Is the conjugate axis in hyperbola just a number?

Question

My maths teacher is teaching hyperbola these days, and when he drew the hyperbola, I was not able to see $b$ (in $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$) in the graph. When I asked about it, all he did was just marked the points $(0,b)$ and $(0,-b)$. I expected a lot more, but sir just said $b$ according to him has an indirect significance ($a^2+b^2=(ae)^2$). I'm still not satisfied. Is $b$ just a number?