The story of zero is a classic story in the history of mathematics. An idea was born; After several places and centuries of circulation, it was refined and spread; It became part of an international mathematical culture. Mathematics is a masterpiece that people all over the world can be proud to share.
Ancient Indian mathematics can be roughly traced back to the papyrus of ancient Egypt and the clay tablets of ancient Babylon, and a fascinating and unsolved problem is the level of contact between these men. There must be suspicions that there was an interaction between ancient Indian and Chinese mathematics, but the magnitude and tendencies of this interaction are probably unknown.
In any case, the ancient Indians were very good at mathematics. Among them, their most important achievement was the development of trigonometry. Much of their work in this area seeped into later Arab culture and into Europe in the 15th century. The contemporary world has benefited greatly from the great ancient Indian triangularists.
The ancient Indians also solved some very wonderful problems with algebraic classes, even though there was no symbology at the time. One of these questions should be attributed to the Bhashkara, also known as Bhaskara or "Bhashkara Teacher", who lived around 1150 AD. For example, there is a problem of finding two integers such that the square of the first number is 61 times less than the square of the second number by 1. In modern notation, this is equivalent to finding two numbers x and y such that 61x² = y² -1. This problem was raised again in 17th-century Europe, and it was a considerable test for mathematicians, and Bhashgara gave the correct solution to the problem. His answer is x = 226 153 980, y = 1 766 319 049, which is hard not to be surprised.
The ancient Indians also left us with many enlightening geometric results, the most notable of which is the Brahmagupta formula for finding the area of a circle with a quadrilateral. A cyclic quadrilateral is a quadrilateral that is attached to a circle, as shown in Figure O-6. Brahmagupta, an astronomer and mathematician in the 7th century AD, said that the area of any quadrilateral with sides of a, b, c, d can be given by the following formula:
其中 s = ½(a + b + c + d), 称为这个四边形的半周长。
To see how it works, consider the rectangle with sides of a and b shown in Figure O-7. Of course, by making the diagonal intersection of the rectangle O the center of the circle, you can make the circumscribed circle of the rectangle. Since a rectangle can be a circle with a quadrilateral, we can use the Brahmagupta formula. Therefore there is
所以有 s − a = (a + b) − a = b 及 s − b = (a + b) − b = a。 因此, 这个矩形的面积是
Of course, we don't need to use such a powerful weapon as the Brahmagupta formula to find that the area of a rectangle is equal to the product of its length and width. It's like using a combine harvester to cut a piece of grass.
However, the following example is not so rudimentary, it is taken from an ancient Indian textbook. Here we are asking for the area of the circumscribed quadrilateral in a circle with side lengths a = 39, b = 60, c = 52, d =25, as shown in Figure O-8. Without the help of the Brahmagupta formula, this would have been very difficult; With the help of the Brahmagupta formula, the answer will soon be found. The half-circumference of the circumference of the circumferent of the circumferent of the circumferent of this circle is
Hence the area is
There is an interesting corollary to the Brahmagupta formula. For Figure O-9, if we slide vertex D to vertex C along the circle, the circle becomes the triangle ABC with the quadrilateral inside. In this transformation, the side length is 0, so the triangle can be seen as a "degenerate" quadrilateral, so its area is
Now, s is the half circumference of △ABC. As some readers may recognize, this formula is Helen's formula for the area of a triangle, named after the ancient Greek mathematician who gave a clever proof of it about 75 AD. Thus, the Brahmagupta formula is an extension of Helen's formula to the circumscribed quadrilateral of the circle. This is a striking example in geometry.
We have already briefly mentioned one of the greatest achievements of ancient Indian mathematics: the introduction of zero within the decimal system. It's impossible to date exactly when this idea was born, but it probably dates back to the middle of the first millennium AD. The documents and inscriptions from this period show zero very clearly, and it looks a lot like the zero we have today. This invention is very useful, not only as a theoretical structure, but also as a computational tool. So, it was because the Indians adopted a digital system that introduced zero that their technology was quickly adopted by the Arabs with whom they interacted. By the end of the first millennium, Arab scholars had written a book on the wonderful "Indian arithmetic."
It was through the Arabs that these ideas eventually flowed westward into Europe. One of the most crucial steps was the Book of Abacus, published by Leonardo of Pisa in 1202. Leonardo is Fibonacci as we know it today, who spent most of his youth in North Africa, where he studied Arabic and studied Arabic mathematics. In this way, he mastered what is now known as the Indo-Arabic numeral system. Fibonacci's book brought these ideas to the academic center of Italy, and from here they soon spread to the European continent.
The story of zero is a classic story in the history of mathematics. An idea was born; After several places and centuries of circulation, it was refined and spread; It became part of an international mathematical culture. Mathematics is a masterpiece that people all over the world can be proud to share.
Artist: [美] 威廉·邓纳姆(William Dunham)译者:冯速
Get a glimpse of the great theorems, puzzles, and arguments that the world of mathematics cannot fail to talk about
Outline the whole picture of mathematics and make the knowledge in the classroom clearer and more understandable
Dunham, a great master of mathematics science, is dedicated to readers who love mathematics and are simply curious about what mathematics is
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Article source: Turing News