An intutive explanation of natural density (asymptotic density)

Question

I was wondering if someone can provide an intuitive explanation to natural density. I understand the concept very basically (pretty much the definition) but I can't seem to understand what natural density is and how it, for example, shows that $\frac{1}{2}$ of the integers are odd. I have tried to understand the limit concept that is used through this post but I am having a little difficult visualizing it. If someone could please show a pictorial representation of asymptotic density or explain it in a more visual context then that would be great.

Thanks

Answer 1

I can think of lots of ways of visualizing this, so here are a couple that spring to mind. They are all essentially the same.

• A crude way Imagine you colour even numbers white and odd numbers black. Now get a big tub, and lots of white and black paint. Then count through the integers, adding a dash of the paint of the corresponding colour, mixing it up as you go. Initially, it's all black (assuming you start with 1), but then you add an equal amount of white, and get a perfectly balanced grey. Then you add more black, and it darkens, but not all the way to black. Keep going. Eventually, the colour stays almost perfectly at the balanced grey - half the paint in there is white, and half is black, and no matter how long you keep going, the colour won't drift away. (That's the limit part.)
• A more accurate way Start building two towers, one for odd numbers, one for evens, one block at a time. Take photos every time, zooming out to get everything in the shot (this is the division by $n$ in the limit). Then after a while, the two towers look the same size, because the percentage difference gets smaller and smaller (specifically, given any small number, after a certain time the percentage difference between the heights is as small as you want - the limit part). Alternatively, build one tower with everything in, and one with just odds. Then the second one ends up looking half the height.

Similarly, if you build one tower with all the square numbers, and the other containing all numbers, the square tower would seem to shrink away, giving a density of 0.