Irrational Numbers Proof
What's the Cliff Note's on this for non-mathematicians??
8 Replies
What's the Cliff Note's on this for non-mathematicians??
It's easy to be irrational but hard to prove it.
PairTheBoard
The article is discussing proof of irrationality via the idea that if a given value is rational then it can differ from another rational number by a minimum amount. For example suppose x=1/13. We know that is a rational number, but suppose we did not. We take another value we know to be rational, say 1/7. We calculate the difference between x and 1/7. Since x is rational, we know that this difference must be m/91 for some m. Therefore if we try to approximate x by a rational number that has 7 in the denominator, we cannot do better than a difference of 1/91. If we want a closer approximation we must use a new rational with a larger denominator. We can make a sequence of rational approximations with ever increasing denominator. The essence of the proof in the article is that if the difference between x and each successive term of the sequence decreases rapidly enough, we can conclude that x is irrational.
Ah, it’s all coming back now. pi is irrational but 22/7 isn’t because it’s just a close approximation of pi. By the way, I recently ordered and quickly returned this book because it was described as a “popular history” so I assumed it was for laymen.
Ah, it’s all coming back now. pi is irrational but 22/7 isn’t because it’s just a close approximation of pi. By the way, I recently ordered and quickly returned this book because it was described as a “popular history” so I assumed it was for laymen.
That is correct concerning what it means for a number to be irrational. Pi is irrational precisely because any fraction a/b, where a and b are integers can only be an approximation to pi, not exactly equal to it. There are many such irrational numbers, in fact more of them than there are rational numbers (If you don’t have a math background this seems silly since there are infinite numbers of both rationals and irrationals, but it can be proven mathematically). When confronted with a new number that arises from consideration of some mathematics, a good question is whether it’s rational or irrational. This is usually extremely difficult to prove. As an example: we know pi is irrational and that e (Euler’s number, the base for natural logarithms) is also irrational. We don’t know if pi+e is rational or irrational. It’s easy to prove that the sum of a rational and an irrational is irrational*, but no such proof exists for the sum of two irrationals. The article linked discusses a method for proving numbers irrational.
*- in case you were curious: let n be rational and x be irrational. Assume that n+x is rational, seeking a contradiction. Let n+x=q where q is rational. Then x=q-n. Since q and n both are rational, their difference q-n also is rational (when we subtract fractions we get another fraction). Hence x is also rational. But x is irrational by hypothesis. This is a contradiction, so our assumption that x+n is rational must be false. Hence x+n is irrational.
To be more precise, algebraic numbers are countably infinite, while transcendental numbers are uncountably infinite. Hence, while both are infinite, transendentals are "infinitely" more infinite than algebraic numbers.
Real numbers are made up of both algebraic and transendental numbers.
Algebraic numbers are those that can be expressed exactly by a polynomial, that is using solely the integers. Because integers can be counted, algebraic numbers are described as countably infinite.
Transcendental numbers are those that cannot be expressed exactly by a polynomial. Cantor I believe was the first person to prove transendental numbers are uncountably infinite.
A real number is either algebraic, or transcendental, but not both. That is, the union of the two sets of these numbers is the empty set.
An irrational number may be algebraic, but all transendental numbers are irrational.
Virtually all transcendental numbers are unknown. The two most notable exceptions are pi (ratio of circumference of circle to its diameter), and e (base of natural logarithm), both of which can be expressed by infinite series (i.e. converging sequence).
The infinite series:
4 -4/3 + 4/5 -4/7 + 4/9 ...
converges to pi, that is this infinite series equals pi even though it is impossible to express pi exactly using finite terms.
Also, the infinite series:
9 + 9/10 + 9/100 + 9/1000 ...
converges and is equal to 10. But because 10 is an algebraic number, it can be expressed exactly using finite terms, that is no infinite sequence of algebraic terms is necessary.
Are physical constants rational or irrational numbers?
In the SI system of measurement, the 7 base constants are axiomatic and have been assigned rational values;
-transition frequency of Caesium-133;
-speed of light;
-Planck constant;
-elementary charge;
-Boltzmann constant;
-Avogadro constant;
-luminous efficacy of 540Thz radiation.
Many well known formulae to derive dinensional measures use pi, as well as physical relationships that have no obvious dependency on geometry or circles. Basically, without pi a whole bunch of stuff would be incalculable.
Gauss derived the Normal Distribution from other known relations, and contains sqrt2), pi, and e, all irrational. This formula also has physical applications, as demonstrated by Maxwell.
Any formulas with polynomials of degree 2 or higher may yield irrational numbers. For example, the Pythogeran theorem x^2 + y^2 = z^2 will always yield an irrational value for z, if the sum x^2 + y^2 is not a perfect square.
I may be wrong, but I believe it is impossible to ascribe an irrational number directly to a physical measurement. This would not apply to an indirect measurenent. For example, if I assume the diameter of a circle is exactly a rational number, that implies the circumference would be irrational, which is an indirect measurement. And vice versa.
