Transcript of maths project
• INVENTION OF MATHS• ALGORITHMS• LOGARITHMS• ARETHMETIC PROGRESSION
INVENTION OF MATHS
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the abstract study of subjects encompassing quantity, structure,space, change, and more; it has no generally accepted definition.
Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. The research required to solve mathematical problems can take years or even centuries of sustained inquiry. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. When those mathematical structures are good models of real phenomena, then mathematical reasoning often provides insight or predictions about nature.
Through the use of abstraction and logical reasoning, mathematics developed from counting, calculation, measurement, and the systematic study of theshapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous argumentsfirst appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Galileo Galilei (1564–1642) said, 'The universe cannot be read until we have learned the language and become familiar with the characters in which it is written. It is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth'. Carl Friedrich Gauss (1777–1855) referred to mathematics as "the Queen of the Sciences". Benjamin Peirce (1809–1880) called mathematics "the science that draws necessary conclusions". David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise." Albert Einstein (1879–1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality".
Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, and the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries, which has led to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.
ALGORITHMS Algorithmic information theory is a subfield of information theory and
computer science that concerns itself with the relationship between computation and information. According to Gregory Chaitin, it is "the result of putting Shannon's information theory and Turing's computability theory into a cocktail shaker and shaking vigorously." \l
Algorithmic information theory principally studies complexity measures on strings (or other data structures). Because most mathematical objects can be described in terms of strings, or as the limit of a sequence of strings, it can be used to study a wide variety of mathematical objects, including integers and real numbers.
This use of the term "information" might be a bit misleading, as it depends upon the concept of compressibility. Informally, from the point of view of algorithmic information theory, the information content of a string is equivalent to the length of the shortest possible self-contained representation of that string. A self-contained representation is essentially a program – in some fixed but otherwise irrelevant universal programming language – that, when run, outputs the original string.
Turing himself was fascinated with how the distinction between software and hardware illuminated immortality and the soul. Identifying personal identity with computer software ensured that humans were immortal, since even though hardware could be destroyed, software resided in a realm of mathematical abstraction and was thus immune to destruction.
An infinite binary sequence is said to be random if, for some constant c, for all n, the Kolmogorov complexity of the initial segment of length n of the sequence is at least n − c. It can be shown that almost every sequence (from the point of view of the standard measure — "fair coin" or Lebesgue measure — on the space of infinite binary sequences) is random. Also, since it can be shown that the Kolmogorov complexity relative to two different universal machines differs by at most a constant, the collection of random infinite sequences does not depend on the choice of universal machine (in contrast to finite strings). This definition of randomness is usually called Martin-Löf randomness, after Per Martin-Löf, to distinguish it from other similar notions of randomness. It is also sometimes called 1-randomness to distinguish it from other stronger notions of randomness (2-randomness, 3-randomness, etc.).
(Related definitions can be made for alphabets other than the set .) Algorithm An algorithm is any well-defined procedure for solving a given class of problems.
Ideally, when applied to a particular problem in that class, the algorithm would yield a full solution. Nonetheless, it makes sense to speak of algorithms that yield only partial solutions or yield solutions only some of the time. Such algorithms are sometimes called "rules of thumb" or "heuristics."
Algorithms have been around throughout recorded history. The ancient Hindus, Greeks, Babylonians, and Chinese all had algorithms for doing arithmetic computations. The actual term algorithm derives from ninth-century Arabic and incorporates the Greek word for number (arithmos ).
Algorithms are typically constructed on a case-by-case basis, being adapted to the problem at hand. Nonetheless, the possibility of a universal algorithm that could in principle resolve all problems has been a recurrent theme over the last millennium. Spanish theologian Raymond Lully (c. 1232–1315), in his Ars Magna, proposed to reduce all rational discussion to mechanical manipulations of symbolic notation and combinatorial diagrams. German philosopher Gottfried Wilhelm Leibniz (1646–1716) argued that Lully's project was overreaching but had merit when conceived more narrowly.
The idea of a universal algorithm did not take hold, however, until technology had advanced sufficiently to mechanize it. The Cambridge mathematician Charles Babbage (1791–1871) conceived and designed the first machine that could in principle resolve all well-defined arithmetic problems. Nevertheless, he was unable to build a working prototype. Over a century later another Cambridge mathematician, Alan Turing (1912–1954), laid the theoretical foundations for effectively implementing a universal algorithm.
Turing proposed a very simple conceptual device involving a tape with a movable reader that could mark and erase letters on the tape. Turing showed that all algorithms could be mapped onto the tape (as data) and then run by a universal algorithm already inscribed on the tape. This machine, known as a universal Turing machine, became the basis for the modern theory of computation (known as recursion theory) and inspired the modern digital computer.
Turing's universal algorithm fell short of Lully's vision of an algorithm that could resolve all problems. Turing's universal algorithm is not so much a universal problem solver as an empty box capable of housing and implementing the algorithms placed into it. Thus Turing invited into the theory of computing the very Cartesian distinction between hardware and software. Hardware is the mechanical device (i.e., the empty box) that houses and implements software (i.e., the algorithms) running on it.
It is a deep and much disputed question whether the essence of what constitutes the human person is at base computational and therefore an emergent property of algorithms, or whether it fundamentally transcends the capacity of algorithms.
In mathematics and computer science, an algorithm
The logarithm of a number is the exponent by which another fixed value,
the base, has to be raised to produce that number. For example, the logarithm of
1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 103 = 10 × 10 ×
10. More generally, if x = by, then y is the logarithm of x to base b, and is written y =
logb(x), so log10(1000) = 3.
Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. These devices rely on the fact—important in its own right—that the logarithm of a product is the sum of the logarithms of the factors:The present-day notion of logarithms comes from Leonhard Euler, who connected them to the exponential function in the 18th century.The logarithm to base b = 10 is called the common logarithm and has many applications in science and engineering. The natural logarithmhas the constant e (≈ 2.718) as its base; its use is widespread in pure mathematics, especially calculus. The binary logarithm uses baseb = 2 and is prominent in computer science.
Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel is a logarithmic unit quantifying sound pressure and voltage ratios. In chemistry, pH and pOH are logarithmic measures for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They describe musical intervals, appear in formulae counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting.In the same way as the logarithm reverses exponentiation, the complex logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant; it has applications in public-key cryptography.
In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 3, 5, 7, 9, 11, 13, … is an arithmetic progression with common difference of 2.
If the initial term of an arithmetic progression is and the common difference of successive members is d, then the nth term of the sequence ( ) is given by:
A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.
The behavior of the arithmetic progression depends on the common difference d. If the common difference is:
Positive, the members (terms) will grow towards positive infinity. Negative, the members (terms) will grow towards negative infinity.
Arithmetic Arithmetic is a branch of mathematics concerned with the addition,
subtraction, multiplication, division, and extraction of roots of certain numbers known as real numbers. Real numbers are numbers with which you are familiar in everyday life: whole numbers, fractions, decimals, and roots, for example.
Early development of arithmetic Arithmetic grew out of the need that people have for counting objects.
For example, Stone Age men or women probably needed to count the number of children they had. Later, one person might want to know the number of oxen to be given away in exchange for a wife or husband. For many centuries, however, counting probably never went beyond the 10 stage, the number of fingers on which one could note the number of objects.
Numbering system The numbering system we use today is called the Hindu-Arabic
system. It was developed by the Hindu civilization of India about 1,500 years ago and then brought to Europe by the Arabs in the Middle Ages (400–1450). During the seventeenth century, the Hindu-Arabic system completely replaced the Roman numeral system that had been in use earlier.
The Hindu-Arabic system is also called a decimal system because it is based on the number 10. The ten symbols used in the decimal system are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Other number systems are possible and, in fact, are also used today. Computers, for example, operate on a binary system that consists of only two numbers, 0 and 1. Our system of time uses the sexagesimal (pronounced sek-se-JES-em-el) system, consisting of the numbers 0 to 60.
Words to Know Associative law: An axiom that states that grouping numbers
during addition or multiplication does not change the final result. Axiom: A basic statement of fact that is stipulated as true
without being subject to proof. Closure property: An axiom that states that the result of the
addition or multiplication of two real numbers is a real number. Commutative law: An axiom of addition and multiplication that
states that the order in which numbers are added or multiplied does not change the final result.
Hindu-Arabic number system: A positional number system that uses ten symbols to represent numbers and uses zero as a place holder. It is the number system that we use today.
Inverse operation: A mathematical operation that reverses the work of another operation; for example, subtraction is the inverse operation of addition.
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