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Gottfried Wilhelm Leibniz was originally accused of plagiarizing Sir Isaac Newton's unpublished work, but is now regarded as an independent inventor of and contributor to calculus.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