Irrational s

7/28/2019 Irrational s

http://slidepdf.com/reader/full/irrational-s 1/16

Irrational Numbers

Danielle M. Tarnow

2 2 1 414 R( ) . ...

3567810

11 12 13 14 15 17

(I wish I knew…)



Archytas’ Method (428-365 BC)

Known since the time of the Babylonians

To approximate R(a), let a1 be your first guess, then:

b1=a/a1 (Here either ai or bi is too big and the other too small)

a2=arithmetic mean of a1 and b1=(a1+b1)/2

b2=a/a2

Continuing, an=(an-1+bn-1)/2 and bn=a/an

The value of R(2) on the Yale Babylonian Tablet is a3

if a1=1½. This method is frequently used by

computers today.



Greek Irrational “Ladders”

The Greek Irrational Ladder for R(2) is:1 1 1/1=13 2 3/2=1.5

7 5 7/5=1.417 12 17/12=1.41666… 41 29 41/29=1.41379… 99 70 99/70=1.41429…

etc.

Which comes from the easily shown identity that:

2x2-y2=(2x+y)2-2(x+y)2=…



The Discovery of Incommensurability

There is no evidence that the Babylonians or the Egyptians knewthat R(2) was irrational (they might have just believed that moreplaces needed to be used).

Hippasus of Metapontom is credited with the discovery of incommensurable ratios (apparently he was drowned for this bythe Pythagoreans). ALOGOS meant “not a ratio” and “not to bespoken.”

The Pythagoreans were the first to realize that incommensurableratios were different in character from commensurable ones.

The proof that R(2) is irrational was probably not originally partof Euclid’s Elements, it was probably added in later editions.(Although the proof was known long before Euclid, according to

Aristotle.)



Eudoxus of Cnidus

(408-355 BC)

Did you know the ratio of thediagonal length of a cube to

the length of its edge is R(3)?

The Greeks did not realize that lines were not discretecollections of units, that’s why they had so much trouble withZeno’s ideas (the discrete vs. the continuous).

Eudoxus created the idea of a MAGNITUDE (which allowedpeople to stay away from the idea of irrational NUMBERS). Thiscaused the view that geometry and algebra were unrelatedand that geometry was the only rigorous way to prove things.

Theatetus investigated and classified the types of incommensurable lengths that can be generated with acompass and straight edge. Fibonacci later showed that therewere types of irrationals that could not be created this way.



Aristotle of Stagira (384-322 BC)

Aristotle gives a reductio ad absurdumproof that R(2) is irrational. This proof,which only works for square roots of

EVEN non-squares, was known long before him.

Here’s how it goes: Let R(2)=a/b where a and b are positive integers that arerelatively prime.

Then, a2=2b2. This means that a2 is even, so a is even and canbe represented by, say, 2c.

So, 4c2=2b2 or 2c2=b2. So, b2 is even, so b is even. If a and bare both even, then they are not relatively prime, thus it is

impossible that R(2) is rational.



Some Approximations

Plato used R(2)≈7/5 because 2≈49/25

Theodorus used R(3)≈7/4 because 3≈49/16

Archimedes, while working on Pi, used the factthat 1351/780 > R(3) > 265/153, which mayhave come from using what will later come to beknown as Heron’s method.



Heron’s Method (10-75)

Heron was probably not the creator of this algorithm.

Where a2 is the rational square nearest to the number and b isthe remainder. Several applications of this for R(3) results inArchimedes’ approximation.

Heron’s way of saying this (for R(720)): “Since 720 has not its side rational, wecan obtain its side within a very small difference as follows. Since the next succeeding square number is 729, which has 27 for its side, divide 720 by 27. This gives 26 2/3. Add 27 to this, making 53 2/3, and take half this or 26 5/6. The side of 720 will therefore be very nearly 26 5/6. In fact, if we multiply 265/6 by itself, the product is 720 1/36, so the difference in the square is 1/36. If we desire to make the difference smaller still than 1/36, we shall take 720 1/36 instead of 729 (or rather we should take 26 5/6 instead of 27) , and by proceeding in the same way we shall find the resulting difference much less than

1/36.”

ab

aa b a

b

a

2 2 1

2



How Ptolemy Got His Approximation (85-165)

Ptolemy gives 103/60+55/602+23/603 for R(3) whichis correct to 6 decimal places.

This comes from the idea that: (a+b)2=a2+2ab+b2

So, if you can find an a and a b such that a2+2ab+b2

is close to 3, then a+b will be close to R(3).

Ptolemy actually uses(a+b+c)2=a2+b2+c2+2ab+2bc+2ac here.



Narayana’s Method (1340-1400)

This Indian method uses the second orderindeterminate equation Nx2+1=y2, with x<y, toapproximate R(N) by y/x.

For N=10, there are many solutions, including x=6and y=19, which yields R(10)≈3.1666…, which iscorrect to 2 decimal places.

Narayana’s solution of 10x2+1=y2 where x=8658and y=227379 gives an approximation for R(10)that is correct to eight decimal places.

Did you know the current day square rootsymbol was first introduced in 1544?



Fermat’s Method of Infinite Descent (1879)

Let R(5)= a/b where a and b are positive integers.

Now, 4<5<9, so 2<R(5)<3. So, 2<a/b<3, which is equivalent to2b<a<3b. So, 0<a-2b<b.

Since R(5)=a/b, 5b2=a2. We can subtract 2ab from both sides,getting 5b2-2ab=a2-2ab. We can factor this, gettingb(5b-2a)=a(a-2b).

So, a/b = (5b-2a)/(a-2b), which is a “more reduced” fraction

than a/b, since a-2b<b. You could do this over and over again,and this is not possible with b being a positive integer, so R(5) isirrational.

This is a number theory equivalent to the natural idea of infinite reproducibilityof figures inside figures (for example, you can keep putting smaller and smallerpentagrams inside of each other). It is only a short step from here to prove that

Phi=(1+R(5))/2 is irrational.



Before the 19th Century…

The Hindus and Arabs worked freely with the irrationals. Theydid not bother to prove that they could operate with irrationals inthese ways.

The decimal system was adopted and in 1696, Wallis identifiedthe rationals as periodic decimal numbers.

It was ASSUMED that functions were also defined for irrationalvalues of x. No one questioned letting x equal an irrationalnumber. It was IMPLICITLY understood that the function valueresulting from substituting an irrational x would be between thevalues obtained by substituting rational values above and belowx.

The concept of CONTINUITY had not been defined in a

satisfactory way.



Julius Wilhelm Richard Dedekind (1831-1916)

•The essence of continuity is the ability to divide all points onthe number line into two groups using one and only one point

•This idea of a CUT came to him while lecturing Calculus

•Not all cuts are created by rational numbers(proof uses something

similar to Fermat’s method of infinite descent)

•From this herigorously develops=, <, >, and the

operations



Georg Ferdinand Ludwig Philipp Cantor (1845-1918)

•Always remembered his childhood in Russia with greatnostalgia and never felt at ease in Germany

•Exchanged thoughts

with Dedekind

•He defined numbers in termsof convergent sequences of

rational numbers, forinstance R(2) is

represented by the “fundamental” sequence:

1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.414213, …

•He also rigorously formalized the operations



Irrational Numbers

Irrational numbers are not just single symbolsor even pairs of symbols…they must berepresented by an infinite collection of

symbols, such as Cantor’s sequences orDedekind’s cuts.

Stoltz (1842-1905) said that every irrationalnumber could be represented as a non-periodic

decimal (and this can be used as a definingproperty of irrationals).

I am currently creating a chronology of thedevelopment of irrational numbers.



References

From the Course Recommended Book List:

•Eves’ An Introduction to the History of Mathematics •Katz’s A History of Mathematics: An Introduction •Kline’s Mathematical Thought From Ancient to Modern Times

Other Books:

•Boyer’s A History of Mathematics•Burton’s The History of Mathematics: An Introduction•Katz’s Using History to Teach Mathematics: An International Perspective •Nahin’s An Imaginary Tale: The Story of R(-1)

From the Course Handouts:

•Two Dedekind Papers•History of Mathematics (by Shank?)

From the Internet:

•http://www-gap.dcs.st-and.ac.uk/~history/

http://www-gap.dcs.st-and.ac.uk/~history/






Date post:	03-Apr-2018
Category:	Documents
Upload:	aslam844
View:	216 times
Download:	0 times

Irrational s

Documents