HUMAN IMPACT AND THE
SOCIETAL STRUCTURING
OF MATHEMATICS
Babylonian Civilization
■ Is a state in Mesopotamia and the most powerful state in ancient world.
■ It is known for their engineering, architecture and was founded more than 4,000
years ago
■ It is also known for their “Hanging Garden” built by Nebuchadnezzar II.
■ The Law Code of Hammurabi
■ They are also great astronomers and studied stars, using this they Calendars
consisting of Lunar Month.
■ Also known as part of Iraq in today’s modern world.
Babylonian Civilization
■ One of the earliest known sites of civilization to develop mathematics.
■ The Babylonian system of mathematics was a sexagesimal (base 60) numeral
system.
■ “Completing the Square”
■ “Pythagorean Triples”
Complete The Square
■ The problem can be visualized geometrically as a process of finding a missing area associated
with a square.
■ The area of a rectangle is 16 and its length exceeds its breadth by 6. What is the breadth?
■ In modern form it is written as x² + 6x = 16 be presented geometrically:
x 6
x x² + 6x x
= 16 units
Complete The Square
■ In doing the process we can slice the rectangle vertically into two parts and distribute it to the
sides to make is look like almost a square.
x² 3x6x3x x² 3x
3x
Complete The Square
■ To complete the square, we can put a small square at the corner to complete the square.
Now that the shape became a square
again we can say that
16 + 9 = 25
x² 3x
6
And to solve for the breadth or the x
we can say that;
6
3x (x+3)² = 25
(x+3)² = 25
√(x+3)² = √25
x+3=5
x=5–3
x=2
Pythagorean Triples
■ A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c².
■ Pythagorean Triples are translated from Plimpton 322 Clay Tablet.
■ In choosing numbers p, q where:
1. p>q
2. p, q are relatively prime
3. p and q have different parity (means p has to be odd and q has to be even, vice versa)
The formulas in getting the triples are:
a = p² - q²
b = 2pq c
c = p² + q² a
b
Pythagorean Triples
■ If p = 2 and q = 1, determine the triples that will satisfy the Pythagorean
Theorem (a² + b² = c².
a = p² - q² b = 2pq c = p² + q²
a = (2)² – (1)² b = 2 •(2)(1) c = (2)² + (1)²
a=4–1 b = 2•2 c=4+1
a=3 b=4 c=5
Pythagorean Triples
This are the
Pythagorean
Triples that can
help us in
determining the
hypotenuse of a
right triangle.
Egyptian Civilization
■ Ruled by a god-king.
■ Also known for their Great Pyramids for being the only wonder of the ancient world that has
survived to the present day.
■ First people to develop known to employ sun dial and water clocks to record time.
■ Is known for its stupendous achievements in a whole range of fields, including art and
architecture, engineering, medicine and statecraft.
■ Egyptians have a process in multiplication and division.
■ They also develop the “Rule of Three” were known today as Ratio and Proportions.
■ “Rule of False Position” is also developed and known today as Regula Falsi.
Ancient Egyptian Multiplication
■ The operation of multiplication is performed by a doubling process.
■ For example, 25 × 9
25 × 9
1 9
2 18
25 4 36 225
8 72
16 144
Ancient Egyptian Division
■ The operation of division was viewed as the opposite of multiplication.
■ For example, 91 ÷ 7
91 ÷ 7
1 7
2 14 91
13
4 28
8 56
Ancient Egyptian Division with
Remainder
■ There are times when division resulted with a remainder. Egyptians used fractional
multiplication to obtained a quotient with fractional part.
■ For example, 35 ÷ 8
1 8
_ _ 2 16
4+4+8 4 32 35
(4.375) _
2 4
_
4 2
_
8 1
Rule of Three
■ Was used when three numerical values were known and a fourth is unknown.
■ For example; Find the number of loaves of bread of strength 45 which are equal to 100
loaves of strength 10.
In modern notation:
45 10
——— = 450 loaves
———
x 100
Rule of False Position
■ Situations involving linear equations in one variable are solved.
■ Known in Europe by its Latin name “Regula Falsi”.
■ For example: An unknown and a ⅐ the unknown equals 19. What is the value of the
unknown?
In modern notation:
x
x + ——— =
19
7
