I have a good friend that I spend a lot of time biking with. Actually we do more talking than biking. For us, biking is more of an excuse for talking than anything else.
We are both retired, but Frank spent forty years teaching math at the University of New Mexico and various other schools around the country. And he has degrees in both philosophy and the philosophy of science as well as multiple degrees in math. So he is a pretty interesting talker. At least to me he is, but my wife and his girlfriend have both been known to accuse us of going on and on “about all your bullshit, forever and ever.” And then they always finish up by reminding us “and nobody is interested in all that crap.”
That may all be true, but Frank and I think we say a few interesting things occasionally. And, I have to say that all Frank’s bullshit about math is endlessly interesting to me. Especially since he has the talent of making some pretty abstract ideas actually understandable to someone like me who has forgotten all the math he once knew. And what I once knew about math was actually very little.
So Frank and I thought we might share some of our occasional discussions about math on this blog, and that maybe, hopefully, a few other people might also find them interesting.
So, here is one of the first ideas Frank tossed at me as we cruised the bike trails of Albuquerque one windy New Mexico day.
Frank said, here’s a fact accepted by all mathematicians but which almost all non mathematicians think is total nonsense. Non-mathematicians universally say, “Of course 1 isn’t the same as .999…. Any idiot knows that .999… is smaller than 1.
So, lets begin by writing out this problem as a mathematician would. And then try to prove that it is true.
1 = .999… (the dot, dot, dot part means the 9s go on forever.)
Below is how a mathematician would try to prove that this is true.
Let’s suppose you accept that 1/3 = .333… Almost everyone accepts this. You probably heard it the first time in grade school, long before your critical faculties developed, and you haven’t questioned it since. I certainly didn’t.
So we can write this as three identical equations.
1/3 = .333…
1/3 = .333…
1/3 = .333…
And now if I add up both the left and right sides of these equations we get:
1/3 + 1/3 +1/3 = .333… + .333… + .333…
or 1 = .999…
Voila! We have now proved that 1 actually does equal .999…
Or, if you are still skeptical, here’s another way we can prove that 1=.999…
.999… must be some number, we don’t know exactly what number because of the fact that the .999…s go on forever. So let’s give it the mathematician’s favorite name which is x.
So, lets write this out as a mathematician would:
Here it is: x = .999…
Now, let’s multiply both sides of this equation by 10, giving us a new equation.
10x = 9.999….
Now, subtract the original equation (x = .999…) from the above equation
10x = 9.999…
x = .999… (subtract this from the above)
9x = 9 (The answer after subtraction)
Now divide both sides by 9, and we get x = 1
But we already know that x=.999… We defined x this way.
We now we see that 1 = .999…
So, once again we have proved that 1 = .999…
So, we have now proved pretty conclusively, using mathematics, which is basically just logic, that even though it doesn’t look like 1 could possibly be the same thing as .9999999999999999…, it actually is exactly the same thing. Or else some logical mistake was made somewhere, but I agreed with Frank that all of the steps viewed individually were absolutely logical.
Well, big deal, I said to Frank, so what? Why would I or anyone else care about this?
Frank hasn’t answered this question yet, but it seems to me that the answer might be something like this.
Not everything that looks like it’s true actually is true. And these simple little proofs about .999… make this apparent. We thought .999… was smaller than 1, but actually it isn’t. Our intuitions were wrong.
And this is exactly why math is so valuable, why it is a great thing to know. It helps us figure out what is really true and separates true things from stuff that only looks true. And that is a very powerful tool to have. Logic trumps intuition every time, if what you are looking for is truth.
And without math there is no science. Math is what makes science work, it is what empowers science. And without science there is no technology and without technology there are no iPhones. And who wants to go back to that?
Or for that matter, without math and science we are back in the hunter-gather world, or maybe the Neolithic world of primitive farms and just the beginnings of towns. Without math and science, we are back in the world of magic where there is no medicine, or air conditioning or even enough food for everyone. Of course all of these things bring a whole new set of problems with them, but with math and science all these problems are solvable. I hope.
But, then again, math can’t do everything, if you happen to be looking for love, not truth, then math doesn’t work quite so well. Actually it doesn’t work at all. But, nothing is perfect.”
Maybe Frank and I will make this an ongoing series about math. We’ll see how it goes.
Post by Fred Hanselmann
Math provided by Fred’s mathematician friend Frank