Things I did not know about until today

[yendi]

Really stupid and persistent math trolls.

(Ganked from trochee; also, all three links refer to the comment threads, not the posts.)

Comments

[blitheandbonny.livejournal.com]

Wow. I have lost absolutely all faith in modern education. I learned that shit in, like, third grade and these (presumable) adults don't get it? Also, now my head hurts.

[xiphias.livejournal.com]

I've had that argument.

With a creationist. Who claimed that the inequality between 1/3 and 0.333... repeating proved the existence of God.

Personally, I find that . . . disturbing . . . since, if such an argument WAS valid, it might suggest that it would be possible to create an argument such that the equality of 1/3 and 0.333.... disproves God.

Actually, I had one of my all-time favorite flamewars with her. Well, it started with her, but it drew in everyone else in the newsgroup.

She commented that some things are so well known that you don't HAVE to point out how you know them, such as "George Washington was the first president of the United States," and "the sky is blue."

I responded, reasonably enough, "But John Hansen was the first President of the United States, and, right now, where I am, the sky is black with little white twinkly things in it."

People were willing to grant me the "John Hansen" thing, after I provided sources, but the "sky is blue" thing went on for WEEKS.

I kept posting things like, "Right now, the sky is grey." And people would say, "No it's not -- it's blue, but you just can't SEE it. The sky is BEHIND the clouds." And I'd say, "When I look up and I'm outside, I see the sky. The sky is now black with twinkly things and a moon in it." And they'd say, "No, it's still blue, but it's just too dark to see that it's still blue."

[lynxreign.livejournal.com]

Since the sky being blue is a condition of our eyes interpreting the difracted and partially absorbed light that passes through the atmosphere, or more generously just the latter half of that, if there isn't enough light to color the sky then it isn't blue. Also, at sunset, the sky can be red and not blue at all thanks to the light having more of its spectrum absorbed.

[xiphias.livejournal.com]

I eventually had to use an "Appeal To Authority" argument, which is an argument I don't usually like to use.

I phoned my mom.

I said, "Hey, Mom, what color is the sky?"

She said, "Hang on. Lemme check."

[jerel.livejournal.com]

Wow, my eight-year-old nephew even knows the sky isn't really blue (light just makes it appear blue on a clear day). And he's EIGHT.

[slipjig.livejournal.com]

Oh, my hell. I swear, if I ever pull a clock tower sharpshooter thing, it's going to be over this. Well, this, or the Monty Hall problem (which is especially touchy because I used to be on the wrong, it-doesn't-matter-if-you-switch side, until my father convinced me otherwise in roughly 15 seconds).

the difference is one of actuality vs. practicality.

[hinj.livejournal.com]

In truth, 0.99999(repeating) != 1.

1 is a rational number in which both the numerator and denominator are identical.

0.999(repeating) is a rational number in which the numerator has an infinite number of nines and the denominator is equal to the numerator + 1. Because of the infinite nature of it, it's really a theoretical concept, but the quotient is clearly not equal to 1 as the numerator and denominator are not equal.

HOWEVER, practically speaking, they are equal. To computers and others who can't think theoretically, they are equal. To those of us who love the crispy sounds our brain makes when we think about it, they will never be equal.

Re: the difference is one of actuality vs. practicality.

[afeldspar.livejournal.com]

I'm ... afraid you're mistaken on a few points.

"0.999(repeating) is a rational number in which the numerator has an infinite number of nines and the denominator is equal to the numerator + 1"

No, that would be "0.999(repeating) / 1.999(repeating)".

"Because of the infinite nature of it, it's really a theoretical concept, but the quotient is clearly not equal to 1 as the numerator and denominator are not equal."

You're employing circular logic. You assume the premise that 1 - 0.999(repeating) is not equal to 0, and then present that premise as leading to the conclusion that 0.999(repeating) is not 1.

Here is a simple proof that 0.999(repeating) is equal to 1.

1) Let X be equal to .999(repeating) which is equal to the infinite series 9/10^1 (that is, 9 over 10-to-the-power-of-1) + 9/10^2 + 9/10^3 + 9/10^4 ....

2) If we multiply each side by 10, we get the equation 10X = 9/10^0 + 9/10^1 + 9/10^2 + 9/10^3 + 9/10^4 ....

3) If we subtract the left side of the equation from step 1 from the left side of the equation from step 2, we get 9X. If we subtract the right side of the equation from step 1 from the right side of the equation from step 2, the only term that is not canceled out by another term is 9/10^0. Thus, 9X = 9/10^0.

4) Simplifying, we get 9X = 9/1, 9X = 9, and finally X = 1. Since we defined X as .999(repeating), we have proved that .999(repeating) = 1.


I suspect that you're making an error of logic that I once made, thinking that 0.999(repeating) isn't equal to 1 because if you go all the way, down to the very end, you'll find an incredibly small but nevertheless real delta which separates the two values. However, this just isn't so. Infinity has no end.

Re: the difference is one of actuality vs. practicality.

[hinj.livejournal.com]

I said:
"0.999(repeating) is a rational number in which the numerator has an infinite number of nines and the denominator is equal to the numerator + 1"

You mistakenly thought I said:
"No, that would be "0.999(repeating) / 1.999(repeating)".

To clarify:

9 divided by 9+1 = 0.9
99 divided by 99+1 = 0.99
999 divided by 999+1 = 0.999
.
.
.
99999(infinite # of 9's) divided by 99999(infinite # of 9's)+1 = 0.99(repeating)

The decimal point appears only in the quotient, not in the numerator of the rational number.


You summarized:
"I suspect that you're making an error of logic that I once made, thinking that 0.999(repeating) isn't equal to 1 because if you go all the way, down to the very end, you'll find an incredibly small but nevertheless real delta which separates the two values. However, this just isn't so. Infinity has no end."

Which I agree with, and I even said it was really a theoretical concept because of the infinite nature.

It's fascinating. I think this is really a religious divide. I'm a theoretical mathematician who loves the epsilons & deltas & infinitudes of all of it. That theoretical fringe is my playground.

I'm also a public school teacher which instantly demonizes me and brands me an idiot not capable of teaching PE (which truly, I'm not capable of teaching, but the Bachelor & Master's degree I have in Mathematics tend to indicate I know something about mathematics, and my students' success indicates I can actually teach). That is NOT a place for the theoretical fringe -- their heads explode. It's fun to watch on occasion, but it is, without a doubt, no place for such religious arguments.

Carry on!

(I once heard a lovely statement about arguing on the internet...it's like getting a gold medal in the special olympics.)

Re: the difference is one of actuality vs. practicality.

[hinj.livejournal.com]

and by the way, your proof also depends on infinity, which can't happen except in theory and the crispy edges of our brains.

Re: the difference is one of actuality vs. practicality.

[afeldspar.livejournal.com]

Well, you must admit it's kind of hard to follow your argument when you say that "To computers and others who can't think theoretically, [.999... and 1] are equal" and then you say that infinity "can't happen except in theory and the crispy edges of our brains." (I'm not even sure what it means for "infinity" to "happen"; it's not so much that infinity "happens" as that it's a feature of the universe. If you've got an infinitely repeating fraction, it's going to have a particular digit at a particular place value, no matter how far you go, and that's what the proof really depends upon.)