Ask Professor Puzzler
Do you have a question you would like to ask Professor Puzzler? Click here to ask your question!
Eighth grader Sandesh asks, "consider y=x*x*x*x*x*...... up to infinity . hence y=x*y which means either y=0 or x=1 how can we explain it?"
First, you need to understand that what you're talking about is called a LIMIT. We can't actually multiply x times itself an infinite number of times, because there isn't enough time in the universe to do that!
What we can say is that we're going to multiply x by itself n times, and n is a really really big number. In fact, we say that n is "approaching infinity."
Now, some things have limits, and some things don't. For example, consider this:
1/2 x 1/2 x 1/2 x ...
If we multiply 1/2 times itself a billion times, we get something really close to 0, and if we multiply it a billion and one times, we get something that's even closer to zero! And if we keep multiplying, we get closer and closer to 0. So we say the LIMIT is 0, or that it converges to zero.
On the other hand, consider this:
2 x 2 x 2 x 2 x ...
If we multiply 2 times itself a billion times, we get an astronomically huge number. But if we multiply it a billion and one times, we aren't getting closer to any particular number, we're just getting even more astronomically huge. So this doesn't really have a limit. We say it diverges.
So, I think the important thing you're missing here is that x * x * x * x ... is only convergent for certain values. it converges if x is between 0 and 1 (inclusive). Otherwise it is divergent. (It's divergent for x < 0 because the product keeps flopping back and forth between negative and positive each time you multiply, so it never settles in on any one value. It's divergent for x > 1 because the product explodes to astronomically large values).
And if it's DIVERGENT, then you aren't allowed to do your cute little substitution. However, if the expression does converge (x is between 0 and 1) then your substitution is a useful way of looking at it. (There are more precise ways of looking at it, that you'll learn someday when you take a calculus class, but this will be sufficient for now).
Here's why your substitution won't work in all cases:
The original equation is: y = xn, where n is approaching infinity.
Rewrite it slightly: y = x * xn - 1
xn - 1 is the part you want to replace with y.
You see, what you're really doing when you substitute is, you've assumed that "the limit as n approaches infinity of xn - 1" is the same as "the limit as n approaches infinity of xn". But this is only true if xnhas a limit! When xn has a limit, then you can do your substitution:
y - xy = 0
y( 1 - x) = 0
y = 0 or x = 1
And this is actually the correct result, for the situation when xn has a limit. Because if xn has a limit, x must be between 0 and 1. If x is any number LESS than 1, then when you multiply it by itself a billion times, you get something really close to zero. We say that its limit is 0. And if x = 1, then y = 1 as well, both of which fit your solution.
The entire study of limits is much more complex than I've made it seem here; I've done my best to keep the math and the terminology fairly simple (hopefully I've succeeded in that!). When you get into a Calculus or Pre-Calculus class, you'll find that there's a lot more to it, but hopefully that'll at least whet your appetite for more!