Comments on: Math Fact of the Week #0 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/ We celebrate all facets of geekiness, including Tech Geek, Science Geek and Gaming Geek... coming to you straight from Boston. Thu, 20 Nov 2008 17:38:09 +0000 http://wordpress.org/?v=2.5.1 By: Carl http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12900 Carl Fri, 25 Aug 2006 12:25:30 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12900 Or, you could just kick someone in the nuts. Or, you could just kick someone in the nuts.

]]> By: Jason http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12881 Jason Fri, 25 Aug 2006 00:45:15 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12881 Sorry! I don't like having to use images everywhere for math, and MathML support isn't widespread enough yet for my liking. You're correct that my clumsy ellipsis are meant to indicate the recurring decimal. I actually like the infinite series proof for this "math fact" myself — simply consider the limit as n -> infinity of the sum for all x = 0 to n of 9 * 10^-x (or, for the LaTeX inclined, $\lim_{n\rightarrow\infty} \sum_{x=0}^{n}9\times10^{-x}$). Sorry! I don’t like having to use images everywhere for math, and MathML support isn’t widespread enough yet for my liking. You’re correct that my clumsy ellipsis are meant to indicate the recurring decimal.

I actually like the infinite series proof for this “math fact” myself — simply consider the limit as n -> infinity of the sum for all x = 0 to n of 9 * 10^-x (or, for the LaTeX inclined, $\lim_{n\rightarrow\infty} \sum_{x=0}^{n}9\times10^{-x}$).

]]> By: Laurence http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12869 Laurence Thu, 24 Aug 2006 19:42:16 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12869 Nevermind... I just got that there are little dots which represent the 99999.... Nevermind… I just got that there are little dots which represent the 99999….

]]> By: Laurence http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12868 Laurence Thu, 24 Aug 2006 19:39:35 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12868 How do you go from x = 0.9 to 10x = 9.9?? If x = 0.9 then 10x = 9... What am I missing?? How do you go from x = 0.9 to 10x = 9.9??
If x = 0.9 then 10x = 9… What am I missing??

]]> By: Dave http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12847 Dave Thu, 24 Aug 2006 14:47:24 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12847 Two columns enter, one column leaves! Two columns enter, one column leaves!

]]> By: Jason http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12840 Jason Thu, 24 Aug 2006 14:03:50 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12840 Oh yeah, I almost forgot to mention: please send me any math questions you may have, and I'll be happy to discuss them in a future Math Fact of the Week. But please note that I am not a mathematician! Oh yeah, I almost forgot to mention: please send me any math questions you may have, and I’ll be happy to discuss them in a future Math Fact of the Week. But please note that I am not a mathematician!

]]> By: Big Remy http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12836 Big Remy Thu, 24 Aug 2006 13:01:08 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12836 You guys do realize that column will now become the arch enemy of the Beer Roster right? You guys do realize that column will now become the arch enemy of the Beer Roster right?

]]> By: Jason http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12832 Jason Thu, 24 Aug 2006 12:46:06 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12832 You are correct that any finite number of 9s after the decimal point results in a real number that is less than one. However, .9... = 1. There is no distinction to be drawn. If it helps, consider some similar examples: 1/9 = 0.1... 1/3 = 0.3... 2/3 = 0.6... Also, where "(x)" is used to indicate the repeating digit: 1/4 = 0.25 = 0.24(9) 1/2 = 0.5 = 0.4(9) etc. You are correct that any finite number of 9s after the decimal point results in a real number that is less than one. However, .9… = 1. There is no distinction to be drawn.

If it helps, consider some similar examples:

1/9 = 0.1…
1/3 = 0.3…
2/3 = 0.6…

Also, where “(x)” is used to indicate the repeating digit:

1/4 = 0.25 = 0.24(9)
1/2 = 0.5 = 0.4(9)

etc.

]]> By: Carl http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12830 Carl Thu, 24 Aug 2006 11:43:31 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12830 0.999...999 < 1, is what I am getting at (and the proofs bear that out). But since you never stop adding 9's, it <i>may as well be</i> equal to 1. I guess that's where I draw the distinction. 0.999…999 < 1, is what I am getting at (and the proofs bear that out). But since you never stop adding 9's, it may as well be equal to 1.

I guess that’s where I draw the distinction.

]]> By: Big Remy http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12817 Big Remy Thu, 24 Aug 2006 03:18:44 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12817 Math is the lesbian sister of Biology Math is the lesbian sister of Biology

]]> By: Russtopher http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12807 Russtopher Wed, 23 Aug 2006 23:42:18 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12807 Did someone say Danica McKellar? I thought I heard someone say Danica McKellar. Did someone say Danica McKellar? I thought I heard someone say Danica McKellar.

]]> By: Colin http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12800 Colin Wed, 23 Aug 2006 21:15:48 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12800 Hehe, I like how Carl complains about the <em>useless</em> grain of sand analogy and then responds with his own useless analogy. Hehe, I like how Carl complains about the useless grain of sand analogy and then responds with his own useless analogy.

]]> By: Jason http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12799 Jason Wed, 23 Aug 2006 20:52:32 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12799 Well, it's not a trick; .9... = 1, by which I mean, they're the same number. I don't see what the speed of light has to do with it, though. If you don't think that .9... = 1, then what do you think? Is it less than 1? If it's less than 1, then what's the result of taking 1 - .9...? Well, it’s not a trick; .9… = 1, by which I mean, they’re the same number. I don’t see what the speed of light has to do with it, though. If you don’t think that .9… = 1, then what do you think? Is it less than 1? If it’s less than 1, then what’s the result of taking 1 - .9…?

]]> By: Carl http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12798 Carl Wed, 23 Aug 2006 20:49:33 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12798 I have been discussing it with J on IM, and he convinced me to post my take on the illusion proven in the equation 0.999... = 1. I contend that while it is absolutely true, it is really just a trick of the way we do arithmetic. Let's start with the speed of light, <i>c</i>. Einstein's Theory of Special Relativity states that for any object of non-zero mass, as velocity approaches c, mass approaches infinity. Einstein indicated that you could accelerate forever, approaching ever higher fractions of the speed of light, and be fine, as long as you never actually hit the speed of light, since at that point you would have infinite mass (and thus require infinite energy, etc). He also pointed out that because of this, it would take an infinite amount of time to accelerate to the speed of light. So, obviously there can be an infinitely large fraction of something without attaining that something, yes? Maybe I'm mixing my mathseses. :) I have been discussing it with J on IM, and he convinced me to post my take on the illusion proven in the equation 0.999… = 1. I contend that while it is absolutely true, it is really just a trick of the way we do arithmetic.

Let’s start with the speed of light, c.

Einstein’s Theory of Special Relativity states that for any object of non-zero mass, as velocity approaches c, mass approaches infinity. Einstein indicated that you could accelerate forever, approaching ever higher fractions of the speed of light, and be fine, as long as you never actually hit the speed of light, since at that point you would have infinite mass (and thus require infinite energy, etc). He also pointed out that because of this, it would take an infinite amount of time to accelerate to the speed of light.

So, obviously there can be an infinitely large fraction of something without attaining that something, yes? Maybe I’m mixing my mathseses. :)

]]> By: Carl http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12797 Carl Wed, 23 Aug 2006 20:24:02 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12797 Then the teacher should have said "for our purposes, it will do," not belabor how right she is and throw me out of her classroom. I'm sorry, High School math shouldn't get a break. Kids who learn shoddy shortcuts early carry them into college, and become Liberal Arts majors. I wish I had become a teacher. My favorite line in class would have been, "NO SHORTCUTS! NO SHODDY WORK!" Okay, my <i>favorite</i> line would have been, "Miss, I'm afraid I need to see you alone after class." But the above would have been a close second. :) Then the teacher should have said “for our purposes, it will do,” not belabor how right she is and throw me out of her classroom.

I’m sorry, High School math shouldn’t get a break. Kids who learn shoddy shortcuts early carry them into college, and become Liberal Arts majors. I wish I had become a teacher. My favorite line in class would have been, “NO SHORTCUTS! NO SHODDY WORK!”

Okay, my favorite line would have been, “Miss, I’m afraid I need to see you alone after class.” But the above would have been a close second. :)

]]> By: Greg http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12796 Greg Wed, 23 Aug 2006 20:22:10 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12796 I for one would like to see Carl punch the author of that article in the head. As well as his 10th grade teacher. I for one would like to see Carl punch the author of that article in the head. As well as his 10th grade teacher.

]]> By: Carl http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12795 Carl Wed, 23 Aug 2006 20:20:08 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12795 Even the ancients weren't too lazy to try. Archmides started off on the right foot: http://en.wikipedia.org/wiki/The_Sand_Reckoner Even the ancients weren’t too lazy to try.

Archmides started off on the right foot:

http://en.wikipedia.org/wiki/The_Sand_Reckoner

]]> By: Colin http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12794 Colin Wed, 23 Aug 2006 20:19:02 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12794 In H.S. science/math, if you want to convey an idea concerning an abstract concept by trying to relate it to something the students might possibly be able to grasp, I think its fine. These aren't scientists publishing papers, after all. It's fine to ease them into it and refine their knowledge/understanding as necessary over time. In H.S. science/math, if you want to convey an idea concerning an abstract concept by trying to relate it to something the students might possibly be able to grasp, I think its fine. These aren’t scientists publishing papers, after all. It’s fine to ease them into it and refine their knowledge/understanding as necessary over time.

]]> By: Carl http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12793 Carl Wed, 23 Aug 2006 20:18:19 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12793 And yes, Miss McKellar is freakin' HAWT. What is it with child actors? They seem to either become drugged out misfits or absolute geniuses. And yes, Miss McKellar is freakin’ HAWT.

What is it with child actors? They seem to either become drugged out misfits or absolute geniuses.

]]> By: Jason http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12792 Jason Wed, 23 Aug 2006 20:18:01 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12792 The funny part is that, not only is the number of grains of sand on the Earth obviously finite, it's not even very hard to estimate! The funny part is that, not only is the number of grains of sand on the Earth obviously finite, it’s not even very hard to estimate!

]]> By: Carl http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12791 Carl Wed, 23 Aug 2006 20:15:10 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12791 No. In science, math... all those subjects, you have to be very careful about bandying words like 'infinite', and 'theory' around. Those words mean something. She wasn't composing a sonnet using 'the infinite depths of my love', she was describing an infinite number. No. In science, math… all those subjects, you have to be very careful about bandying words like ‘infinite’, and ‘theory’ around. Those words mean something.

She wasn’t composing a sonnet using ‘the infinite depths of my love’, she was describing an infinite number.

]]> By: Colin http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12790 Colin Wed, 23 Aug 2006 20:12:33 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12790 I think you just missed her point and were simply trying to be an ass. I think you just missed her point and were simply trying to be an ass.

]]> By: Carl http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12789 Carl Wed, 23 Aug 2006 20:10:27 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12789 To be fair, I would have had the same argument. I actually had an argument that I felt I won once, even though it got me suspended from math class for a few days. My teacher at the time (10th grade, iirc) contended that "all the grains of sand on Earth" was a good example of an infinite number. To which I replied, most vociferously, that it was definitely not. "Yes, it is," she said. "You could never count them all." Gloating, I said something like, "That doesn't mean it's infinite. If the Earth has a size, and is composed of atoms, and we can count the atoms in a rock or a tree, it follows that we can count all the atoms that comprise the planet. So we can count the grains of sand." She wasn't too happy. I swear, some teachers are incredible people who take their work and their subjects seriously. Others just make you want to vomit. To be fair, I would have had the same argument. I actually had an argument that I felt I won once, even though it got me suspended from math class for a few days.

My teacher at the time (10th grade, iirc) contended that “all the grains of sand on Earth” was a good example of an infinite number. To which I replied, most vociferously, that it was definitely not.

“Yes, it is,” she said. “You could never count them all.”

Gloating, I said something like, “That doesn’t mean it’s infinite. If the Earth has a size, and is composed of atoms, and we can count the atoms in a rock or a tree, it follows that we can count all the atoms that comprise the planet. So we can count the grains of sand.”

She wasn’t too happy. I swear, some teachers are incredible people who take their work and their subjects seriously. Others just make you want to vomit.

]]> By: Colin http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12788 Colin Wed, 23 Aug 2006 20:09:10 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12788 As I pointed out to Jay, Danica McKellar does a decent job at explaining it. And, she aint bad to look at either. http://www.danicamckellar.com/mathematics.html Her 0.9... proof is about 1/4 ways down the page. As I pointed out to Jay, Danica McKellar does a decent job at explaining it. And, she aint bad to look at either.

http://www.danicamckellar.com/mathematics.html

Her 0.9… proof is about 1/4 ways down the page.

]]> By: Jason http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12787 Jason Wed, 23 Aug 2006 20:00:57 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12787 Well the author of _Polymathematics_ is a math teacher at a private high school — something tells me that his rant (yes, he bills it as such) is a reaction to the sheer number of times he's had to "prove" that .9... = 1 to a group of argumentative students! :^) Well the author of _Polymathematics_ is a math teacher at a private high school — something tells me that his rant (yes, he bills it as such) is a reaction to the sheer number of times he’s had to “prove” that .9… = 1 to a group of argumentative students! :^)

]]> By: Carl http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12786 Carl Wed, 23 Aug 2006 20:00:47 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12786 By the way, in no way do I disagree with the whole 0.999... = 1 (even though it seems more illusion than reality), I just feel like punching the author of that post in the head after I read it. So I offer up less punch-inducing fare to help understand the concept. By the way, in no way do I disagree with the whole 0.999… = 1 (even though it seems more illusion than reality), I just feel like punching the author of that post in the head after I read it. So I offer up less punch-inducing fare to help understand the concept.

]]> By: Carl http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12785 Carl Wed, 23 Aug 2006 19:55:49 +0000 http://www.bostongeek.com/2006/08/23/math-fact-of-the-week-0/#comment-12785 Neat trick. Of course, the author on the blog descends into maddening self-indulgence on how it "isn't obvious" to him how 0.9 repeating is always less than 1, and then goes into heavy talk about him being right (he admits at the end that it's a rant). I find the gloating a little reminiscent of some of my arguments with J, so please allow me to add links with less... inflammatory style and more easily digested explnations: http://mathforum.org/library/drmath/view/55746.html http://mathforum.org/library/drmath/view/55748.html http://mathforum.org/dr.math/faq/faq.0.9999.html Neat trick. Of course, the author on the blog descends into maddening self-indulgence on how it “isn’t obvious” to him how 0.9 repeating is always less than 1, and then goes into heavy talk about him being right (he admits at the end that it’s a rant).

I find the gloating a little reminiscent of some of my arguments with J, so please allow me to add links with less… inflammatory style and more easily digested explnations:

http://mathforum.org/library/drmath/view/55746.html
http://mathforum.org/library/drmath/view/55748.html
http://mathforum.org/dr.math/faq/faq.0.9999.html

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