Penguins

Ancient Mathematics Concepts

1. The one I can find about mathematics concept, problem, or solution in the ancient time that still used in the present time.

a. Pythagorean Theorem

Pythagorean Theorem was found in the ancient time, and today it is still used. The theorem is as follows:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

This is usually summarized as follows:

The square of the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides

If we let c be the length of the hypotenuse and "a and "b are the lengths of the other two sides, the theorem can be expressed as the equation:

b. Calculus

In science and engineering, algebra is not enough to solve problems. Therefore calculus is used. Calculus builds on algebra, trigonometry, and analytic geometry and includes two major branches, differential calculus and integral calculus. In more advanced mathematics, calculus is usually called analysis and is defined as the study of functions.

The ancient period introduced some of the ideas of integral calculus, but does not seem to have developed these ideas in a rigorous or systematic way. Calculating volumes and areas, the basic function of integral calculus, can be traced back to the Egyptian Moscow papyrus (c. 1800 BC), in which an Egyptian successfully calculated the volume of a pyramidal frustum. From the school of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which prefigures the concept of the limit, to calculate areas and volumes while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble integral calculus. The method of exhaustion was later used in China by Liu Hui in the 3rd century AD in order to find the area of a circle. It was also used by Zu Chongzhi in the 5th century AD, who used it to find the volume of a sphere.

Gottfried Wilhelm Leibniz was originally accused of plagiarizing Sir Isaac Newton's unpublished work, but is now regarded as an independent inventor of and contributor to calculus.

Leibniz and Newton are usually both credited with the invention of calculus. Newton was the first to apply calculus to general physics and Leibniz developed much of the notation used in calculus today. The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, second and higher derivatives, and the notion of an approximating polynomial series. By Newton's time, the fundamental theorem of calculus was known.

When Newton and Leibniz first published their results, there was great controversy over which mathematician deserved credit. Newton derived his results first, but Leibniz published first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the Royal Society. This controversy divided English-speaking mathematicians from continental mathematicians for many years, to the detriment of English mathematics. A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. Today, both Newton and Leibniz are given credit for developing calculus independently. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "the science of fluxions".

Up to now Calculus is still to be studied at the high schools and universities, and mathematicians around the world continue to contribute to its development.

2. The one I can find about mathematics concept, problem, or solution in the ancient time that no used in the present time.

a. The Egyptian Number System

In the ancient time Egyptian has known number system. It based on the proof that persons, animals, or plants pictorial signs is written on monument wall stone a long time ago.

The Egyptians used a written numeration that was changed into hieroglyphic writing, which enabled them to note whole numbers to 1,000,000. It had a decimal base and allowed for the additive principle. In this notation there was a special sign for every power of ten. For I, a vertical line; for 10, a sign with the shape of an upside down U; for 100, a spiral rope; for 1000, a lotus blossom; for 10,000 , a raised finger, slightly bent; for 100,000 , a tadpole; and for 1,000,000, a kneeling genie with upraised arms.

This hieroglyphic numeration was a written version of a concrete counting system using material objects. To represent a number, the sign for each decimal order was repeated as many times as necessary. To make it easier to read the repeated signs they were placed in groups of two, three, or four and arranged vertically.

Example:

In writing the numbers, the largest decimal order would be written first. The numbers were written from right to left.

Example

Egyptians method of multiplication is fairly

clever, but can take longer than the modern day method. This is how they would have multiplied 5 by 29 *1 29

2 58

*4 116

1 + 4 = 5 29 + 116 = 145

When multiplying they would began with the number they were multiplying by 29 and double it for each line. Then they went back and picked out the numbers in the first column that added up to the first number (5). They used the distributive property of mult

iplication over addition.

29(5) = 29(1 + 4) = 29 + 116 = 145

The way they did division was similar to their multiplication. For the problem 98/7, they thought of this problem as 7 times some number equals 98. Again the problem was worked in columns. 1 7

2 * 14

4 * 28

8 * 56

2 + 4 + 8 = 14 14 + 28 + 56 = 98

This time the numbers in the right-hand column are marked which sum to 98 then the corresponding numbers in the left-hand column are summed to get the quotient.

So the answer is 14. 98 = 14 + 28 +

56 = 7(2 + 4 + 8) = 7*14

But, the Egyptian number system is not used anymore in the present days.

3. The one I can find about mathematics concept, problem, or solution for the present time that no relationship with the ancient term of mathematics

At Recent, we often use a negative num

ber, but our ancients don’t. because they were never know what negative number, neither its functions.

Some mathematicians in the 17th century discovered that negative numbers did have their uses. Provided they didn't worry about what negative numbers meant, and more particularly what the square roots of negative numbers meant,

they found that they could solve some very tricky equations, like cubic and quartic equations. What's more, although the intermediate steps of a calculation may have involved negative numbers, the solution often came out as a real, positive number which was exactly what they wanted.

Since then mathematicians and scientists have found all sorts of uses for negative numbers. We now recognize that in many cases a negative answer can be a real, meaningful solution and can be thought of in terms of direction. For instanc

e, if I wanted to calculate how many steps forward Robert has taken, and the answer is -5, then it means he has taken 5 steps backwards. The first person to recognize the link between negative numbers and direction was John Wallis, a mathematician in the 17th century. He was the first to come up with the idea of a number line as a geometrical representation of the number system. Confusingly however, he also thought that negative numbers were larger than infinity!

Nowadays we use negative numbers just like any other numbers without even a second thought. Their troubled history shows how the simple mathematical principles we take for granted have taken thousands of years to develop. Physical meaning has given way to algebraic utility, but negative numbers and their derivatives have turned out to have all kinds of practical applications. Take the square root of -1 for example - it seems meaningless in itself, but many calculations in science and engineering wouldn't be possible without it.

References

http://en.wikipedia.org/wiki/Pythagorean_theorem
http://en.wikipedia.org/wiki/Calculus
http://www.math.wichita.edu/history/topics/num-sys.html#egypt
http://nrich.maths.org/public/viewer.php?obj_id=5747

THE FOUNDATIONS OF MATHEMATICS

THE FOUNDATIONS OF MATHEMATICS

Arranged by:
LENNY PUSPITA DEWI
05301244122






MATHEMATICS EDUCATION
FACULTY OF MATHEMATICS AND NATURAL SCIENCE
YOGYAKARTA STATE UNIVERSITY
2008

Mathematics is a logical science, cleanly structured, and well-founded. Here we look at those foundations.

A. What is a "foundation" for mathematics?
We discuss some of the ways the word "foundation" is used in relation to mathematics.
· Branch Foundations
Each branch or field of mathematics may have its own foundational elements, special to the field. These may be the fundamental concepts investigated in the field, fundamental results on which most other results are based, or pervasive methods.
· Fundamental Concepts
There are some mathematical concepts which pervade not just one branch of mathematics but the whole of mathematics. An obvious example is the concept of a function or that of a homomorphism. Variation in these concepts may either provide different ways of doing mathematics (e.g. category theory), or may lead to different kinds of mathematics (e.g. constructive mathematics).

B. Logical Foundations for Mathematics
The methods of mathematics are deductive, and logic therefore has a fundamental role in the development of mathematics. Suitable logical frameworks in which mathematics can be conducted can therefore be called logical foundation systems for mathematics.
· Paradigms
There are many alternative logical foundations for mathematics. They differ in some cases just in detail or strength. In other cases the differences are more fundamental, possibly representing radically divergent views on the nature of mathematics.

· Formal
The formal aspect concerns the mathematical theorems which can be proven using the foundation system. Foundation systems are partially ordered according to their proof-theoretic strength.
· Logical
In some cases it is possible to separate out a part of the system which is concerned with logic and independent of matters ontological.
· Ontological
Ontology is an important part of semantics, and differences both paradigmatic and in detail, is likely to be reflected in or caused by ontology.
· Dimensions
We consider five different characteristics or dimensions of logical foundation systems, sometimes clearly separated, other times not so. The formal the semantic, the logical, the ontological and the conceptual.
· Semantic
Gödel's first incompleteness theorem guarantees that truth and theorem hood do not coincide in any foundation system adequate for mathematics. It is therefore desirable to have an account of the intended meaning of the language independent of the definition of formal derivability.
· Conceptual
Over and above the logical and ontological features which determine the strength of the system there is likely to be some conceptual apparatus which provides the first stages in developing mathematics.

MATHEMATICS FIGURE

MATHEMATICS FIGURE
(klik disini untuk lihat lebih banyak lagi..)

THALES

• Masterpiece:
first mathematics.
formulating theorem / proposisi.
Thales theorem.

• Pupil:
Aristoteles


Al Khawarizmi (Al Goritm)

Al-Khawarizmi. born before 800 M and was die on 847 M. His Family name is Abu Abdullah Muhammad Ibnu Musa. He Define that we know as al-goritm.

For further chronological mathematic history, follow this link. make sure it sounds Click!

ACTIVITY REPORT

ACTIVITY REPORT AND MY RESULT IN SEARCHING, COLLECTING, STUDYING, ANALYZING, DISCUSING ABOUT HISTORY OF MATHEMATICS
DURING THE TIME

Arranged by:
LENNY PUSPITA DEWI
05301244122

MATHEMATICS EDUCATION
FACULTY OF MATHEMATICS AND SCIENCE
YOGYAKARTA STATE UNIVERSITY
2008

A. INTRODUCTION

Study coverage is called as mathematics history is especially in the form of investigation to coming from new finding in mathematics, in smaller scope in the form of investigation to standard mathematics notation and method in ancient time.
Before modern epoch and global spread over knowledge, the examples was written from new mathematics development have reaching is only in some place. Ancient mathematics article which have been found is Plimpton 322 (numerical Mathematics Babylonia of year 1900 is Pre-Christian), numerical Egypt mathematics Moscow mathematics sheet of year 1850 is pre-Christian), Rhind mathematics sheet (numerical Egypt mathematics of year 1650 is pre-Christian) and Shulba Sutra (numerical India mathematics of year 800 is pre-Christian).
All pertinent articles give all minds to what habit known as Pythagoras Theorem, what seen to be as result of most widespread and ancient mathematics development after elementary arithmetic and geometry.


B. ACTIVITY
Activity in studying history of mathematics is with searching article about mathematics history in internet. Besides also read book related to mathematics history, for example Howard Eves composition book.

C. RESULT
Some activity sample result in studying mathematics history shall be as follows:
1. Pythagoras Theorem and Triple Pythagoras
Habit assume Pythagoras as theorem inventor in right triangle which now by [is] totally referred [as] by the name of him, that hypotenuse square from right triangle [is] the amount of other square two sides. we have see this theorem [is] recognized [by] Babylonia people [at] a period of/to Hamurabi, more than last 1000 year, but first verification from this best theorem is which have been given by Pythagoras. There are still a lot of conjectures as Pythagoras evidence admits of to assay and in general evidence might be the pertained dissection type.
But Eudemian summary looking into triangle theorem come from Pythagoras followers. Because verification from this theorem needs knowledge about some nature of from parallel line, Pythagoras antecedent followers are also looked into meritorious in developing that theory.
Since Pythagoras epoch, many difference evidence for the theorem of Pythagoras. In secondary printing; mould book which entitle "The Pythagoras Proposition", E.S. Lcomis have collected and classify 370 evidences from famous theorem.

2. ARAB MATHEMATICS
Islam mathematician give good contribution which is either in geometry algebra area which its top there are resolving of geometry from cubic equation by Omar Kayyam.
Some Islam mathematicians show attention to undetermined. Analysis given by a evidence for the theorem of expressing that not possible to be found two positive integer which cubic amount is equal to rank three from an integer third.
Al-Karkhi is the first Arab writer which makes and prove theorem yielding amounts from square and cubic from n the first original number.

D. CONCLUSION
During the time we have studied mathematics. Learnt history of mathematics, we get motivation to continue to learn mathematics.

ROLE OF PYTHAGORAS IN MATHEMATICS DEVELOPMENT

ROLE OF PYTHAGORAS IN MATHEMATICS DEVELOPMENT

Arranged by:

LENNY PUSPITA DEWI

05301244122

MATHEMATICS EDUCATION

FACULTY OF MATHEMATICS AND SCIENCE

YOGYAKARTA STATE UNIVERSITY

2008

A. Introduction

I am sure that all of us know about Pythagoras with his popular theorem which side’s comparison of a right angled of triangle. If c is the symbol of sideways, and two sides other are a and b, so c² = a² + b².

Of course we guess that Pythagoras is the true scientist (more than his popular as philosopher). But all of your guess will be controvertible when you read this statement, that Pythagoras is very belief in superstitions, moreover he make his superstitious himself, of course with the philosophy principles. As a philosopher, he has many follower they name is Pythagorean.

Pythagoras is familiar as a mathematician. He gives important contribution for science development (especially mathematics).

B. Role of Pythagoras in Mathematics Development

Mathematics has an important role in the human’s life. Science and technology progress which very quick now, it caused by mathematics role. Can say that main base of science and technology is mathematics.

The discovery of paramount importance from Pythagoras is in the right angled of triangle equals quadrate of it’s sideways, moreover uncomparison theorem content nullifying again all Pythagoras philosophy. Because adequate insufficient arithmetical theory hit uncomparison, hence these matters progressively assure all mathematicians at the time that geometry have to be compiled separately with arithmetical.

And since then geometry have big influence to erudite method and philosophy. Deductive reasoning of axiomatic become main key in comprehending knowledge. This bring consequence, mathematics shall no longer study objects which directly can be arrested by human being senses. Mathematics substances are object think having the character of abstract. And become pure mathematics predominate.

For example, geometry deal with exact circles however is careful we were drawing by using meter, immanent of irregularity and imperfection. These prove view that any reasoning of exact only by reference to different ideal object with object sensory. Farther this bring view in philosophy that mind more especial than senses, and mind object more real compared to sensory perception objects.

Mystique doctrine which is concerning relation between eternity and time even also get support from pure mathematics, mathematics objects, like number, real even if, etemal in character and do not burst by time. Endless objects that way conception as the infinite mind. Hence do not surprise if emerging Plato doctrine that God is geometry expert. Different rationalistic religions with apocalyptic religion, since Pythagoras, and especially since Plato, have fully predominated by mathematical method and mathematics.

Mathematics combination and theology, beginning from Pythagoras, have inculcated characteristic at philosophy which have religion pattern of Greek, in the middle ages and modern-day till Immanuel Kant. But starting Plato era and Descartes happened circumstantial solidarity between reasoning and religion, between moral aspiration and logic attitude glorifying all which is eternal. This matter do not get out of influence predominate pure mathematics of that time.

BROTHER AND SISTER

ITS ME
and my brother
we love mathematics, my brother was an undergraduate of FMIPA UGM.