Riemann prose, part 3


I left off with the sentence about there being a lot of irrational numbers. This lets me segue into big numbers. Really big numbers! The ones that you can’t count on one hand big!

Say we have the equation:
x(n+1) = x(n) + 1
If we start at n = 0, and x0 = 0; then x1 = 0 + 1 = 1.

If we turn around and put 1 back into the formula,
x2 = x1 + 1 = 1 + 1 = 2
Put 2 back into the formula, we get 3! Where will the madness end!!?

x(n+1) = x(n) + 1 is unbounded. It just keeps getting bigger without limit.


(Unbounded line going to infinity and beyond.)

By definition, this is what we (some of us; me) call “infinity” (∞). Infinity as a math concept was pretty much handed to us by Zeno around 450 BC. The symbol ∞, representing infinity, was popularized by John Wallis somewhere around the 1650’s AD. Euler, in the 1700’s, used “i” to mean “infinity”, kind of confusing things. So, let’s just ignore him.

There are different kinds of infinity, unless you consider the car brand, then there’s only one kind. But, for numbers, we have an infinite number of values between 0 and 1, giving us an infinite number of both rational and irrational numbers. There’s an infinite number of counting numbers (positive integers going from 1 to infinity). There’s an infinite number of prime numbers. We could say there’s an infinite number of negative integers, but maybe that’s just the positive integers facing the wrong direction on the graph.

Consider this,
x(n+1) = x(n) + 1
goes to infinity. (0, 1, 2, 3…)

x(n+1) = x(n) + 2
goes to infinity twice as fast. (0, 2, 4, 6…)

They both go to infinity, so does this mean that
x(n) + 2 = x(n) + 1
if we allow n (number of steps) to get big enough? Sadly, no, but it would be cool if it did.

This just means that infinity isn’t just one number, and we have to look at each formula on a case-by-case basis to determine what infinity is doing there.

x(n+1) = x(n) + 1
is unbounded. It just keeps getting bigger.

How about: y = 1/x ?

As x gets bigger, y gets smaller. y will approach 0 without ever actually getting there. Because x is unbounded and has no ultimate top value, 1/x will keep getting infinitely smaller. Conversely,

y = 1/0

Is undefined because this is in fact infinity, and thus has no upper value.

Does this mean that having infinity in a formula is a bad thing? Depends.

y = 1/((2 + 1/x))

As mentioned above, 1/x approaches 0 as x gets bigger. If we assume that x has gotten big enough, then:

y => 1/(2 + 0) => 1/2

Y converges to 0.5 as x approaches infinity. It doesn’t actually reach 1/2, since 1/x never actually gets to 0, but it comes close enough as to make no nevermind.

x = 0.5; y = 1/(2+2) = 0.25
x = 1; y = 1/(2 + 1/1) = 0.33
x = 2; y = 1/(2 + 0.5) = 0.40
x = 3; y = 1/(2 + 0.33) = 0.42
x = 4; y = 1/(2 + 0.25) = 0.44
x = 10; y = 1/(2 + 0.1) = 0.476
x = 100; y = 1/(2 + 0.01) = 0.498

So, infinity, as with fire, is our friend as long as we take precautions against being burned.

We’ll be seeing infinity showing up a lot as we go along, as well as convergence (in fact, we already saw (non-)convergence when we were introduced to the Riemann Zeta function).

 

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