# Complexity of the Newton's method approximation of $\frac{R}{b}$

I am trying to understand a particular concept in this lecture. The lecture goes on to describe that to compute a division of $$\frac{a}{b}$$, you have to:

1. Compute high-precision representation of $$\frac{1}{b}$$ first (that is the floor of $$\frac{R}{b}$$ where $$R$$ is a large value that is easy to divide by).

2. Multiply the high-precision representation of $$\frac{1}{b}$$ by $$a$$ then divide by $$R$$? (The lecture doesn't state this part).

In the process of using Newton's method to estimate $$\frac{R}{b}$$ (see page 3), the lecture states (in the beginning of page 4) that 'division requires multiplication of different-sized numbers at each iteration'. This leads to the a complexity of $$\theta(d^\alpha)$$ for some $$\alpha \geq 1$$, as argued in page 4.

How's that the case? In the example given on page 3, division requires multiplication of d-sized numbers on each iteration leading to a complexity of $$\theta(\log{d} * d^\alpha)$$. So, how exactly is the statement above, in bold, true?