1. Egyptian and Babylonian Period (2000 B.C.- 500 B.C.) Introduction to early numeral systems Simple arithmetic Practical geometry Decimal and Sexagesimal numeral systems Sources: Ahmes (Rhind) papyrus; Moscow papyrus; Babylonian tablets ( There are 110 problems in both of these papyri and these are very simple numerical problems) From Egypt we have only two papyrus: One of them the Rhind papyrus that, I will talk about, was copied around the 17th century BC by a scribe named Ahmes.

2. Ahmes papyrus is about 32 cm wide and 200 cm long, which is the most valuable source of information we have about Egyptian Mathe- matics. The text begins by stating that the scribe "Ahmes" is writing it (in about 1600 BC, and is the earliest named individual in the history of mathematics) but that he has copied it from "ancient writings," which probably go back to at least 2000 BC.

4. Although there is some strictly practical mathematics on the papyrus, including calculations needed for surveying, building, and accounting, some of which involve Egyptian fractions, many of the problems in the Rhind papyrus take the form of arithmetic puzzles. One of these is: “Seven houses contain seven cats. Each cat kills seven mice. Each mouse had eaten seven ears of grain. Each ear of grain would have produced seven hekats of wheat. What is the total of all of these?”

5. Solution: 7 houses +7^2 (49 cats) + 7^3 (343 mice) + 7^4 (2,401 ears of grain) + 7^5 (16,807 hekats (a measure) of grains) = 7+ 7^2 +7^3+7^4+7^5 = (1-7^6) / (1-7) = 19 607 Let us examine Problem 21 of the Rhind Papyrus : Complete 2 / 3 and 1 / 15 to 1.

6. In modern terms, this asks for a fraction x such that 2 / 3 + 1 / 15 + x = 1 If we multiply each fraction by 15, we obtain 10 + 1 + y = 15. This is called the "red auxiliary" equation since Ahmes wrote this equation in red ink as : “complete 10 and 1 to 15“. Now the answer to the red auxiliary equation is 4.

7. Other important source of Egyptian mathematics from this period is the Moscow papyrus. The Moscow papyrus contains only about 25, mostly practical, examples. The author is unknown. It was purchased by V. S. Golenishchev and sold to the Moscow Museum of Fine Art. Origin: 1700 BC It is 518 cm long and about 7.5 cm wide.

8. Problem 14. Volume of a frustum. The scribe directs one to square the numbers two and four and to add to the sum of these squares the product of two and four. Multiply this by one third of six. "See, it is 56; you have found it correctly." What the student has been directed to compute is the number :

10. Here is the modern version of the picture and a perspective drawing: The general formula for a frustum was evidently known to the Egyptians. It is:

11. Taking b1=0 we get This was also known by Egypts. Egyptian Number System The Egyptians used a base 10 number system. ( This is common in many ancient cultures, due to the fact that we have 10 fingers. ) The had special symbols for 1, 10, 100, 1,000, 10,000, 100,000, and 1,000,000 as seen in the table below.

12. The normal direction of writing was right to left. Primarily only unit fractions (1/n) existed in the Egyptian number system. Such fractions were written by placing a "mouth" above the number representing the denominator :

13. Some examples of Egyptian hieroglyphic numbers 1275 BC : Battle of Kadesh took place between the forces of the Egyptian Empire under Ramses II and the Hitit Empire under Muvatalli II Both sides claimed victory

14. Ramses II

16. Greek Mathematics Period (500 B.C.-A.D. 500) Development of deductive geometry (Thales, Pythagoras) Start of number theory, Discovery of irrational numbers, Geometric solution of Quadratic equations (Pythagorean school) Systematization of deductive logic (Aristotle 340 B.C.) Geometry of conic sections (Apollonius,225 B.C.) Axiomatic development of geometry, Algebraic identities (Euclid,300 B.C) Germ of the Integral calculus (Archimedes, 225 B.C)