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Blast into Math! Analatic nummer theora: ants, ghosts and giants

7 Analytic number theory:

ants, ghosts and giants

So far, our proofs have been based on algebra. Doing algebra is like playing with building blocks. We can
move the blocks here and there, we can stack them and rearrange them, but the blocks remain blocks.
We can’t stretch or squish them without breaking them. Doing analysis is more like playing with dough.

Dough is squishy and can be stretched out thin to make a pizza crust, rolled and twisted to make bread
sticks, or folded to make a croissant. Building blocks can’t be stretched or squished without breaking
them. They can be moved into different positions, but the blocks themselves don’t change. In analysis,
we work with mathematical quantities which are changing. The world is changing too, and it is useful to
understand how it’s changing; we use analysis to do this.When we use analysis to understand numbers,

we are doing analytic number theory.

7.1 Sequences: mathematical ants

Analysis is the study of limits. A limit is an abstract goal. It is an abstract goal, because it may never
actually be reached. In mathematics, we use analysis to to understand how something is changing and
where it is going: what its limit is. To understand limits, we need help from sequences. A sequence is a set
with a specific order. Because a sequence is an ordered set of numbers, we can imagine that a sequence

is like an infinite trail of mathematical ants marching on the number line.

Definition 7.1.1 A sequence is an infinite list of elements indexed by the natural numbers. We write a
sequence as

∞
{x n } n=1 .

The element x n is the n element of the sequence, which we may also call the n term in the sequence.
th
th
The natural numbers, in their natural order, is a sequence. This sequence is

∞
{n} n=1 ,

so the n term in the sequence is the n natural number. For example, the first term is 1, the second
th
th
term is 2, the third term is 3 and so forth. Imagine the sequence as a trail of ants on the number line.
The first ant is at 1. The next ant is at 2. Which way are the ants going? In this example, the sequence
ants are marching tirelessly to the right on the number line and will never stop.

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