 0 0 1 0 0 0 H

20001 0 1 1 0 0 1 0 1 e

20002 0 1 1 0 1 1 0 0 l

20003 0 1 1 0 1 1 0 0 l

20004 0 1 1 0 1 1 1 1 o

20005 0 0 1 0 0 0 0 1 !



– The macros for minimum and maximum values of signed char are

CHAR_MIN and CHAR_MAX defined in header file limits.h.

CHAR_MIN is equal to -128(-27) and CHAR_MAX is equal to

127(27-1).

– The macro UCHAR_MAX, defined in the header file limits.h,

specifies the maximum value of unsigned short. It is equal to

255(28-1).

• CHAR_MAX = 127 0b0111 1111

• CHAR_MIN = -128 0b1000 0000

• UCHAR_MAX = 255 0b1111 1111



• Character Constants

A character constant, stored as an integer, can be written as one character

within a pair of single quotes like „x‟. A character constant can be

assigned to the variable of type char. For example,

> char c = ’x’

> c

x



• Escape CharactersEscape Code Description

\a (alert) Produces an audible or visible alert. The active position shall

not be changed.

\b (backspace) Moves the active position to the previous position on the

current line.

\f (form feed) Moves the active position to the initial position at the start

of the next logical page.

\n (new line) Moves the active position to the initial position of the next

line.

\r (carriage return) Moves the active position to the initial position of the

current line.

\t (horizontal tab) Moves the active position to the next horizontal

tabulation position on the current line.

\v (vertical tab) Moves the active position to the initial position of the next

vertical tabulation position.

\\ (backslash) Produces a backslash character \.

\’ (single quote) Produces a single quote character „.

\” (double quote) Produces a double quote character “.

\? (question mark) Produces a question mark character ?.



In notebook for next time

(do not turn in for grading!)

Read and take notes: numeric escape codes,

first two paragraphs on p.57



Strings of characters

• A string of characters is a sequence of zero or more

characters enclosed in double quotes, such as “xyz”.

– Strings can be empty!

• Strings in C are represented as arrays of char

– The last character must be ‘\0’ (ASCII zero)

• Using an array of characters to define a string variable.

char str1[6] = ”abcde” // The last one is '\0'

char str2[] = ”This is a string.”



Strings of characters

• Printing a string variable: use format specifier %s

char str1[6] = "abcde" // The last one is '\0'

...

printf("The string is %s \n", str1);

More about strings in Ch.12



3.9 Floating-Point Types

float, double, long double

The float data type uses 32 bits (4 bytes) for its storage.

• The minimum and maximum values of float data type are defined as

macros FLT_MIN and FLT_MAX, respectively, in header file float.h

The double data type uses 64 bits(8 bytes) as its storage.

• DBL_MIN and DBL_MAX, also in float.h

The long double should have at least as many bits as double

• LDBL_MIN and LDBL_MAX, also in float.h

Let‟s quickly find out!



Not in text

• In C99, %lf and %f are equivalent

– both can be used interchangeably for float

and double!

• For long double, %Lf should be used

– As long as long double and double are

equivalent, %lf will work too, however...



Below is a list of metanumbers for floating-point numbers and their

mathematical equivalent.

Meta-numbers

Metanumbers Mathematical Representation

-0.0 0-

+0.0 0+

-Inf -

+Inf +

NaN Not-a-Number (Invalid value)



> 0.0/0.0

nan

> 1.0/0.0

inf

> -1.0/0.0

-inf

> sqrt(4)

2.000

> sqrt(-4)

nan

Examples for NaN and Inf



Sample Problem:

The same as in Ch.2!

Two changes in the solution …



/* File: acceldef.c */

#include <stdio.h>

/* gravitational acceleration */

#define M_G 9.81

int main() {

/* declare variables */

double a; /* acceleration */

double mu; /* friction coefficient */

double m; /* mass */

double p; /* external force */

/* Initialize variables */

mu = 0.2;

m = 5.0;

p = 20.0;

/* Calculate the acceleration */

a = (p-mu*m*M_G)/m;

/* display output */

printf("Acceleration a = %f (m/s^2)\n", a);

return 0;

}



3.10 Complex Types

Read over lightly for now, more in the lab



We‟ve covered the following sections of the text:

--end of 3.8

--3.9

--skipped 3.10

Solve in notebook for next time (do not turn

in for grading!):

• End-of-chapter question 5 (a through l)



int i, *p;

i = 10;

p = &i; // p contains the address of i

3.11 Pointer Data Type

– A pointer is defined as a variable which contains the address of another variable or dynamically allocated memory.

– With an asterisk „*‟ in front of the variable names, the variables of pointer type can be declared similar to variables of other data types. The unary operator „&‟ gives the “address of a variable”.

Example:

Diagram not in text: “Two worlds”



p = &i; // p now stores the address of i

If the address of i is 0x00E81E20, the value of p is 0x00E81E20 as shown below.



How to declare a String using a Pointer

char *s = ”This is a string”;



3.12 typedef

For convenience, the C programmer can define her/his own data

types, e.g.

typedef unsigned int size_t; //used by sizeof

typedef char[5] DNA_t;

The new data types can be used just like the built-in ones, e.g.

DNA_t seq1, seq2;

DNA_t seq3 = "GATA";

DNA_t seq4 = {'G', 'A', 'T', 'A', '\0',};



3.13 Determining the programming data

model of a computer

Use sizeof, to find the number of Bytes used

internally for each data type (as shown in week 2‟s lab)



3.14 Initialization

– The declaration of a variable may be accompanied by an initializer that specifies the value of the variable should have at the beginning of its lifetime.

– The initializer for a scalar must be a single expression, such as

int i = 6/3;

double d = 12.345;

char c = ’a’;

– An array of character type may be initialized by a character string literal (constant).

char str1[20] = ”this is a string.”;



3.15 Introduction to Formatted

Input and Output

• printf()Precisely formatted output is accomplished using the output function

printf. The printf function has following form

printf( format-control-string, arguments );

– Format-control-string: Using specifications to describe output

format. Each specification begins with a percent sign (%), ends

with conversion specifier and is enclosed in quotation marks.

– arguments: correspond to each conversion specification in format-

control-string.



Table 3.5 Format-control-strings for printf().

Argument Type Format-Control-String

binary number “%b” in Ch only

char “%c”

signed char, short, int “%d”

unsigned char, short, int “%u”

long long “%lld”

unsigned long long “%ulld”

octal number “%o”

hexadecimal number “%x”

float “%f”

double “%f” or “%lf”

string “%s”

pointer “%p”



Printing multiple numerical values in a

single statement

printf(”integer is %d, floating-point number is %f”,

10, 12.34);

Output is:

integer is 10, floating-point number is 12.340000

Use multiple format specifiers.

Each format specifier corresponds to an argument.



Precision of Floating-Point Numbers

Specifies the number of digits to be printed after the decimal point .

The precision typically takes the form of a period (.) followed by

an decimal integer.

For example, the format “%.2f” specifies 2 digits after the decimal point.

printf(”%.2f”, 12.1234);

12.12

printf(”%6.2f”, 12.5678);

12.57

printf(”%.20f”, 0.2);

0.20000000000000001110

The fractional part after the specified precision number is rounded up.

A floating-point number may not be represented exactly.



Precision of Floating-Point Numbers

If represented internally on 4 Bytes, floats have about 7“real” digits of precision .

If represented internally on 8 Bytes, doubles have about 15“real” digits of precision .



Field width of numbers

The field width is the size of a field in which data are printed.

An integer width is inserted between % and the conversion specifier

to indicate the field width

For example, “%6d” specifies the field width of 6 for an integer,

“8.4” specifies a field width of 8 with 4 digits after the decimal point.

printf(”%6d”, 12);

12

printf(”%8.4f”, 5.12345);

5.1234



Function scanf()Precise formatting input is accomplished using the input function

scanf. The scanf function has following form

scanf( format-control-string, arguments );

– Format-control-string: Using specifications to describe input

format. Each specification begins with a percent sign(%), ends

with conversion specifier and is enclosed in quotation marks. The

format-control-string is similar to format-control-string discussed

in printf function.

– arguments: pointers to variables in which the input value will be stored.



Table below lists the format-control-string of different argument types for

function scanf().

Argument Type Format-Control-String

binary number “%b” in Ch only

char “%c”

short “%hd”

unsigned short “%uhd”

int “%d”

unsigned int “%u”

long long “%lld”

unsigned long long “%ulld”

float “%f”

double “%lf”

string “%s”

pointer “%p”



/* File: scanfc.c for input and output example */

#include <stdio.h>

int main() {

int num;

double d;

printf("Please input one integer and one floating-point number\n);

scanf("%d%lf",&num, &d);printf("Your input values are %d and %f\n“, num, d);

return 0;

}



Problem Statement:

For the system shown in Figure1(a), the external force p is expressed as a

function of time t,

p(t) = 4(t-3)+20 when t >= 0

write a program to calculate the acceleration a according to the input values

for m, , and t.



/* File: accelio.c */

#include <stdio.h>

#define M_G 9.81

int main() {

double a, mu, m, p, t;

printf("***** Acceleration Calculator *****\n\n");

printf("Please enter value for mass in kilogram\n");

scanf("%lf", &m);

printf("mass is %lf (kg)====\n\n", m);

printf("Please enter value for friction coefficient\n");

scanf("%lf", &mu);

printf("friction coefficient is %lf\n\n", mu);

printf("Please enter value for time in second\n");

scanf("%lf", &t);

printf("time is %lf (s)\n\n", t);

p = 4*(t-3)+20;

a = (p-mu*m*M_G)/m;

printf("Acceleration a = %f (m/s^2)\n", a);

return 0;

}

Program:

Using formatted

input function

scanf.



Execution:

Please enter value for mass in kilogram

5

mass is 5.000000 (kg)

Please enter value for friction coefficient

0.2

friction coefficient is 0.200000

Please enter value for time in second

2

time is 2.000000 (s)

Acceleration a = 1.238000 (m/s^2)



SKIP 3.15.3 and 3.15.4

Redirecting

SKIP 3.16

Homework for Ch.3:

Problems 12, 17, 18, 19

Chapter 3: Number Systems, Scalar Types, I/O · There are only two possible values for each digit:...

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