Challenging Problems in Algebra (eBook)
Mathematics is one of many oldest sciences in human history. In ancient times, Mathematics was defined while the science of numbers and shapes. Mathematics, like other branches of science, has evolved over time; it is no longer possible to explain it in several sentences. What I have to state now is going to be words that emphasize its various aspects, as opposed to describe mathematics. In one aspect, mathematics is an art like painting and music. The great majority of mathematicians perform it being an art. Out of this standpoint, the fact a work done, a developed theory works in one way or another other than mathematics does not concern them much. What matters to them may be the depth of the job done, the novelty of the methods used, the aesthetic value and the usefulness of mathematics in itself. Mathematics, in another aspect, is just a language. If the objective of science is the universe; If it’s to know, rule and direct everything in the universe, we should be able to read the book of nature. The book of nature is written in the language of mathematics, with the highly cited words of Galile; its letters are shapes of geometry. In order to understand and interpret them, we have to know the language of mathematics. In another aspect, mathematics is definitely an intellectual game like chess.
Some mathematicians also view it as a game. Mathematics is merely a tool for the user. After entering it, we understand and perceive what mathematics is inside our knowledge and in the direction of our interest. Mathematics has become far beyond the dimensions any human can rule. Therefore, I do not genuinely believe that those that handle mathematics are far more than we understand and perceive it from mathematics compared to the blind touched net understands and perceives the elephant. The phrase mathematics, for initially, BC. It had been utilized by the members of the Pythagorean school in the 550s. His entry into the written literature, with Plato BC. It absolutely was in the 380s. The phrase meaning is “what must be learned”, that’s, information. In the years before these dates, instead of the word mathematics, words which means that geometry, equivalent to it in geometry or old languages were used.
It’s difficult to express anything definite about where and how mathematics started. If we take documents which are not centered on archaeological findings that require interpretation, but open enough to require interpretation, We are able to say so it started between 3000 and 2000 in Egypt and Mesopotamia. Based on Heredotus (485-415 BC), mathematics started in Egypt. You may already know, 97% of the Egyptian lands aren’t suitable for agriculture; It’s the 3% portion that provides life to Egypt and forms the Nile delta. Therefore, these lands are really valuable. However, by the end of the floods brought on by the Nile river every year, the boundaries of the landowners’lands become obscure. Considering that the landowners also pay taxes in proportion to the land they own, after each flood, the “geometricists” of the state, who are responsible for these works, should arrived at take the necessary measurements and provide the landowners just as much land as they’d in the previous year. Herodotus says that geometry has begun to emerge as a result of the measurements and calculations. Another opinion about the birth of mathematics is usually the one put forward by Aristotle (384-322 BC). In accordance with Aristotle, mathematics was born in Egypt. However it was born out from the boredom of clergymen and priests, not the need for measurement-calculation due to Nile floods. At that time, the sole intellectual class of countries such as for instance Egypt was the priest class. Since the livelihood with this class is supplied by the public or their state, they have much time and energy to share with intellectual pursuits. To help keep them busy, they invented geometry and arithmetic, the mathematics of the period, in the same way others invented games like chess, bridge, and go&hellip ;.These two views may be true; priests wished to simplify the work of the geometric, or they discovered how to calculate the areas of some geometric shapes such as for instance triangular and trapezoidal to check on that the distribution was fair, and in this way generated the birth of geometry.
We will divide the written history of mathematics into five periods. The first period will soon be Egypt and Mesopotamia; this period BC In 2000s BC. It’ll cover a period of 1500-2000 years between 500s. The next period, BC. 500-M.S. It’ll cover a period of 1000 years, called the Greek Mathematics period, between 500 years. The 3rd term, M.S. It will cover a 1200-year period from the 500’s until the beginning of calculus and will mainly cover European mathematics in the Hind, Islam and Renaissance era. The fourth semester will cover the classical mathematics era, called the golden age of mathematics, dating from 1700-1900. The time scale we are residing in, dating back to the early 1900s, called the age of modern mathematics, could be the fifth period. I will attempt to provide details about the development of mathematics for the reason that period, contributing mathematicians, the place of mathematics in social life and the basic options that come with mathematics for the reason that period.
We will start the first semester with Egyptian mathematics. Written documents about ancient Egyptian mathematics and generally Egyptian history – I don’t mean the remains of archaeological works – are almost nonexistent. There are two main reasons for this. The very first is that the ancient Egyptians wrote the writing on papyrus; The next reason could be the 3 big fires of the Alexandria libraries, the final of these fires happened throughout the conquest of Egypt by 641 Muslims, the written documents disappeared. Papyrus is the leaves of a reddish, reed type plant growing in the Nile delta, an average of 15-25 meters long and 30-50 inches wide. These leaves were used to publish text rather than paper after cutting, joining, pressing and undergoing some simple operations. Words in western languages such as “Paper”, “papier” are derived from the term papyrus. The typical lifespan of a papyrus is 300 years; 300 years later, it’s flaky due to moisture, heat and similar reasons. Up to now, two papyrus linked to mathematics appear to own been hidden under exceptional circumstances. The main resources of our understanding of Egyptian mathematics are these two papyri. The first of the papyrus is really a 6-meter long and 35-cm wide papyrus called the Ahmes (or Rhind) papyrus. This papyrus, BC. You’re a puree written in 2000s, BC. It is just a copy published by a “mathematician” named Ahmes in the 1650s. This papyrus was bought by the Irish antiquarian H. Rhind in the 1850s, now in the British museum. This papyrus is a book written to instruct math. In the introduction part, following a few exercises given to teach operations with fractional numbers, 87 questions are shown making use of their solutions. These are the kind of questions people can encounter in everyday life, such as for instance sharing account, interest calculation, or finding the location of some geometric shapes. This is more or less our 8th grade mathematics. The next papyrus, referred to as the Moscow papyrus and now in the Moscow museum, is also BC. It is just a booklet written in the 1600s. This papyrus contains 25 questions. These questions are of the kind of questions in the Ahmes papyrus, aside from the two. When it comes to other two questions, one may be the calculation of the amount and section of the surface of the sphere part cut by way of a plane. One other could be the question of finding the amount of a pyramid cut by way of a plane. Both questions were solved correctly. These two questions are accepted because the pinnacle of Egyptian mathematics. The Egyptians realized that the area of the circle was proportional to its diameter and found the number of pi to be 4x (8/9) squared, ie 256/81 = 3.16. It’s understood that Egyptian mathematics has remained at this level for 2000 years and has not made any significant progress.
B.C. 600s would be the years when the Persians started initially to dominate the middle east. B.C. By the 550s, Persians are the sole rulers of the whole middle east, including Anatolia and Egypt. The Persians organize three trips to Greece between 500-480 BC; They captured Athens in 480, but burned it, per year later, in 479, Greeks expelled the Persians from Greece. This date BC. 479 could be the date that has been accepted as the beginning of Greek civilization. This date is the start of a really bright period in science, art and literature. Greek mathematics actually started earlier than this period. Two people, Tales (624-547 BC) and Pythagoras (569-475 BC), are considered to be the daddy of Greek mathematics. Tales Milet (Aydın) was also born. It is known he visited Egypt, stayed there for some time and learned geometry in Egypt. While in Egypt, it’s described in books where he calculates the height of the truly amazing pyramid by measuring the size of the shadow of the truly amazing pyramid, multiplying this number by the ratio of its length to the length of the present shadow. After returning to Tales Milet, he taught them geometry by forming an organization around him to instruct what he learned. It’s assumed that abstract proof predicated on reasoning, that is not centered on mathematics – experimental verification, entered into Tales. Additionally, Tales is the one who is considered the initial philosopher in human history. He was born on the island of Pythagoras Samos (Samos), another father of Greek mathematics. Pythagoras stayed with Tales for a while, visited Egypt following his advice, learned geometry there, visited Egyptian temples, learned religious information, and was taken fully to Babylon by capturing the Persians during the occupation of Egypt by the Persians. it is known. During his 5 years in Babylon, he learned mathematics, music and religious information, and after time for Samos, he created a school and tried to instruct the people he gathered around. For political reasons, BC. He left 518 Samos, settled in southern Italy, in the town of Crotone, where he created a semi-mystical-semi-scientific, cult-like school. The senior people of this school called “mathematics” live together and they are connected to one another with oath. The second group consists of students attending school. Pythagoras school is based on number cult. According for them, everything could be reduced to numbers; It posseses an unusually perfect harmony among numbers, and harmony is just a reflection of the divine harmony. Known numbers for that day are integers indicating the plurality such as for instance 1,2,3,…; and kes, ¾,… will be the fractional numbers that indicate the ratio of the part to the whole. The emergence of irrational numbers with the theorem known as the Pythagorean theorem (the square of the proper sides of a right triangle equals the square of the hypotenuse) put the Pythagorean school in a strong crisis. The discovery of irrational numbers is the first major crisis of mathematics. Most of the members of the Pythagorean school were massacred by way of a raid led by way of a big cyber named Cylon. Pythagoras saved his life, but after many years he died. Pythagoras’thoughts, the Pythagorean school lived for many years under this or that name. As may be understood from this information, Egyptian and Mesopotamian mathematics are the foundation of Greek mathematics.