Concept of Zero
The introduction of zero into the decimal system in 13th century was the most significant achievement in the development of a number system, in which calculation with large numbers became feasible. Without the notion of zero, the descriptive and prescriptive modeling processes in commerce, astronomy, physics, chemistry, and industry would have been unthinkable. The lack of such a symbol is one of the serious drawbacks in the Roman numeral system. In addition, the Roman numeral system is difficult to use in any arithmetic operations, such as multiplication.
The notion of zero was introduced to Europe in the Middle Ages by Leonardo Fibonacci who translated from Arabic the work of the Persian (from Usbekestan province) scholar Abu Ja'far Muhammad ibn (al)-Khwarizmi. The word "algorithm," Medieval Latin 'algorismus', is a contamination of his name and the Greek word arithmos, meaning "number, has come to represent any iterative, step-by-step procedure. Khwarizmi in turn documented (in Arabic, in the 7th century) the original work of the Hindu mathematician Ma-hávíral as a superior mathematical construction compared with the then prevalent Roman numerals which do not contain the concept of zero. When these scholarly treatises were being translated by European accountants, they translated 1, 2, 3,.. upon reaching zero, they pronounced, "empty", Nothing! The scribe asked what to write and was instructed to draw an empty hole, thus introducing the present notation for zero.
Dividing by Zero Can Get You into Trouble: If we persist in retaining such errata in our educational texts, an unwitting or unscrupulous person could utilize the result to show that 1 = 2 as follows:
(a).(a) - a.a = a2 - a2
for any finite a. Now, factoring by a, and using the identity
(a2 - b2) = (a - b)(a + b) for the other side, this can be written as:
a(a-a) = (a-a)(a+a)
dividing both sides by (a-a) gives
a = 2a
now, dividing by a gives
1 = 2, Voila!
This result follows directly from the assumption that it is a legal operation to divide by zero because a - a = 0
Emergence of Calculus
In the course of developing a precise mapping of the lunar eclipse, Aryabhatta was obliged to introduce the concept of infinitesimals - i.e. tatkalika gati to designate the infinitesimal, or near instantaneous motion of the moon, and express it in the form of a basic differential equation. Aryabhatta's
Trigonometry
Trigonometry is the branch of mathematics that deals with triangles, circles, oscillations and waves; it is absolutely crucial to much of geometry and physics. You will often hear it described as if it was all about triangles, but it is a lot more interesting than that. For one thing, it works with all angles, not just triangles. For another, it describes the behaviour of waves and resonance, which are at the root of how matter works at the most fundamental level. They are behind how sound and light move, and there are reasons to suspect they are involved in our perception of beauty and other facets of how our minds work - so trigonometry turns out to be fundamental to pretty much everything. Any time you want to figure out anything to do with angles, or turning, or swinging, there's trigonometry involved.
Tan
Trigonometry basics are often taught in middle school or junior high school as part of the basic math curriculum. More in depth trigonometry is sometimes taught in high school either as a separate course or as part of a precalculus course. The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics such as Fourier analysis and the wave equation, which are in turn essential to many branches of science and technology. Spherical trigonometry studies triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. It is fundamental to astronomyand navigation. Trigonometry on surfaces of negative curvature is part of Hyperbolic geometry.
Ancient Algebra Is divided Into Two Main Parts:-
Hindu Algebra
Arabic Algebra
The successors of the Greeks in the history of mathematics were the Hindus of India. The Hindu civilization dates back to at least 2000 B.C. Their record in mathematics dates from about 800 B.C., but became significant only after influenced by Greek achievements. Most Hindu mathematics was motivated by astronomy and astrology. A base ten, positional notation system was standard by 600 A.D. They treated zero as a number and discussed operations involving this number.
The Hindus introduced negative numbers to represent debts. The first known use is by Brahmagupta about 628. Bhaskara (b. 1114) recognized that a positive number has two square roots. The Hindus also developed correct procedures for operating with irrational numbers.
They made progress in algebra as well as arithmetic. They developed some symbolism which, though not extensive, was enough to classify Hindu algebra as almost symbolic and certainly more so than the syncopated algebra of Diophantus. Only the steps in the solutions of problems were stated; no reasons or proofs accompanied them.
The Hindus recognized that quadratic equations have two roots, and included negative as well as irrational roots. They could not, however, solve all quadratics since they did not recognize square roots of negative numbers as numbers. In indeterminate equations the Hindus advanced beyond Diophantus. Aryabhata (b. 476) obtained whole number solutions to ax ± by = c by a method equivalent to the modern method. They also considered indeterminate quadratic equations.
In the 7th and 8th centuries the Arabs, united by Mohammed, conquered the land from India, across northern Africa, to Spain. In the following centuries (through the 14th) they pursued the arts and sciences and were responsible for most of the scientific advances made in the west. Although the language was Arabic many of the scholars were Greeks, Christians, Persians, or Jews. Their most valuable contribution was the preservation of Greek learning through the middle ages, and it is through their translations that much of what we know today about the Greeks became available. In addition they made original contributions of their own.They took over and improved the Hindu number symbols and the idea of positional notation. These numerals (the Hindu-Arabic system of numeration) and the algorithms for operating with them were transmitted to Europe around 1200 and are in use throughout the world today.Like the Hindus, the Arabs worked freely with irrationals. However they took a backward step in rejecting negative numbers in spite of having learned of them from the Hindus.In algebra the Arabs contributed first of all the name. The word "algebra" come from the title of a text book in the subject, Hisab al-jabr w'al muqabala, written about 830 by the astronomer/mathematician Mohammed ibn-Musa al-Khowarizmi. This title is sometimes translated as "Restoring and Simplification" or as "Transposition and Cancellation." Our word "algorithm" in a corruption of al-Khowarizmi's name.The algebra of the Arabs was entirely rhetorical.They could solve quadratic equations, recognizing two solutions, possibly irrational, but usually rejected negative solutions. The poet/mathematician Omar Khayyam (1050 - 1130) made significant contributions to the solution of cubic equations by geometric methods involving the intersection of conics.Like Diophantus and the Hindus, the Arabs also worked with indeterminate equations.
Hindu Algebra
Arabic Algebra
Hindu Algebra
The Hindus introduced negative numbers to represent debts. The first known use is by Brahmagupta about 628. Bhaskara (b. 1114) recognized that a positive number has two square roots. The Hindus also developed correct procedures for operating with irrational numbers.
They made progress in algebra as well as arithmetic. They developed some symbolism which, though not extensive, was enough to classify Hindu algebra as almost symbolic and certainly more so than the syncopated algebra of Diophantus. Only the steps in the solutions of problems were stated; no reasons or proofs accompanied them.
The Hindus recognized that quadratic equations have two roots, and included negative as well as irrational roots. They could not, however, solve all quadratics since they did not recognize square roots of negative numbers as numbers. In indeterminate equations the Hindus advanced beyond Diophantus. Aryabhata (b. 476) obtained whole number solutions to ax ± by = c by a method equivalent to the modern method. They also considered indeterminate quadratic equations.
Arabic Algebra
Arithmatic
The gradual development of Hindu-Arabic numerals independently devised the place- value concept and positional notation, which combined the simpler methods for computations with a decimal base and the use of a digit representing zero. This allowed the system to consistently represent both large and small integers. This approach eventually replaced all other systems. In the early 6th century AD, the Indian mathematician Aryabhata incorporated an existing version of this system in his work, and experimented with different notations. In the 7th century, Brahmagupta established the use of zero as a separate number and determined the results for multiplication, division, addition and subtraction of zero and all other numbers, except for the result of division by zero. His contemporary, the Syriac bishop Severus Sebokht described the excellence of this system as "...valuable methods of calculation which surpass description". The Arabs also learned this new method and called it hesab.
The basic arithmetic operations are addition, subtraction, multiplication and division, although this subject also includes more advanced operations, such as manipulations of percentages, square roots,exponentiation, and logarithmic functions. Arithmetic is performed according to an order of operations. Any set of objects upon which all four arithmetic operations (except division by zero) can be performed, and where these four operations obey the usual laws, is called a field.
