Are there any instances of significant progress deriving from mathematical 'silliness'?

Question

Last night I thought I'd be silly finding the eigenvalues of a $2\times2$ matrix $A$ with real components. Instead of calculating $\det(A-\lambda I)=0$ I tried to compute the determinant by subtracting $\lambda$ in silly ways.

For example I computed

$$\begin{align} det \left(\begin{bmatrix} a&b\\ c&d \end{bmatrix}-\lambda \begin{bmatrix} 0&1\\ 1&0 \end{bmatrix} \right)&=0 \\ ad-(c-\lambda)(b-\lambda)&=0 \\ ad-(bc-b\lambda-c\lambda+\lambda^2)&=0 \\ ad-bc+b\lambda+c\lambda-\lambda^2&=0 \\ \lambda^2-\det(A)+\lambda(b+c)&=0 \end{align}$$

Firstly is there a name for $b+c$? I know that $tr(A)=a+d$

Secondly, are there any good examples of interesting maths having been derived from 'silliness'? I appreciate that "silly" is hard to define but I am referring to a whimsical or carefree maths with no real hope of anything arising from it. I would be especially interested to hear about mathematics that has been done deliberately incorrectly and yet new ideas are formed or questions arise (in the sense of my example)

edit To attempt a clarification. My silliness led to me wanting to know about the 'trace' of $b+c$ a question I had not set out to investigate. I understand the question is vague and so might need closing on that grounds.

I hope this is an interesting question for the community but I shall not hasten to remove it should it not fit site standards.

Thanks in advance for any help.

Answer 1

Like @Dan Brumleve in the comments, I was immediately reminded of John Conway, in particular of this article which you may find of interest. Its introduction reads:

John Horton Conway claims to have never worked a day in his life. This adaptation from the biography Genius at Play shows how serious advances such as the surreal numbers can spring out of fun and games.

... although upon consulting Wikipedia, it seems the surreals were not entirely unknown before.