some dumb math questions (1 Viewer)

[andrew markworthy]

Lying awake with a (for me) rare bout of insomnia the other night, I started to think about math formulae (normally a good way to induce unconsciousness). However, I got to the formula for an area of circle, which is of course pi times the square of the radius, and had a thought. We know that pi is an infinite decimal, so any calculation made using pi must by definition be inaccurate. Okay, if you are using pi to 20 decimal places this will be by a miniscule amount, but nonetheless it is still inaccurate.

Dumb question - does this mean that in theory there could be a more accurate method of calculating the area of a circle and we just haven't figured it out yet, or are we 'doomed' to be 'inaccurate' by using pi?


And talking of infinity - are all infinite groups equal? All whole numbers form an infinite set. But there again, so are all possible pairs of whole numbers. In e.g. a finite set of 1000 numbers, there are a lot more possible pairings of whole numbers (1+2, 1+3, 1+4 ... 999+1000) than there are single numbers (1,2,3 ... 1000). Of course all infinite sets are by definition infinite, but there is a strong intuitive sense that the set of e.g. all possible pairs of whole numbers is a lot bigger than the single numbers. Please does anyone know a good (non-technical) source that can explain this?

 

[TerryS]

Consider this:

pi is the ratio of the circumference to the diameter, C/d. Both of these numbers are measurable to nearly exact values.

This means the area is actually A=(C/d) * r^2 or A=(C/2r) * r^2 or simplified further A=Cr/2.

pi is simply a convenience to avoid measuring the circumference, and the radius is needed either way. pi to 2 or 5 digits is usually a close enough approximation to avoid making things complex.

 

[BrianW]

I think you mean to say "imprecise" when you use the work "inaccurate." Precision and accuracy are not the same thing.

3.14159 is more precise than 3. But which number is more accurate? Well, if you're talking about the number of sides of a triangle, 3 is the more accurate number.

Pi will yield the most accurate result possible, period. The precision required for the task at hand will determine the number of decimal places you use in your calculation. I think it was the French government who legislated that pi is equal to 3.0, just to make architecture easier. Needless to say, it didn't work. Unfortunately, pi is irrational, and we just have to deal with it. But if you want to examine a really weird constant, take a look at e.
No, they are not. For instance, there are far more irrational numbers than rational numbers on the number line, even though there is an infinite number of both. Your intuition about whole number pairings outnumbering individual whole numbers is correct.

I remember someone mentioning how he used to confuse his teacher by asking how an infinite container wouldn't encompass and contain the whole Universe. His teacher shouldn't have been teaching math if he couldn't figure out the difference between infinite and unbounded. An infinitely tall tin can, for instance, will contain an infinite amount of whatever you care to put in it, yet it encompasses an insignificant volume of the Euclidean Universe itself. There are indeed different classes of infinity that cannot be equated to one another, so your first instinct in this regard is correct.

I'll try to find some links for you and get back.

 

[Andrew W]

Search on cardinality of infinite sets to learn about how some sets are "more infinite" than other sets.

 

[Haggai]

Not so: whole number pairings CAN be put into a one-to-one correspondence with whole numbers. A demonstration of this is essentially given within this link at Wikipedia. Masochists who want this spelled out in more detail are invited to ask me. I'm always happy to inflict mathematical pain on others who are willing to try to brave through it.

 

[BrianW]

Haggai is right. I humbly stand corrected.

 

[Henry Gale]

It works, really! I was slipping out of my chair by the end of your second paragraph.

 

[Ken Chan]

What if the radius of the circle is 2/sqrt(pi) -- two over the square root of pi? Then the area of the circle is exactly 4.

At this point, you might think you cannot accurately draw a circle with that radius, and you may be right. But it's no more difficult than drawing a circle with a radius of exactly 2.

If you're doing a math problem on a whiteboard it's very easy: draw a shape that is approximately a circle. From a point near the average center, draw a single stroke (ideally as straight as possible) to the edge; try not to go over. Label that stroke: 2/sqrt(pi)

 

[Cees Alons]

Well, all correct answers are already given above.
So I'll just add this remark.

Pi isn't "inaccurate': we know exactly how "big" it is. We're just unable to express it accurately in our decimal system used to represent numbers.

But (for instance) you could represent it exactly and correctly in we had a numbering system using circles (circumferences and/or areas).


Cees

 

[andrew markworthy]

Many thanks for these responses guys, that have very much cleared my mind of a lot of fuzzy thinking.

 

[Chu Gai]

So that's the reason for those crop circles!

 

[Cees Alons]

Of course! Everything the aliens (any aliens) do makes sense. Unlike our doings.


Cees

 

[brentl]

"It works, really! I was slipping out of my chair by the end of your second paragraph. "



Brent

 

[Greg.K]

Mmmmmm. Pie.

 

[Chris Bardon]

If only we could get March 14th (Pie Day) declared a national holiday... All you have to do to celebrate is to eat pie-how hard can that be?

As for the infinity question, it really depends on what branch of mathematics you're dealing with. I know that in Engineering, we usually only have one infinity to deal with, but pure math guys have "special" infinities.

It's a little like thinking about quantum physics though. Infinity plus Infinity is infinity. The infinite set of real numbers have an infinite number of combinations. Infinity plus one-still infinity. If it doesn't make your head hurt, you're not really understanding it.

If all else fails though, just transform things to the frequency domain-that usually simplifies things!