Rational numbers vs. Irrational numbers

88 Pages·2013·1.61 MB·English

Rational numbers vs. Irrational
numbers
by
Nabil Nassif, PhD
in cooperation with
Sophie Moufawad, MS
and the assistance of Ghina El Jannoun, MS and Dania Sheaib, MS
American University of Beirut, Lebanon
An MIT BLOSSOMS Module
August, 2012
Rational numbers vs. Irrational numbers
“The ultimate Nature
of Reality is
Numbers”
A quote from Pythagoras (570-495 BC)
Rational numbers vs. Irrational numbers
“Wherever there is
number, there is
beauty”
A quote from Proclus (412-485 AD)
Rational numbers vs. Irrational numbers
Traditional Clock plus Circumference
1
1 min = of 1 hour
60
Rational numbers vs. Irrational numbers
An Electronic Clock plus a Calendar
Hour : Minutes : Seconds
dd/mm/yyyy
1
1 month = of 1year
12
1
1 day = of 1 year (normally)
365
1
1 hour = of 1 day
24
1
1 min = of 1 hour
60
1
1 sec = of 1 min
60
Rational numbers vs. Irrational numbers
TSquares: Use of Pythagoras Theorem
Rational numbers vs. Irrational numbers
Golden number ϕ and Golden rectangle
√ √
2 1 + 5 1 1 − 5
Roots of x − x − 1 = 0 are ϕ = and − =
2 ϕ 2
Rational numbers vs. Irrational numbers
Golden number ϕ and Inner Golden spiral
Drawn with up to 10 golden rectangles
Rational numbers vs. Irrational numbers
Outer Golden spiral and L. Fibonacci
(1175-1250) sequence
F = {
︸︷1︷︸, ︸︷1︷︸, 2, 3, 5, 8, 13..., fn, ...} : fn = fn−1+fn−2, n ≥ 3
f1 f2
1 1
n n−1
fn = √ (ϕ + (−1) )
n
5 ϕ
Rational numbers vs. Irrational numbers
Euler’s Number e
1 1 1
s3 = 1 + + + = 2.6666....66....
1! 2 3!
1 1 1
s4 = 1 + + + = 2.70833333...333....
2 3! 4!
1 1 1 1
s5 = 1 + + + + = 2.7166666666...66....
2 3! 4! 5!
.............................
1 1 1 1 1
lim {1 + + + + + .... + } = e = 2.718281828459........
n→∞ 2 3! 4! 5! n!
e is an irrational number discovered by L. Euler (1707-1783), a limit of a
sequence of rational numbers.
Rational numbers vs. Irrational numbers