Irrational Numbers Outline
A Brief History
• Both the Egyptians and Babylonians calculated tables of square roots, approximations being
used in many cases so the problems of irrational numbers was not appreciated (3, pg 180)
• Hippasus of Metapontom is credited with the discovery of incommensurable ratios (4, pg 3)
• The invention of irrational numbers attempts to compute plane areas bounded by curves or by
incommensurable straight lines, which forced mathematicians to invent the concept of
continuity (1, pg 13)
• The Pythagoreans were the first to realize that incommensurable ratios were different in
character than the commensurable ones (4, pg 3)
• The most important discovery (in the last decades of the 5th century) ascribed to the
Pythagoreans was the discovery of the irrational by means of incommensurable line
segments. This discovery may have been the result of their interest in the geometric mean,
which led to the study of the ratio of the side and the diagonal of the square, and that this
ration could not be expressed by rationals (2, pg 42)
• The present theory of irrational numbers was developed by Dedekind and Wierstrass and
follows Eudoxus' mode of thought almost literally (2, pg 46)
• Finding the squre root of 2 marks one of the cardinal turning points in the history of math
(1, pg 38)
Different Methods for Approximating
• Archytas' Method (428-365BC) to estimate root(a)
Let a1 be guess, b(i)=a/a(i) and a(i)=[ a(i-1)+b(i-1) ]/2. See supplement
• Greek irrational ladders (homework 2)
• Plato used 2 ≈ 7 / 5 Q 2 ≈ 49/25 (4, pg6)
3 ≈ 7 / 4 Q 3 ≈ 49/16 (4, pg 6)
•
Theodorus used
•
•
Archimedes used 1351/780 > 3 > 265/153 (4, pg 6)
Ptolemy gives 3 ≈ 103/60 + 55/60 2 + 23/60 3 which comes from
( a + b + c ) 2 = a2 + b2 + c2 + 2ab + 2bc + 2ac (4, pg 8)
Narayana's Method (1340-1400) uses the inderterminate equation Nx 2 + 1 = y 2 with x<y to
approximate sqrt(N) by y/x (4, pg 9)
i.e., for N=10, x=6 and y=19 is a solution, which is correct to 2 places.
x=8658 and y=227379 is correct to 8 places
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Irrational Numbers, Page 1
•
The Babylonians realized that not every rational number in their tables had a tabulated square
root, and the approximated by means of the rules
b2
e
a 2 + b2 = a + = a 2 + 2 ab2 or written n 2 +e ≈ n +
2a
2n
The first is used later by Heron of Alexandria, and the second is totally wrong and
dimensionally impossible (1, pg 38)
Proving Root 2 is Irrational
• It is impossible to tell who first proved that the square root of two was irrational, but
Pythagoreans by the end of the 5th century knew that it was irrational, and approximated it
with successive solutions of the equations 2 x 2 − y 2 = ±1 (1, pg 62)
• The proof of root 2 being irrational was probably added to Euclid's Elements in later editions
(4, pg 3)
• Fermat's Method of Infinite Descent (1879) proved sqrt(5) is irrational (4, pg 10)
1. The Development of Mathematics by Bell
2. A Concise History of Mathematics by Struik
3. Numbers: Their History and Meaning by Flegg
4. http://rpimath.topicities,com/irrationals/Irrationals.ppt
Irrational Numbers, Page 2