Suppose that you have a circle with a circumfrence of 1. What would the diameter be (1/3.14… ad infinitum)? Would it also be an irrational* number? Is any byproduct of Pi an irrational number?
Pi is supposed to be a number of infinite non-repetative length, correct? But if you divide Pi by 1 to determine the diameter, wouldn’t the resultant number be a much longer non-repetive number than Pi itself? So how can a number of infinite length be “longer” than another number of infinite length? Is there any relation in this to negative numbers being “less” than zero?
Suppose this circle with a circumfrence of 1 is surrounded by an equilateral triangle which contacts the circle on each side. What is the relation of the legs of the triangle to the circumfrence of the circle? The diameter of the circle?
Suppose that this self-same triangle / circle drawing had the distance measured from the intersection of two legs to it’s opposite point on the opposing leg. What is the relation of this line to the circumfrence? The diameter?
Suppose that you drew a line from the mid-point of one of the legs of the triangle to the mid-point of one of the other legs of the triangle. What is the relationship of this line to the circumfrence? The diameter?
–Tim
*Irrational does mean a number of infinite length, right?
Okay, okay … A rational number is any number that can be represented as one integer divided by another. Examples are 1, 1/2, 0.0124 (= 124/10000) and 1/3. Irrational numbers are ones that can’t.
But 1/3 = 0.3333…, which appears to be an example of “a number of infinite length” in the terms of the OP.
I’ll pick the one I can answer. Irrational numbers “are the set of all nonterminating, nonrepeating decimals,” according to this math book in my hands. A rational number can be expressed as a ratio of two numbers (hence the name). Thus one-third (1/3) expressed as a decimal (0.3333…) does not terminate, but it repeats, so it is rational.
Pi is irrational because it cannot be expressed as the ratio of two integers. In decimal form, it is nonterminating and nonrepeating.
The general assertion is far from obvious. Thus Legendre had to seperately prove that pi and pi^2 were irrational (after all, proving that sqrt(2) is irrational hardly establishes that 2 is as well). That pi is transcendental however greatly restricts the possibilities.
Offhand, I’m not sure it’s been proved whether, say, pi^pi is rational or not. Granted, every mathematician on the planet would be gobsmacked if such a “simple” case turned out to be rational, but there’s plenty in this sort of area that awaits proof.
Sigh…that should be ratio of two integers (like I wrote in my second paragraph).
[sub]I can’t believe FOUR posts snuck in before me! You see, I open a lot of threads at once, and don’t always remember to refresh before submitting a reply. :wally [/sub]
It’s been proven, quite abundantly. [PI] belongs to a class known as transcendental numbers, a subset of the irrational numbers. The very definition of a transcendental number is that it cannot be the root of any polynomial equation with rational coefficients. This means, trivially, that for [PI][sup]n[/sup]=k, k and n cannot both be rational.
Actually, Some Guy, that doesn’t demonstrate anything about pi[sup]pi[/sup] being rational or irrational, since both the base and exponent are transcendental. 2[sup]sqrt(2)[/sup] is irrational, and if you raise this number to the power of sqrt(2), you get a rational number (2, of course), so a transcendental raised to an irrational can be rational. I’m not aware if the status of pi[sup]pi[/sup] is known. e[sup]pi[/sup] is transcendental, but, to my knowledge, it has not been proven that pi[sup]e[/sup] is transcendental.
Rational/Irrational/Transendental has already been covered. As to the other questions:
If by relationship, you mean ratio, then the ratio does not depend upon the size of the circle. If the circle has a radius of r, then the length of the side of the triangle will be 2r*sqrt(3). The ratio of the side of the triangle to the circumfrence (2pi*r) will be sqrt(3)/pi. The ratio of the side to the diameter (2r) will be sqrt(3).
This line is the height of the triangle. The height of the triangle is 3r. Details left to the reader.
The length of a line drawn from the center of one leg to the center of another leg of an equalateral triangle is half the length of a leg.
Worth a reminder that we are talking about pure mathematics here. If you have a mathematical circle of diameter exactly 1, then it’s circumference is exactly pi (infinite non-repeating decimal.)
In the real world, however, there is no such circle. The circle is drawn with a pencil, and however thin the tip, the line of the circle has some width. You can measure the diameter as being very very close to 1, but you will never be “exactly” one – no measurement could be that precise. You get down to measuring less than the width of an electron, and you’ve still got a margin of error that is HUGE in terms of the pure maths.
The short answer is, it can’t be longer, they are exactly equal. Infinity is very very weird (to put it mildly). There are indeed different infinities, but the ones that come up here are all of the exact same kind, and they are all exactly equal. pi’s decimal representation is infinite length, and so is it’s reciprocal.
BTW, what makes you think the reciprocal of a number must be longer? The reciprocal of 3.3333… (3 and 1/3) is 0.3, much shorter. (First person to mention trailing zeros gets a free smack.) Offhand, I don’t think there is any particular relationship between the decimal representation lengths of a number and it’s reciprocal.