Insertion sort is a simple sorting algorithm, a comparison sort in which the sorted array (or list) is built one entry at a time. It is much less efficient on large lists than more advanced algorithms such as quicksort, heapsort, or merge sort. However, insertion sort provides several advantages:
Simple implementation
Efficient for (quite) small data sets
Adaptive, i.e. efficient for data sets that are already substantially sorted: the time complexity is O(n + d), where d is the number of inversions
More efficient in practice than most other simple quadratic (i.e. O(n2)) algorithms such as selection sort or bubble sort: the average running time is n2/4,[citation needed] and the running time is linear in the best case
Stable, i.e. does not change the relative order of elements with equal keys
In-place, i.e. only requires a constant amount O(1) of additional memory space
Online, i.e. can sort a list as it receives it.
Sunday, April 4, 2010
SELECTION SORT
Selection sort is a sorting algorithm, specifically an in-place comparison sort. It has O(n2) complexity, making it inefficient on large lists, and generally performs worse than the similar insertion sort. Selection sort is noted for its simplicity, and also has performance advantages over more complicated algorithms in certain situations.
Greeting.h: interface for the Greeting class.
//////////////////////////////////////////////////////////////////////
#if !defined(AFX_GREETING_H__34246B3E_5382_401E_9265_7B6DFEFA39AF__INCLUDED_)
#define AFX_GREETING_H__34246B3E_5382_401E_9265_7B6DFEFA39AF__INCLUDED_
#if _MSC_VER > 1000#pragma once#endif // _MSC_VER > 1000
class Greeting {public:void PrintMessage();Greeting();virtual ~Greeting();
private:char* message;};
#endif // !defined(AFX_GREETING_H__34246B3E_5382_401E_9265_7B6DFEFA39AF__INCLUDED_)
Select Greeting.cpp file and edit it. Enter a line
message="Hello World!\n";
inside the Greeting class constructor
Greeting::Greeting(){message = "Hello World\n";}
Enter line of code inside the PrintMessage function
printf(message);
void Greeting::PrintMessage(){ printf(message);}
Enter a line #include <> under existing #include lines
#include "stdafx.h"#include "Greeting.h"#include
Your Greeting.cpp file should be like this:
Greeting.cpp: implementation of the Greeting class.
//////////////////////////////////////////////////////////////////////
#include "stdafx.h"#include "Greeting.h"#include
//////////////////////////////////////////////////////////////////////// Construction/Destruction//////////////////////////////////////////////////////////////////////
Greeting::Greeting(){ message = "Hello World\n";}
Greeting::~Greeting(){}
void Greeting::PrintMessage(){ printf(message);}
Select HelloWorld.cpp file and enter the following two line of code in main function.
Greeting gr;gr.PrintMessage();
Now your HelloWorld.cpp file must be like this:
// HelloWorld.cpp : Defines the entry point for the console application.
#include "stdafx.h"#include "Greeting.h"
int main(int argc, char* argv[]){Greeting gr;gr.PrintMessage();return 0;}
Selection sort is a sorting algorithm, specifically an in-place comparison sort. It has O(n2) complexity, making it inefficient on large lists, and generally performs worse than the similar insertion sort. Selection sort is noted for its simplicity, and also has performance advantages over more complicated algorithms in certain situations.
Greeting.h: interface for the Greeting class.
//////////////////////////////////////////////////////////////////////
#if !defined(AFX_GREETING_H__34246B3E_5382_401E_9265_7B6DFEFA39AF__INCLUDED_)
#define AFX_GREETING_H__34246B3E_5382_401E_9265_7B6DFEFA39AF__INCLUDED_
#if _MSC_VER > 1000#pragma once#endif // _MSC_VER > 1000
class Greeting {public:void PrintMessage();Greeting();virtual ~Greeting();
private:char* message;};
#endif // !defined(AFX_GREETING_H__34246B3E_5382_401E_9265_7B6DFEFA39AF__INCLUDED_)
Select Greeting.cpp file and edit it. Enter a line
message="Hello World!\n";
inside the Greeting class constructor
Greeting::Greeting(){message = "Hello World\n";}
Enter line of code inside the PrintMessage function
printf(message);
void Greeting::PrintMessage(){ printf(message);}
Enter a line #include <> under existing #include lines
#include "stdafx.h"#include "Greeting.h"#include
Your Greeting.cpp file should be like this:
Greeting.cpp: implementation of the Greeting class.
//////////////////////////////////////////////////////////////////////
#include "stdafx.h"#include "Greeting.h"#include
//////////////////////////////////////////////////////////////////////// Construction/Destruction//////////////////////////////////////////////////////////////////////
Greeting::Greeting(){ message = "Hello World\n";}
Greeting::~Greeting(){}
void Greeting::PrintMessage(){ printf(message);}
Select HelloWorld.cpp file and enter the following two line of code in main function.
Greeting gr;gr.PrintMessage();
Now your HelloWorld.cpp file must be like this:
// HelloWorld.cpp : Defines the entry point for the console application.
#include "stdafx.h"#include "Greeting.h"
int main(int argc, char* argv[]){Greeting gr;gr.PrintMessage();return 0;}
bubble sort
Bubble sort
Bubble sort is a simple sorting algorithm. It works by repeatedly stepping through the list to be sorted, comparing each pair of adjacent items and swapping them if they are in the wrong order. The pass through the list is repeated until no swaps are needed, which indicates that the list is sorted. The algorithm gets its name from the way smaller elements "bubble" to the top of the list. Because it only uses comparisons to operate on elements, it is a comparison sort.
// Example Program in C++ // to sort an array
// using bubble sort
#include
void main(void)
{
int temp, i, j;
int ar[10];
cout<<"enter elements of array:\n";
for(i=0; i<10; i++)
cin>>ar[i];
// sorting is done here
for(i=0;i<10;i++)
for(j=0; j<(10-1); j++)
if (ar[j]>ar[j+1])
{
temp=ar[j];
ar[j]=ar[j+1];
ar[j+1]=temp;
}
// till here
for(i=0; i<10; i++)
cout<
cout<
Bubble sort is a simple sorting algorithm. It works by repeatedly stepping through the list to be sorted, comparing each pair of adjacent items and swapping them if they are in the wrong order. The pass through the list is repeated until no swaps are needed, which indicates that the list is sorted. The algorithm gets its name from the way smaller elements "bubble" to the top of the list. Because it only uses comparisons to operate on elements, it is a comparison sort.
// Example Program in C++ // to sort an array
// using bubble sort
#include
void main(void)
{
int temp, i, j;
int ar[10];
cout<<"enter elements of array:\n";
for(i=0; i<10; i++)
cin>>ar[i];
// sorting is done here
for(i=0;i<10;i++)
for(j=0; j<(10-1); j++)
if (ar[j]>ar[j+1])
{
temp=ar[j];
ar[j]=ar[j+1];
ar[j+1]=temp;
}
// till here
for(i=0; i<10; i++)
cout<
cout<
CS-10
sorting
In computer science and mathematics, a sorting algorithm is an algorithm that puts elements of a list in a certain order. The most-used orders are numerical order and lexicographical order. Efficient sorting is important to optimizing the use of other algorithms (such as search and merge algorithms) that require sorted lists to work correctly; it is also often useful for canonicalizing data and for producing human-readable output. More formally, the output must satisfy two conditions:
The output is in nondecreasing order (each element is no smaller than the previous element according to the desired total order);
The output is a permutation, or reordering, of the input.
Since the dawn of computing, the sorting problem has attracted a great deal of research, perhaps due to the complexity of solving it efficiently despite its simple, familiar statement. For example, bubble sort was analyzed as early as 1956.[1] Although many consider it a solved problem, useful new sorting algorithms are still being invented (for example, library sort was first published in 2004). Sorting algorithms are prevalent in introductory computer science classes, where the abundance of algorithms for the problem provides a gentle introduction to a variety of core algorithm concepts, such as big O notation, divide and conquer algorithms, data structures, randomized algorithms, best, worst and average case analysis, time-space tradeoffs, and lower bounds.
In computer science and mathematics, a sorting algorithm is an algorithm that puts elements of a list in a certain order. The most-used orders are numerical order and lexicographical order. Efficient sorting is important to optimizing the use of other algorithms (such as search and merge algorithms) that require sorted lists to work correctly; it is also often useful for canonicalizing data and for producing human-readable output. More formally, the output must satisfy two conditions:
The output is in nondecreasing order (each element is no smaller than the previous element according to the desired total order);
The output is a permutation, or reordering, of the input.
Since the dawn of computing, the sorting problem has attracted a great deal of research, perhaps due to the complexity of solving it efficiently despite its simple, familiar statement. For example, bubble sort was analyzed as early as 1956.[1] Although many consider it a solved problem, useful new sorting algorithms are still being invented (for example, library sort was first published in 2004). Sorting algorithms are prevalent in introductory computer science classes, where the abundance of algorithms for the problem provides a gentle introduction to a variety of core algorithm concepts, such as big O notation, divide and conquer algorithms, data structures, randomized algorithms, best, worst and average case analysis, time-space tradeoffs, and lower bounds.
Sunday, September 20, 2009
MY AUTHYOBIOGRAPHY
MY AUTHOBIOGRAPHY
NAME: ROBERT L. BOLABON
AGE: 21 YRS. OLD
SEX: MALE
WIEGHT: 47 KILOS
HIEGHT: 5’6
BIRTHDAY: AUGOST 19, 1988
ADDRESS: UPPER STA CRUZ BRGY. ACACIA BUHANGIN DAVAO CITY
MOBILE NO. 09103098683
EDUCATIONAL BACKGRAUND
ELEMENTARY: STO NIŇO ELEMENTARY SCHOOL
YEAR GRADUATED: 2001
HIGHSCHOOL: F.BUSTAMANTE NATIONAL HIGH SCHOOL
YEAR GRADUATED: 2005
COLLEGE: PHILIPPINE COLLEGE OF TECHNOLOGY
FAMILY BACKGRAUND
MOTHERS NAME: CARLOSA BOLABON
OCCUPATION: FACTORY WORKER
FATHERS NAME: IRENIO BOLABON SR.
OCCUPATION: FARMER
THERE ADDRESS: UPPER STA.CRUZ BRGY. ACACIA BUHANGIN DAVAO CITY
NAME: ROBERT L. BOLABON
AGE: 21 YRS. OLD
SEX: MALE
WIEGHT: 47 KILOS
HIEGHT: 5’6
BIRTHDAY: AUGOST 19, 1988
ADDRESS: UPPER STA CRUZ BRGY. ACACIA BUHANGIN DAVAO CITY
MOBILE NO. 09103098683
EDUCATIONAL BACKGRAUND
ELEMENTARY: STO NIŇO ELEMENTARY SCHOOL
YEAR GRADUATED: 2001
HIGHSCHOOL: F.BUSTAMANTE NATIONAL HIGH SCHOOL
YEAR GRADUATED: 2005
COLLEGE: PHILIPPINE COLLEGE OF TECHNOLOGY
FAMILY BACKGRAUND
MOTHERS NAME: CARLOSA BOLABON
OCCUPATION: FACTORY WORKER
FATHERS NAME: IRENIO BOLABON SR.
OCCUPATION: FARMER
THERE ADDRESS: UPPER STA.CRUZ BRGY. ACACIA BUHANGIN DAVAO CITY
REACTION REPORT
INVENTION THAT CHANGE THE WORLD
Objectives:
To be able to show to the people how invention changes the world.
Summary:
In this stage of our technology, many high-tech gadgets and machine were invented. Those inventions are very useful to the people for their daily activity, for fun and work People used machine and gadgets for the better their daily activity ,Gadgets and machine are useful not only to the adult, but also to the young people.
One of the most useful gadgets is the computer. Computers are very useful to the people. They use the computer for their business and work, personal usage and etc. The computer used also to the student, professional, businessman, businesswoman and etc.
The computer help their to their storage information, to communicate to other people, give their information and the other usage that computer can give to the people.
When the computer was invented, it was a start for a new future for the people. The computers have a big distribution of the changes of the world. Because the computers other machine and gadgets are invented. The scientists and inventors can explore their imagination through the computer. The machine and gadgets are improving because of the computer many ideas were built because of the information that are storage in the computer.
Because of there the world are improving, There are more gadgets and machine were invented. The jobs of the people get easy. The transportation and communication became faster. The technologies are improving and the world is changing.
Knowledge:
The computers have the biggest contributions of the changes of the world. A devices that are useful to everyone. Because of the computer the peoples work became easy. Computers are the reasons, of why the technology becomes high-tech.
Conclusion:
The world is changing because. Many machine and gadgets are invented and improving. There are robots were invented, and the computer is still the influence of the inventions.
Objectives:
To be able to show to the people how invention changes the world.
Summary:
In this stage of our technology, many high-tech gadgets and machine were invented. Those inventions are very useful to the people for their daily activity, for fun and work People used machine and gadgets for the better their daily activity ,Gadgets and machine are useful not only to the adult, but also to the young people.
One of the most useful gadgets is the computer. Computers are very useful to the people. They use the computer for their business and work, personal usage and etc. The computer used also to the student, professional, businessman, businesswoman and etc.
The computer help their to their storage information, to communicate to other people, give their information and the other usage that computer can give to the people.
When the computer was invented, it was a start for a new future for the people. The computers have a big distribution of the changes of the world. Because the computers other machine and gadgets are invented. The scientists and inventors can explore their imagination through the computer. The machine and gadgets are improving because of the computer many ideas were built because of the information that are storage in the computer.
Because of there the world are improving, There are more gadgets and machine were invented. The jobs of the people get easy. The transportation and communication became faster. The technologies are improving and the world is changing.
Knowledge:
The computers have the biggest contributions of the changes of the world. A devices that are useful to everyone. Because of the computer the peoples work became easy. Computers are the reasons, of why the technology becomes high-tech.
Conclusion:
The world is changing because. Many machine and gadgets are invented and improving. There are robots were invented, and the computer is still the influence of the inventions.
Sunday, September 13, 2009
hexadicemal number
In mathematics and computer science, hexadecimal (also base-16, hexa, or hex) is a numeral system with a radix, or base, of 16. It uses sixteen distinct symbols, most often the symbols 0–9 to represent values zero to nine, and A, B, C, D, E, F (or a through f) to represent values ten to fifteen.
Its primary use is as a human-friendly representation of binary coded values, so it is often used in digital electronics and computer engineering. Since each hexadecimal digit represents four binary digits (bits)—also called a nibble—it is a compact and easily translated shorthand to express values in base two.
Uses
|
In digital computing, hexadecimal is primarily used to represent bytes. Attempts to represent the 256 possible byte values by other means have led to problems. Directly representing each possible byte value with a single character representation runs into unprintable control characters in the ASCII character set. Even if a standard set of printable characters were devised for every byte value, neither users nor input hardware are equipped to handle 256 unique characters. Most hex editing software displays each byte as a single character, but unprintable characters are usually substituted with a period or blank.
[edit] Representing hexadecimal
In situations where there is no context, a hexadecimal number might be ambiguous and confused with numbers expressed in other bases. There are several conventions for unambiguously expressing values. In mathematics, a subscript (itself written in decimal) is often used on each number explicitly giving the base: 15910 is decimal 159; 15916 is hexadecimal 159 which is equal to 34510. Some authors prefer a text subscript, such as 159decimal and 159hex.
In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:
- In URLs, character codes are written as hexadecimal pairs prefixed with
%:http://www.example.com/name%20with%20spaceswhere%20is the space (blank) character, code 20 hex, or 32 decimal. - In XML and XHTML, characters can be expressed as hexadecimal using the notation
. Color references are expressed in hex prefixed with#:#FFFFFFwhich gives white.[1] - *nix (UNIX and related) shells, and likewise the C programming language, which was designed for UNIX (and the syntactical descendants of C[2]) use the prefix
0x:0x5A3for numeric constants. Character and string constants may express character codes in hexadecimal with the prefix\xfollowed by two hex digits:'\x1B'(specifies the Esc control character),"\x1B[0m\x1B[25;1H"is a string containing 11 characters (not including an implied trailing NUL).[3] To output a value as hexadecimal with the printf function family, the format conversion code%Xor%xis used. - In the Unicode standard, a character value is represented with
U+followed by the hex value:U+20ACis the Euro sign (€). - In MIME (e-mail extensions) quoted-printable encoding, characters that cannot be represented as literal ASCII characters are represented by their codes as two hexadecimal digits (in ASCII) prefixed by an equal to sign
=, as inEspa=F1ato send "España" (Spain). (Hexadecimal F1, equal to decimal 241, is the code number for the lower case n with tilda in the ISO/IEC 8859-1 character set.) - In Intel-derived assembly languages, hexadecimal is indicated with a suffixed H or h:
FFhor05A3H. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit:0FFh - Other assembly languages (6502, AT&T, Motorola), Pascal, and some versions of BASIC (Commodore) and Forth use
$as a prefix:$5A3. - Some assembly languages (Microchip) use the notation
H'ABCD'(for ABCD16). - Ada and VHDL enclose hexadecimal numerals in based "numeric quotes":
16#5A3# - Verilog represents hexadecimal constants in the form
8'hFF, where 8 is the number of bits in the value and FF is the hexadecimal constant. - Modula 2 and some other languages use # as a prefix:
#05A3 - The Smalltalk programming language uses the prefix
16r:16r5A3 - Postscript indicates hex with prefix
16#:16#5A3. Binary data (such as image pixels) can be expressed as unprefixed consecutive hexadecimal pairs:AA213FD51B3801043FBC... - Common Lisp use the prefixes
#xand#16r. - QuickBASIC, FreeBASIC and Visual Basic prefix hexadecimal numbers with
&H:&H5A3 - BBC BASIC and Locomotive BASIC use
&for hex.[4] - TI-89 and 92 series uses a
0hprefix:0h5A3 - Notations such as
X'5A3'are sometimes seen, such as in PL/I. This is the most common format for hexadecimal on IBM mainframes (zSeries) and midrange computers (iSeries) running traditional OS's (zOS, zVSE, zVM, TPF, OS/400), and is used in Assembler, PL/1, Cobol, JCL, scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well. - Donald Knuth introduced the use of a particular typeface to represent a particular radix in his book The TeXbook.[5] There, hexadecimal representations are written in a typewriter typeface: 5A3
There is no universal convention to use lowercase or uppercase for the letter digits, and each is prevalent or preferred in particular environments by community standards or convention.
The choice of the letters A through F to represent the digits above nine was not universal in the early history of computers. During the 1950s, some installations favored using the digits 0 through 5 with a macron character ("¯") to indicate the values 10-15. Users of Bendix G-15 computers used the letters U through Z. Bruce A. Martin of Brookhaven National Laboratory considered the choice of A-F "ridiculous" and in 1968 proposed in a letter to the editor of the ACM an entirely new set of symbols based on the bit locations, which did not gain much acceptance.[6]
[edit] Verbal and digital representations
Not only are there no digits to represent the quantities from ten to fifteen—so letters are used as a substitute—but most Western European languages also lack a nomenclature to name hexadecimal numbers. "Thirteen" and "fourteen" are decimal-based, and even though English has names for several non-decimal powers: pair for the first binary power; score for the first vigesimal power; dozen, gross, and great gross for the first three duodecimal powers. However, no English name describes the hexadecimal powers (corresponding to the decimal values 16, 256, 4096, 65536, ...). Some people read hexadecimal numbers digit by digit like a phone number: 4DA is "four-dee-eh". However, the letter 'A' sounds similar to eight, 'C' sounds similar to three, and 'D' can easily be mistaken for the 'ty' suffix: Is it 4D or forty? Other people avoid confusion by using the NATO phonetic alphabet: 4DA is "four-delta-alpha". Similarly, some use the Joint Army/Navy Phonetic Alphabet ("four-dog-able"), or a similar ad hoc system.
Systems of counting on (digits) have been devised for both binary and hexadecimal. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 1023 on ten fingers. Another system for counting up to FF (255) is illustrated on the right; it seems to be an extension of an existing system for counting in twelves (dozens and grosses), that is common in South Asia and elsewhere.
[edit] Signs
The hexadecimal system can express negative numbers the same way as in decimal: –2A to represent –42 and so on.
However, some prefer instead to express the exact bit patterns used in the processor and consider hexadecimal values best handled as unsigned values. This way, the negative number –42 can be written as FFFF FFD6 in a 32-bit CPU register, as C228 0000 in a 32-bit FPU register or C045 0000 0000 0000 in a 64-bit FPU register (assuming certain representation schemes, twos-complement in the 32-bit non-FPU instance.)
[edit] Real numbers
As with other numeral systems, the hexadecimal system can be used to represent rational numbers, although recurring digits are common since sixteen (10h) has only a single prime factor (two):
| 1⁄2 | | 0.8 | 1⁄6 | | 0.2AAAAAAAA... | 1⁄A | | 0.1999999999... | 1⁄E | | 0.1249249249... |
| 1⁄3 | | 0.5555555555... | 1⁄7 | | 0.2492492492... | 1⁄B | | 0.1745D1745D... | 1⁄F | | 0.1111111111... |
| 1⁄4 | | 0.4 | 1⁄8 | | 0.2 | 1⁄C | | 0.1555555555... | 1⁄10 | | 0.1 |
| 1⁄5 | | 0.3333333333... | 1⁄9 | | 0.1C71C71C71... | 1⁄D | | 0.13B13B13B1... | 1⁄11 | | 0.0F0F0F0F0F... |
For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system: Counting in base 3 is 0, 1, 2, 10 (three). Thus, whether dividing one by two for binary or dividing one by sixteen for hexadecimal, both of these fractions are written as 0.1. Because the radix 16 is a perfect square (4²), fractions expressed in hexadecimal have an odd period much more often than decimal ones, and there are no cyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix; thus, when using hexadecimal notation, all fractions with denominators that are not a power of two result in an infinite string of recurring digits (such as thirds and fifths). This makes hexadecimal (and binary) less convenient than decimal for representing rational numbers since a larger proportion lie outside its range of finite representation.
All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hexadecimal number with a finite number of digits has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hexadecimal: That is, decimal 0.1 corresponds to the infinite recurring representation 0.199999999999... in hexadecimal. However, hexadecimal is more efficient than bases 12 and 60 for representing fractions with powers of two in the denominator (e.g., decimal one sixteenth is 0.1 in hexadecimal, 0.09 in duodecimal, 0;3,45 in sexagesimal and 0.0625 in decimal).
| Decimal base Prime factors of the base: 2, 5 | Hexadecimal base Prime factors of the base: 2 | ||||
| Fraction | Prime factors of the denominator | Positional representation | Positional representation | Prime factors of the denominator | Fraction |
| 1/2 | 2 | 0.5 | 0.8 | 2 | 1/2 |
| 1/3 | 3 | 0.3333... = 0.3 | 0.5555... = 0.5 | 3 | 1/3 |
| 1/4 | 2 | 0.25 | 0.4 | 2 | 1/4 |
| 1/5 | 5 | 0.2 | 0.3 | 5 | 1/5 |
| 1/6 | 2, 3 | 0.16 | 0.2A | 2, 3 | 1/6 |
| 1/7 | 7 | 0.142857 | 0.249 | 7 | 1/7 |
| 1/8 | 2 | 0.125 | 0.2 | 2 | 1/8 |
| 1/9 | 3 | 0.1 | 0.1C7 | 3 | 1/9 |
| 1/10 | 2, 5 | 0.1 | 0.19 | 2, 5 | 1/A |
| 1/11 | 11 | 0.09 | 0.1745D | B | 1/B |
| 1/12 | 2, 3 | 0.083 | 0.15 | 2, 3 | 1/C |
| 1/13 | 13 | 0.076923 | 0.13B | D | 1/D |
| 1/14 | 2, 7 | 0.0714285 | 0.1249 | 2, 7 | 1/E |
| 1/15 | 3, 5 | 0.06 | 0.1 | 3, 5 | 1/F |
| 1/16 | 2 | 0.0625 | 0.1 | 2 | 1/10 |
| 1/17 | 17 | 0.0588235294117647 | 0.0F | 11 | 1/11 |
| 1/18 | 2, 3 | 0.05 | 0.0E38 | 2, 3 | 1/12 |
| 1/19 | 19 | 0.052631578947368421 | 0.0D79435E50 | 13 | 1/13 |
| 1/20 | 2, 5 | 0.05 | 0.0C | 2, 5 | 1/14 |
| 1/21 | 3, 7 | 0.047619 | 0.0C3 | 3, 7 | 1/15 |
| 1/22 | 2, 11 | 0.045 | 0.0BA2E8 | 2, B | 1/16 |
| 1/23 | 23 | 0.0434782608695652173913 | 0.0B21642C8590 | 17 | 1/17 |
| 1/24 | 2, 3 | 0.0416 | 0.0A | 2, 3 | 1/18 |
| 1/25 | 5 | 0.04 | 0.0A3D70 | 5 | 1/19 |
| 1/26 | 2, 13 | 0.0384615 | 0.09D8 | 2, B | 1/1A |
| 1/27 | 3 | 0.037 | 0.097B425ED | 3 | 1/1B |
| 1/28 | 2, 7 | 0.03571428 | 0.0924 | 2, 7 | 1/1C |
| 1/29 | 29 | 0.0344827586206896551724137931 | 0.08D3DCB | 1D | 1/1D |
| 1/30 | 2, 3, 5 | 0.03 | 0.08 | 2, 3, 5 | 1/1E |
| 1/31 | 31 | 0.032258064516129 | 0.08421 | 1F | 1/1F |
| 1/32 | 2 | 0.03125 | 0.08 | 2 | 1/20 |
| 1/33 | 3, 11 | 0.03 | 0.07C1F | 3, B | 1/21 |
| 1/34 | 2, 17 | 0.02941176470588235 | 0.078 | 2, 11 | 1/22 |
| 1/35 | 5, 7 | 0.0285714 | 0.075 | 5, 7 | 1/23 |
| 1/36 | 2, 3 | 0.027 | 0.071C | 2, 3 | 1/24 |
| Algebraic irrational number | In decimal | In Hexadecimal |
| √2 (the length of the diagonal of a unit square) | 1.41421356237309... (≈ 1.414) | 1.6A09E667F3BCD... (≈ 1.6A) |
| √3 (the length of the diagonal of a unit cube, or twice the height of an equilateral triangle of unit side) | 1.73205080756887... (≈ 1.732) | 1.BB67AE8584CAA... (≈ 1.BB) |
| √5 (the length of the diagonal of a 1×2 rectangle) | 2.2360679774997... (≈ 2.236) | 2.3C6EF372FE95... (≈ 2.3C) |
| φ (phi, the golden ratio = (1+√5)⁄2) | 1.6180339887498... (≈ 1.618) | 1.9E3779B97F4A... (≈ 1.9E) |
| Transcendental irrational number | In decimal | In Hexadecimal |
| π (pi, the ratio of circumference to diameter) | 3.1415926535897932384626433 8327950288419716939937510... (≈ 3.1416) | 3.243F6A8885A308D313198A2E0 3707344A4093822299F31D008... (≈ 3.243F) |
| e (the base of the natural logarithm) | 2.7182818284590452... (≈ 2.718) | 2.B7E151628AED2A6B... (≈ 2.B7E) |
| τ (the Thue–Morse constant) | 0.412454033640... | 0.6996 9669 9669 6996 ... |
| Number | In decimal | In Hexadecimal |
| γ (the limiting difference between the harmonic series and the natural logarithm) | 0.5772156649015328606... (≈ 0.577) | 0.93C467E37DB0C7A4D1B... (≈ 0.93C) |
[edit] Powers
Possibly the most widely used powers, powers of two, are easier to show using the 16th base. The first sixteen powers of two are shown below.
| 2x | value |
|---|---|
| 20 | 1 |
| 21 | 2 |
| 22 | 4 |
| 23 | 8 |
| 24 | 10hex |
| 25 | 20hex |
| 26 | 40hex |
| 27 | 80hex |
| 28 | 100hex |
| 29 | 200hex |
2A ( ) | 400hex |
2B ( ) | 800hex |
2C ( ) | 1000hex |
2D ( ) | 2000hex |
2E ( ) | 4000hex |
2F ( ) | 8000hex |
210 ( ) | 10000hex |
Since four squared is sixteen, powers of four have an even easier relation:
| 4x | value |
|---|---|
| 40 | 1 |
| 41 | 4 |
| 42 | 10hex |
| 43 | 40hex |
| 44 | 100hex |
| 45 | 400hex |
| 46 | 1000hex |
| 47 | 4000hex |
| 48 | 10000hex |
This also makes tetration easier when using two and four since 32 = 10hex, 42 = 10000hex and 52 = 10000000000hex.
[edit] Binary conversion
Most computers manipulate binary data, but it is difficult for humans to work with the large number of digits for even a relatively small binary number. Although most humans are familiar with the base 10 system, it is much easier to map binary to hexadecimal than to decimal because each hexadecimal digit maps to a whole number of bits (410). This example converts 11112 to base ten. Since each position in a binary numeral can contain either a 1 or 0, its value may be easily determined by its position from the right:
- 00012 = 110
- 00102 = 210
- 01002 = 410
- 10002 = 810
Therefore:
| 11112 | = 810 + 410 + 210 + 110 |
| = 1510 |
With surprisingly little practice, mapping 11112 to F16 in one step becomes easy: see table in Uses. The advantage of using hexadecimal rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hexadecimal, it is trivial to regard the binary string as 4-digit groups and map each to a single hexadecimal digit.
This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
| 010111101011010100102 | = 26214410 + 6553610 + 3276810 + 1638410 + 819210 + 204810 + 51210 + 25610 + 6410 + 1610 + 210 |
| = 38792210 |
Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently, and converted directly:
| 010111101011010100102 | = | 0101 | 1110 | 1011 | 0101 | 00102 |
| = | 5 | E | B | 5 | 216 | |
| = | 5EB5216 | |||||
The conversion from hexadecimal to binary is equally direct.
The octal system can also be useful as a tool for people who need to deal directly with binary computer data. Octal represents data as three bits per character, rather than four.
[edit] Converting from other bases
[edit] Division-remainder in source base
As with all bases there is a simple algorithm for converting a representation of a number to hexadecimal by doing integer division and remainder operations in the source base. Theoretically this is possible from any base but for most humans only decimal and for most computers only binary (which can be converted by far more efficient methods) can be easily handled with this method.
Let d be the number to represent in hexadecimal, and the series hihi-1...h2h1 be the hexadecimal digits representing the number.
- i := 1
- hi := d mod 16
- d := (d-hi) / 16
- If d = 0 (return series hi) else increment i and go to step 2
"16" may be replaced with any other base that may be desired.
The following is a JavaScript implementation of the above algorithm for converting any number to a hexadecimal in String representation. Its purpose is to illustrate the above algorithm. To work with data seriously however, it is much more advisable to work with bitwise operators.
function toHex(d) {
var r = d % 16;
var result;
if (d-r == 0)
result = toChar(r);
else
result = toHex( (d-r)/16 ) + toChar(r);
return result;
}
function toChar(n) {
const alpha = "0123456789ABCDEF";
return alpha.charAt(n);
}
[edit] Addition and multiplication
It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value and then performing multiplication and addition to get the final representation. I.e. to convert the number B3AD to decimal one can split the conversion into D (1310), A (1010), 3 (310) and B (1110) then get the final result by multiplying each decimal representation by 16p, where 'p' is the corresponding position from right to left, beginning with 0. In this case we have 13*(160) + 10*(161) + 3*(162) + 11*(163), which is equal 45997 in the decimal system.
[edit] Tools for conversion
Most modern computer systems with graphical user interfaces provide a built-in calculator utility, capable of performing conversions between various radixes, generally including hexadecimal.
In Microsoft Windows, the Calculator utility can be set to scientific calculator mode, which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 (octal) and 2 (binary); the bases most commonly used by programmers. In Scientific Mode, the on screen numeric keypad includes the hexadecimal digits A through F which are active when "Hex" is selected. The Windows Calculator however only supports integers.
[edit] Cultural
[edit] Etymology
The word "hexadecimal" is strange in that hexa is derived from the Greek έξ (hex) for "six" and decimal is derived from the Latin for "tenth". It may have been derived from the Latin root, but Greek deka is so similar to the Latin decem that some would not consider this nomenclature inconsistent. However, the word "sexagesimal" (base 60) retains the Latin prefix. The earlier Bendix documentation used the term "sexadecimal". Donald Knuth has pointed out that the etymologically correct term is "senidenary", from the Latin term for "grouped by 16". (The terms "binary", "ternary" and "quaternary" are from the same Latin construction, and the etymologically correct term for "decimal" arithmetic is "denary".)[7] Schwartzman notes that the pure expectation from the form of usual Latin-type phrasing would be "sexadecimal", but then computer hackers would be tempted to shorten the word to "sex".[8] Incidentally, the etymologically proper Greek term would be hexadecadic (although in Modern Greek deca-hexadic (δεκαεξαδικός) is more commonly used).
[edit] Common patterns and humor
Hexadecimal is sometimes used in programmer jokes because certain words can be formed using only hexadecimal digits. Some of these words are "dead", "beef", "babe", and with appropriate substitutions "c0ffee". Since these are quickly recognizable by programmers, debugging setups sometimes initialize memory to them to help programmers see when something has not been initialized. Some people add an H after a number if they want to show that it is written in hexadecimal. In older Intel assembly syntax, this is sometimes the case. "Hexspeak" may be the forerunner of the modern web parlance of "1337speak"
An example is the magic number in FAT Mach-O files and java class file structure, which is "CAFEBABE". Single-architecture Mach-O files have the magic number "FEEDFACE" at their beginning. "DEADBEEF" is sometimes put into uninitialized memory. Microsoft Windows XP clears its locked index.dat files with the hex codes: "0BADF00D". The Visual C++ remote debugger uses "BADCAB1E" to denote a broken link to the target system.
Two common bit patterns often employed to test hardware are 01010101 and 10101010 (their corresponding hex values are 55h and AAh, respectively). The reason for their use is to alternate between off ('0') to on ('1') or vice versa when switching between these two patterns. These two values are often used together as signatures in critical PC system sectors (e.g., the hex word, 0xAA55 which on little-endian systems is 55h followed by AAh, must be at the end of a valid Master Boot Record).
The following table shows a joke in hexadecimal:
3x12=36
2x12=24
1x12=12
0x12=18
The first three are interpreted as multiplication, but in the last, "0x" signals Hexadecimal interpretation of 12, which is 18.
Another joke based on the use of a word containing only letters from the first six in the alphabet (and thus those used in hexadecimal) is...
- If only DEAD people understand hexadecimal, how many people understand hexadecimal?
In this case, DEAD refers to a hexadecimal number (57005 base 10), not the state of being no longer alive. Obviously, DEAD normally should not be written in all-caps (as in the preceding) as it makes it stand out, thus ruining the riddle.
A Knuth reward check is one hexadecimal dollar, or $2.56.
[edit] Primary numeral system
Similar to dozenal advocacy, there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts usually propose pronunciation and/or symbology.[9] Sometimes the proposal unifies standard measures so that they are multiples of 16.[10][11][12]
An example of unifying standard measures is Hexadecimal time which subdivides a day by 16 so that there are 16 "hexhours" in a day.[12]
[edit] Key to number base notation
Simple key for notations used in article:
| Full Text Notation | Abbreviation | Number Base |
|---|---|---|
| binary | bin | 2 |
| octal | oct | 8 |
| decimal | dec | 10 |
| hexadecimal | hex | 16 |
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