# Chapter 3: real numbers

12 December 2022

## Exercise 3.2

Medium. Assuming the eventual periodicity of [the continued fractions $$[1; \overline 2]$$ and $$[5; 3, \overline{1, 2}]$$], show that they must represent [$$\sqrt2$$ and $$7 - \sqrt3$$] respectively. (Hint: Find a quadratic equation that these quantities must satisfy, and use the quadratic formula.)

### Solution

The idea is to “unravel” the continued fraction. In the first case, we have \begin{aligned} x &= 1 + (2 + (2 + \dots )^{-1})^{-1} \newline &= 1 + (1 + [1 + (2 + \dots)^{-1}])^{-1} \newline &= 1 + (1 + x)^{-1}, \end{aligned} which gives the quadratic equation $$x^2 - 2 = 0.$$ In the second case, left to the reader, we obtain the quadratic equation $$x^2 - 14x + 46 = 0.$$ Resolving the plus/minus appropriately (noting, for instance, that $$[1; \overline2] > 0,$$) we obtain the desired.

## Exercise 3.3

Easy. The first step in the Eudoxan theory was to supply a criterion as to when a length ratio $$a : b$$ would be greater than another such ratio $$c : d.$$ This criterion is that some positive integers $$M, N$$ exist such that $$Ma > Nb$$ and $$Nd > Mc.$$

Can you see why this works?

### Solution

Rearranging gives $$\dfrac{a}{b} > \dfrac{N}{M} > \dfrac{c}{d}.$$

## Exercise 3.4

Hard. Generalize rigorously the preceding length ratios to real numbers, defining the sum and product.

# FIX THIS TEXT, S.

### Solution

RP is essentially asking for a construction of the reals. The nicest way to do this, in my opinion, involves defining a real number as an equivalence class of Cauchy sequences of rationals. This appears as an exercise in Chapter 2 of Baby Rudin and probably somewhere in any decent real analysis book you consult.

Having shown that this construction is well-defined, the sum and product of two real numbers (=sequences) can simply be taken termwise without issue, since limits preserve sums and products.