My favorite class as an undergraduate mathematics major was real analysis. Some of the basic elements are taught in calculus courses, but calculus courses provide just enough theory to use calculus as a tool. Analysis allows students to immerse themselves in the deeper underpinnings of the abstract foundations of math, where we find a world that’s both beautiful and at times surprising.

As one very, very basic example, consider the following question: What’s the next number larger than 0? Suppose I claimed I had such a number. Call it x. If we take x and divide it by 2, we have a number that’s even closer to 0. From this, we conclude there’s no next larger number than 0.

Now consider the following experiment. Take a piece of paper and put two points on it. Label one point 0 and the other point 1. Now imagine a line segment connecting the two points. We can associate each point on the line segment with a number between 0 and 1. One-half would correspond to the point halfway between 0 and 1, one-tenth would correspond to the point one-tenth of the way from 0 to 1, and so on.

Suppose we now draw the line segment. As we start to draw, we must leave the point 0, and in doing so the pencil must pass over a next point on the line segment – the next number larger than 0. But it doesn’t, because there is no next number larger than 0. How can that be?

As one very, very basic example, consider the following question: What’s the next number larger than 0? Suppose I claimed I had such a number. Call it x. If we take x and divide it by 2, we have a number that’s even closer to 0. From this, we conclude there’s no next larger number than 0.

Now consider the following experiment. Take a piece of paper and put two points on it. Label one point 0 and the other point 1. Now imagine a line segment connecting the two points. We can associate each point on the line segment with a number between 0 and 1. One-half would correspond to the point halfway between 0 and 1, one-tenth would correspond to the point one-tenth of the way from 0 to 1, and so on.

Suppose we now draw the line segment. As we start to draw, we must leave the point 0, and in doing so the pencil must pass over a next point on the line segment – the next number larger than 0. But it doesn’t, because there is no next number larger than 0. How can that be?