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Round Numbers:

If square numbers are the areas of squares whose dimensions are integers (the origin of the term square, I think), then shouldn't round numbers be pi, pi*4, pi*9, and so on?

Patrick McKenzie (mail):
Mathematicians, they're so silly sometimes. The triangular numbers start out with 1, 3, 6, 10, 15, 21, 28, when any third grader knows you can't divide 10 or 28 by three.
7.14.2005 4:23am
Anonymous Law Student:
Why do we drive on a parkway and park on a driveway?

Why is it a shipment when it goes by car, and cargo when it goes by a ship?
7.14.2005 6:04am
Lou Wainwright (mail):
And why are perfectly nice everyday numbers like 5, 11, and 49 called 'odd' but crazy ones like 7259.21890310570987152 or 0.0000000001832734633 called 'rational'?
7.14.2005 8:47am
STROBE (mail):
How can a round pi remain round when
connected to real square number like 4 or 9?

And what happens when a nice round pi gets cubed?

Maybe it's something only Mobius or Klein would know. . .
7.14.2005 9:30am
Proud Generation Y Slacker:
Isn't any number multiplied by pi a round number? You're making a square pi. Maybe you should instead raise Euler's number to the power of pi, then put it on your new chair. You know what you'd get, right?
7.14.2005 10:11am
George (mail):
Speaking of pi, check out this news:
Man recites pi from memory to 83,431 places
http://www.msnbc.msn.com/id/8456677/
7.14.2005 10:12am
Scott Moss (mail) (www):
I was about to say, "but to be analogous to square numbers, wouldn't a round number have to be the area of a circle with an integer radius" -- but then I realized that's what you were doing, so never mind.
7.14.2005 10:14am
alkali (mail):
It seems to be that round numbers would be the area of circles with integer diameters, not radii: hence, pi*1/4, pi*4/4, pi*9/4, etc.
7.14.2005 10:18am
Arthur (mail):
The round numbers are 0 and 8, and a case could be made for 6 and 9.
7.14.2005 10:23am
J Fielek (mail) (www):
Come now... every one knows that pies are round.

Cobblers are square.
7.14.2005 10:25am
DR:
You've got it a little wrong. Round numbers would be the areas of circles with integer circumferances!

So round numbers would be 0.25 / pi, 1/pi, 2.25 / pi, etc.
7.14.2005 11:22am
spacemonkey (mail) (www):
Mmmmm, pi.
7.14.2005 11:32am
Marc W:
J Fielek wrote that everyone knows pies are round.

I always thought Pi R Square

BTW, What's brown, runs off a cliff, and is equivalent to the axiom of choice?
7.14.2005 11:55am
JRL:
1/9=.11111, 2/9=.22222, 3/9=.33333, . . ., 9/9=.99999???
7.14.2005 12:04pm
Marc W:
Yes, JRL.

9/9 = .999999...

The key to remember is that there are infinitely many 9's following the decimal place. If you add them up, you'll get 9/10 + 9/100 + 9/1000 + .... That's a geometric series, which adds up to 1.

Thus .9999... is equal to 1, (It is, literally, just another way of expressing the number. So there is no contradiction or inconsistency.
7.14.2005 12:36pm
Dantheman (mail):
Pi are round. Cornbread are square.
7.14.2005 12:37pm
Edward Lee (www):
And why are perfectly nice everyday numbers like 5, 11, and 49 called 'odd' but crazy ones like 7259.21890310570987152 or 0.0000000001832734633 called 'rational'?

You got me. Also puzzling is how transcendental numbers got their name, when in fact almost all real numbers are transcendental.

Although I hear nowadays it's not politically correct to call them transcendental -- they prefer the term "polynomially challenged".
7.14.2005 12:42pm
Edward Lee (www):
Isn't any number multiplied by pi a round number? You're making a square pi. Maybe you should instead raise Euler's number to the power of pi, then put it on your new chair. You know what you'd get, right?

You'd get 23.1407.
7.14.2005 12:45pm
Proud Generation Y Slacker:
No, Ed, you'd get a chair e^pi.
7.14.2005 12:50pm
big dirigible (mail) (www):
The term "round numbers" is not available for this abuse, as it's already taken.
7.14.2005 1:55pm
Doug Sundseth (mail):
The numbers that Prof. Volokh refers to are circular; circular is a subset of round.

Thus, pi/6, 4*pi/3, 9*pi/2, and 32*pi/3 are also round.

As are (pi^2)/32, (pi^2)/2, and 81*(pi^2)/32.

You need stop with the two-dimensional thinking.

ps. I've used volumes (or "volumes", if you prefer) rather than surface areas as they seem more analogous to the "square numbers" case.

pps. I'll leave the "volumes" of hyper-spheres of n greater than 3 as an exercise for someone else.
7.14.2005 2:05pm
Nate Johnston (mail) (www):
I believe you're engaging in circular logic here. Ha!

But really, a round number is defined in mathematics as "A round number is a number that is the product of a considerable number of comparatively small factors" (citation: Hardy, G. H. "Round Numbers." Ch. 3 in Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, pp. 48-57, 1999. Hat tip: MathWorld)

In popular culture, a round number is one of the ones that has nice round zeroes in all places other than the most significant one. That's the one the Wikipedia goes with.

The etymological base in visual analogy, seen in the association of trailing zeroes to a "round" number, is mirrored in the actual definition of a square number, which is a number that is a square of two other numbers, for the obvious reason that then you can lay out the given units in a square formation (i.e. 9 is a square number because 3^2 = 9, ergo you can arrange your 9 widgets three by three on a table). Wikipedia backs me up.
7.14.2005 2:58pm
Marc W:
I wouldn't necessarilly trust Wikipedia.

One of my best friends is Desmond Devlin, a writer for Mad Magazine. Wikipedia has an entry for him. At one point it said that he was born during the 1800s in "what is now Rome Italy." It also linked him romantically to Abraham Lincoln's son. They've since corrected the entry to one which is more accurate (and boring). But bottom line is I don't have much faith in it.
7.14.2005 3:13pm
Christine Hurt (mail) (www):
My head started hurting at "circular is a subset of round."
7.14.2005 3:45pm
triticale (mail) (www):
Pi is equal to three for small values of pi and large values of three.

In particular, when one is calculating the length of steel bar to be used to bent the master link for a chain sling, it is necessary to introduce a correction factor for stretch. Much imperical review shows that setting pi equal to three provides the correction across the common bar size to bend centerline diameter range.

Sorry if this is too serious for the thread.
7.14.2005 4:02pm
triticale (mail) (www):
Cobblers may be square, but hatters are irrational.
7.14.2005 4:03pm
Splunge (mail):
I expect everyone actually knows this, but just in case...

Rational numbers are numbers that can be expressed as ratios of integers. Got nothing to do with the homonym referring to the use of reason.
7.14.2005 4:59pm
Joan of Argghh! (mail):
Everyone knows that Cornbread R square and pie R round.
7.14.2005 5:30pm
duglmac (mail):
If square objects are square in shape, and cirlce objects are round in shape, then it seems ambiguous by your logic whether the numbers you refer to should be 'circle' numbers or 'round' numbers, as it is not clear whether the 'square' in 'square' numbers is from the object name or the name of shape of the object.
7.14.2005 6:31pm
Proud Generation Y Slacker:
Doug:

We're intelligent, but not experienced.
7.14.2005 6:59pm
John Armstrong (mail):
I'd say "round" is really "rounded". As in, "to the nearest" nth power of ten.

Edward Lee: transcendental numbers transcend the nicely-behaved algebraic numbers. Really, the term comes from a transcendental extension of a field, which transcends all fields attainable by algebraic extension (adjoining formal solutions to polynomial equations).

As for "almost all", in a technical sense every "nice" property is false for almost all objects which may or may not have this property. Almost all rational numbers are not integers. Almost all algebraic numbers are irrational. Almost all real numbers are not algebraic. Almost all complex numbers are not real. Almost all Cn functions are not Cn+1. I could go on for hours.
7.14.2005 7:49pm
Marc W:
Splunge,

Don't spoil the fun.
7.14.2005 7:50pm
Jeffrey King (mail) (www):
Sorry for this complex tangent, the whole thing will show I am either odd or square. Even if a fraction of you think me irrational, while quite negative and mean, it would not be imaginary. Please limit your criticism; for my contribution may not be integral to previous comments, it seems prime time I submit my real thoughts on this issue. I was positive that by the time I got round to the meat of this meme, it wouldn't be so average. (I'm a little frustrated I could not work in SOHCAHTOA)
7.14.2005 9:02pm