equities - Why does the adjusted closing price take into account dividends?

I'm trying to get an intuition as to why the adjusted closing price includes a dividend adjustment:

$$\begin{equation} 1 - \frac{dividend}{close} \end{equation}$$

I understand why the adjusted closing price "undoes" the effect of a split on a stock. A stock split hasn't changed the economics of a company so the adjustment effectively removes the split so that the adjusted closing prices before and after the split can be compared.

A dividend does change the underlying economics however so why does the adjusted price account for it? Lastly, the formula also doesn't make sense.