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11 Cards in this Set

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describe the Egyptian hieroglyphic writing system and its usage
read left to right, base 10, symbols only for powers of 10 (ie, if 10 = b, 1 = a, 23 is written aaabb), used for monuments, contained a character for 0 (as in the 0th floor). Dates to 3000 BCE.
describe the Egyptian hieratic writing system and its usage
read left to right, base 10, a ciphered system with symbols for 1,...9, used in everyday situations, 0 is not a number in this system. Dates to 3000 BCE
where did historians learn about the Egyptians' knowledge of math?
From the Rhind papyrus and Moscow Papyrus, both inscribed around 1850 BCE
In the Egyptian number system, how were fractions represented?
Unit fractions such as 1/3 were represented as <span style="text-decoration: overline">3</span>, and fractions such as 2/3 were represented by a 3 with two bars on top (html skillz fail here).
In the Egyptian system, how would they multiply 18*12?
They do all their multiplication by picking one number and applying repeated doubling.
1...12
2...24
4...48
8...96
16...192
As 18 = 2 + 16 we get
16*18 = 16(2+16), so looking at our table,
18 = 24+192 = 216, which is correct
How did the Egyptians do division? For instance, 18/12
They would represent 18/12 as 18*1/12, and then begin doubling as required.
1...1/12
2...1/6
4...1/3
8...2/3 (or 1/2+1/6)
16...4/3 (or 1 and 1/3, a doubling of the above simplification of 2/3)
Thus, 18 = (2+16) so
1/12*18 = 1/12(2+16)
=1/6+4/3 = 9/6 = 1.5 as rqd.
name the two ways Egyptians solved algebraic equations
they solved them using 1. modern day algebraic procedures and 2. false position
how would you use false position to solve x + 1/4x = 15.
As done in class, false position uses an intentionally bad guess to get rid of fractions. Then, they scale up the incorrect answer to get the solution. Example:
x + 1/4x = 15, guess 4.
4 + 1 = 5 ≠ 15
However, 5*3 = 15,
thus for x=4*3 = 12,
x + 1/4x = 5*3 = 15, giving us the solution x = 12.
This method shows that the Egyptians understood order of operations.
what is the Egyptian formula for area of a circle based on the diameter? what does this tell us about their approximation for pi?
The Egyptian area formula is (8/9*d)^2.
Area of a circle is pi*r^2 = 1/4*pi*d^2.
The Egyptian formula (8/9d)^2
= 64/81d^2 = 1/4*256/81*d^2.
Thus, the Egyptian approximation for pi works out to 256/81 ≈ 3.16. Historians believe this formula arises from computing the area of an octagon inscribed in a square.
describe the Babylonian numbering system
base 60 system, uses place value, left to right, used multiplication tables extensively.
Example: 29,703 written as 3+15*60+8*60^2.
Babylonians would write 29703 as 3,15,08.
The fraction 3/4 = 45/60 is written (0;45).
What did Babylonians know about triangles?
Babylonians knew the Pythagorean theorem, and have tables of Pythagorean triples.