Hi again, internets — it’s me, Jason. For some crazy reason, I decided I wanted to write a regular column for BostonGeek, and the the best idea for a column I had was a weekly math fact. I really, really like math. :^)
I know that a lot of people, even some pretty geeky folks, get anixious when it comes to math. Maybe this column will show those people how much fun math can be; at least, I hope it will.
So, without further ado, here’s the very first Math Fact of the Week.
This week’s fact comes to us from the excellent Polymathematics blog.
Many people don’t believe this when first see it, but it’s true! It’s an artifact of how we represent numbers.
Repeating Decimals
The bar that floats over the 9 in the “.9″ above (which is, incidentally, called a viniculum) is used by mathematicians to represent a repeating decimal; (also known as a recurring decimal). Another way to represent a repeating decimal is with the use of an ellipsis, although this can leave an ambiguity about exactly which digits of the decimal expansion repeat.
Most of the repeating decmials that we encounter in our daily lives are due to the fact that we use a decimal number system — that is, a number system in which there are ten possible digits that can be used to build the representation of a number. Some numbers, such as 1/3, simply can’t be represented in our decimal number system without a recurring decimal. Every rational number can be written as either a finite or repeating decimal, but that’s a topic for another week.
So Why Does .9… = 1 Anyway?
The easiest proof to follow is the following algebraic one:
Let x = 0.9…, then 10x = 9.9… and 10x – x = 9x = 9; but 9x = 9 ⇔ x = 1. QED.
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
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.
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! :^)
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.
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.
I think you just missed her point and were simply trying to be an ass.
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.
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!
And yes, Miss McKellar is freakin’ HAWT.
What is it with child actors? They seem to either become drugged out misfits or absolute geniuses.
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.
Even the ancients weren’t too lazy to try.
Archmides started off on the right foot:
http://en.wikipedia.org/wiki/The_Sand_Reckoner
I for one would like to see Carl punch the author of that article in the head. As well as his 10th grade teacher.
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.
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.
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…?
Hehe, I like how Carl complains about the useless grain of sand analogy and then responds with his own useless analogy.
Did someone say Danica McKellar? I thought I heard someone say Danica McKellar.
Math is the lesbian sister of Biology
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.
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 guys do realize that column will now become the arch enemy of the Beer Roster right?
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!
Two columns enter, one column leaves!
How do you go from x = 0.9 to 10x = 9.9??
If x = 0.9 then 10x = 9… What am I missing??
Nevermind… I just got that there are little dots which represent the 99999….
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, $\lim_{n\rightarrow\infty} \sum_{x=0}^{n}9\times10^{-x}$).
Or, you could just kick someone in the nuts.