Math 460 exam 1 part B

State two related significant historical mathematical problems that cannot be solved related to the material covered in this test that we discussed in class. State your problems in self-contained form. (e.g., Construct with straightedge and compass the angle bisector for an arbitrary angle).

1) Prove that the diagonal and the side of a square are commensurable. 2) Prove that 2 is rational.

Briefly describe the four main mathematical features of the Babylonian numeration system.

1) The system lacked the notion of complete cipherization. The numerical notation used two-wedge characters ( and to represent the digits 1-59. 2) It is a partial positional numeration system with base 60 (Sexagesimal). The value of each digit depends on its position on the number. 3) It lacked the concept of zero as a number. Therefore, it did not have a symbol for zero. 4) The system lacks a sexagesimal point and, therefore, it is difficult to distinguish the fractional part of a number from its whole number.

____________ was the author of the Rhind papyrus.

Ahmes

Scholars claim that Babylonians knew how to solve quadratic equations. To what extent is this statement true?

Babylonians knew a procedure to find, when existed, the positive solution of a quadratic equation. However, they did not consider either negative numbers or zero as solutions.

Briefly explain each of the four stages in the evolution of zero described by the authors of our textbook, Berlinghoff & Gouvêa.

Berlinghoff & Gouvêa proposed four stages in the evolution of zero: 1) zero as a place holder, 2) zero as a number, 3) zero as an algebraic tool, and 4) zero as a generalized abstract additive element of rings, fields, and integral domains. 4 1) Zero began its life as a space, or place holder, a symbol for something skipped or an empty space in the Babylonians (before 700 BC) numeration system. 2) By the 9th century A.D., the Hindus began thinking of zero (sunya, the absence of quantity) as a number on its own right and Indian mathematicians began performing arithmetical operations involving zero. 3) Thomas Harriot and Rene Descartes used zero in manner that will change the theory of equations. Harriot proposed solving equations by moving all the terms of an equation to the one side of the equal sign transforming the equation to the form [some polynomial] = 0. Descartes used extensively Harriot's principle for solving equations: If the product of two numbers is zero, then one of them must be zero. 4) Zero was generalized to be the identity additive element of rings and fields. In addition, Harriot's principle was generalized to characterize the algebraic structure known as integral domain.

________, _________, and ________ were three brilliant Indian mathematicians who treated zero as a number.

Brahmagupta, Mahavira, and Bhaskara

_________ was the author of the book in which the symbols =, >, and < were used for the first time.

Harriet

_________ and _______ were two European mathematicians whose work transformed the theory of equations by treating zero as a number.

Harriot and Descartes

Our current method for writing numbers is called

Hindu-Arabic Styste

The _________ developed the concept of zero as a number and influenced European mathematics.

Hindus

______________ was the mathematician who introduced the use of the symbol · for multiplication.

Leibniz

Is the Mayan numeration system a true vigesimal positional numeration system? Explain your response.

No, a true vigesimal positional numeration system has base 20 and, therefore, the positional values are 20 0 , 201 , 20 2 , 20 3 , etc. The positional values of the Mayan numeration system, on the other hand, are 20 0 , 201 , 18(20), 18( 20 2 ), etc.

___________ was the school of thought that believed that all human affairs can be explained in terms of intrinsic properties of [whole] numbers.

Pythagorean school

__________________ discovered that there exits irrational numbers.

Pythagoreans

_________ was the mathematician who used for the first time the symbols +, - and = (in an elongated way) in an English book (The Whetstone of Witte)?

Recorde

______________ was the mathematician who popularized the use of decimal fractions in Europe.

Stevin

Discuss the two historical and mathematical significances of the discovery that the diagonal and the side of a square are incommensurable.

The Greeks believed that any two given segments were always commensurable. Many geometric theorems (e.g., the theory of proportion and similarity of triangles) were based on this incorrect assumption. The discovery of incommensurable segments caused Greeks to rebuild theories based on such assumption. The belief that any two segments were commensurable implied that all numbers were rational, that is, any number could be represented as the quotient of two natural numbers. This supported their philosophy that all human affairs can be explained in terms of natural numbers. Their whole philosophy collapsed with the discovery of incommensurable segments (that is, irrational numbers).

______________________________ is the most known ancient Chinese mathematical book that dates back to 100 B.C.

The Nine Chapters of the Mathematical Art

Explain the following statement "Egyptian arithmetic was essentially additive."

To perform basic arithmetic operations, Egyptians used addition. For example, multiplication and division were performed by a series of additions involving doubling and halving.

The device or notation where a single symbol or mark is used to represent a collection of like symbols in a numeration system is called

cipherization

The five main documents from which we know about Egyptian mathematics are:

the Rhind papyrus, the Moscow papyrus, the Berlin papyrus, the Cairo Mathematical papyrus, and the leather roll.

The four stages in the development of a numeration system are:

to the use of one-to-one correspondences as a counting tool, the notion of grouping, the principle of position, and the notion of cipherization.