[Math] Examples of back of envelope calculations leading to good intuition

Question

Some time ago, I read about an "approximate approach" to the Stirling's formula in M.Sanjoy's Street Fighting Mathematics. In summary, the book used a integral estimation heuristic from spectroscopy

$$\int_{\mathbb{R_{\ge 0}}} f(x) dx \approx \max(f) * (\text{point where}\ \frac{1}{2} \max(f)\ \text{is achieved}) $$

to estimate the Gamma function with $f(x) = f_t(x) = x^{t}e^{-x} $. This leads to the estimate

$$\Gamma(n) = \int_{\mathbb{R}_{\ge 0}} x^{n}e^{-x} dx \approx \sqrt{8 n} \left(\frac{n}{e}\right)^n$$

which is an extremely good estimate (the "proportionality constant" $\sqrt{8}$ is correct to within 10% with correct order of growth.) This heuristic was very helpful in understanding the growth of the actual formula $\Gamma(n) \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$.

I think approximations of this sort is useful because

• It gives a sense of what the answer "ought" to be.
• When the approximation deviates from the actual answer, it's interesting to think about which part of the approximation failed.

Another "back-of-envelope calculation" is the calculation for the Prime Number Theorem in Courant and Robbins, What is Mathematics?

My Question. I am looking for similar instances in mathematics where "back-of-envelope calculations" such as the above leading to good intuition in mathematics.

For the purpose of my question, let's require that the calculation addresses questions in pure mathematics (so, no physics, engineering, etc. since there seems to already be plenty of literature on this).

Edit: as per helpful feedback from Peter LeFanu Lumsdaine, I removed two requirements: "Does not require anything beyond, say undergraduate mathematics" and "Does not formalize into a rigorous proof."

Edit 2 (as per helpful discussion in the comments): part of what I am interested in is how people use various techniques to compute/approximate objects of interest. For instance, I think we can all agree that the use of integral approximation demonstrated above is quite creative (if not, a nonstandard way of approaching Stirling). In response to Meow, topological invariants for "similar" (homotopy equivalent, homeomomorphic, etc) mostly amounts to the "same sort" of argument, so I would count that as "one" approximation argument unless there is a particular example where the heuristic argument is highly nontrivial.

Answer 1

Finding the primitive of logarithm

Finding a primitive amounts to calculating an integral. Calculating an integral amounts to measuring the area under a curve. What is the area under the curve of $\ln$? We want to calculate the value: $$\int_1^x \ln(t) \text{d}t$$

What do we know about function $\ln$? We know that logarithm is an increasing function which goes to infinity, and we know that logarithm is a "slow" function.

How does "slow" translate on the back of an envelope calculation, and how does it help us estimate the area under the curve?

On the back of our envelope, we will write the following thing: the curve of $\ln$ is flat. It's a horizontal line.

The curve is so flat that we can say: for a very large $x$, for almost every value $x_2 < x$, $\ln(x_2) \approx \ln(x)$. In other words, the graph of $\ln$ is made of two parts:

• a short vertical line that goes from $\ln(1) = 0$ to $\ln(1+\varepsilon) = \ln(x)$;
• a long horizontal line that goes from $\ln(1+\varepsilon) = \ln(x)$ to $\ln(x) = \ln(x)$.

Calculating the area under the curve becomes easy: it's the area of a rectangle. Thus: $$\int_1^x \ln(t) \text{d}t \approx x \ln(x)$$

We have a candidate for our primitive! A possible primitive for $\ln$ is function $F$ given by: $F(x) = x \ln(x)$.

How close was our approximation? We can check our result by taking the derivative of $F$: $$F'(x) = \ln(x) + 1$$

We're off by a constant term! Constant terms are easily removed. A correct primitive of $\ln$ is function $G$ given by: $$G(x) = x \ln(x) - x$$