Ancient Mathematics

Egyptian Geometry: Approximation

Approximating the Area of a Quadrilateral

• Egyptian Formula• A = 1.4 (a+c)(b+d)• Where a,b,c,d are the lengths of the

consecutive sides• Is the formula correct? When will this be

correct? Prove!

Approximating the area of a circle

• Example of Egyptian problems: A round field of diameter 9 khet. What is its area?

• Solution: Take away 1/9 of the diameter, namely 1; the remainder is 8; it makes 64. Therefore it contains 64 setat of land.

• The formula here is A = (8d/9)^2.• If this was the formula, what was the

approximate value of ∏, pi.

Babylonian: Approximating the value of ∏

• The circumference of a circle was found by taking three tomes its diameter. What was their value for pi?

Approximating the area of a trapezoid

• A = ½ (b + b’)h• b and b’ are the parallel sides; h is the height.• Is this formula correct?

Approximating the Volume of a Truncated Pyramid

• V = h/3 ( a^2 + ab + b^2)• Where h is the height and a and b are the

parallel side• Is this correct? Compare it the formula V = h/3

( B + b + (√Bb))

Approximating the area of a square prism

• V = h/3 x a^2• Is this correct? Prove

Word Problems

• Refer to the examples to be given (pp.61-63)

Babylonian Mathematics: A tablet of reciprocals

• A pair of numbers whose product is 60

4 15

5 12

6 10

8 7;30

9 6;40

10 6

12 5

16 3;45

18 3;20

Analysis

• 8 7; 30• -the left side is the quotient; the right is the

reciprocal based on the sexagesimal system

Complete the table

• Rule: product must be 50

12

4

7

8

9

11

13

The Babylonian Treatment of Quadratic Equations

• Their formula was

• Prove the formula

Plimpton 322: A tablet concerning number triplets

• X^2 + y^2 = Z^2 (similar to a^2 + b^2 = c^2)• Z^2 – y^2 = x^2 ( divide by x)• If α = z/x and β= y/x• α ^2 – β^2 =1 (factor out)• 1 to be represented as (m/n)(n/m)• Apply APE, α = ½ (m/n + n/m); β = ½ ( m/n – n/m)• The formula are α = (m^2 + n^2 )/2mn; β =

(m^2-n^2)/2mn• But y = β x, z = α x and x = 2mn• X= 2mn, y = m^2-n^2; z = m^2 + n^2

Verify the formula: X= 2mn, y = m^2-n^2; z = m^2 + n^2

The Cairo Mathematical Papyrus

• Solve x^2 + y^2 = 169; xy = 60