# nLab real square root function

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

The different types of square root partial functions on the real numbers that satisfy the functional equation $f(x)^2 = x$ on some subset of the real numbers.

## Definition

This following definition of a principal square root function comes from a joint proof of the existence of a principal square root function in the non-negative real numbers in constructive mathematics by Madeleine Birchfield here and François G. Dorais here. Most of the following text has been copied from the two sources.

Let us define the real numbers to be a Cauchy complete Archimedean ordered field, since that is the minimum requirement for which the inverse function theorem is true.

There exists a square root function $\mathrm{sqrt}:(0, \infty) \to \mathbb{R}$ defined by

$\mathrm{sqrt}(x) \coloneqq e^{\frac{1}{2} \ln(x)}$

Let us define continuous functions $f_n:[0, \infty) \to [0, \infty)$:

$f_n(x) = \begin{cases} 1/2^n & \mathrm{when}\; x \leq 1/4^n \\ e^{\frac{1}{2} \ln(x)} & \mathrm{when}\; x \geq 1/4^n \\ \end{cases}$

As stated, that requires knowing whether $x \leq 1/4^n$ or $x \geq 1/4^n$, but it is possible to work around this by patching three functions together:

• $f^{0}_k:[0,1/4^k) \to \mathbb{R}$ is the linear function $f(x) = 2^k x$,
• $f^{+}_k:(0,\infty) \to \mathbb{R}$ is defined as $\min(2^k x,\exp(\frac{1}{2} \ln(x)))$.

Since these functions agree on their overlap, and their domains comprise all of $[0, \infty)$ we do get a total function $f_n:[0, \infty) \to [0, \infty)$ as a result.

Now the sequence of functions $(f_n)_{n=0}^\infty$ so defined converges uniformly on any bounded interval to a continuous function $\sqrt{(-)}:[0, \infty) \to [0, \infty)$ called the principal square root function. It is easily seen that $(\sqrt{x})^2 = x$ and $\sqrt{x^2} = x$.

The principal square root function is used to define the Euclidean metric in Euclidean spaces.

## Other square root functions

According to (Richman 2010), given the existence of a principal square root function, there are an uncountable number of functions that satisfy the functional equation $f(x)^2 = x$ on some subset of the real numbers. Each of these could be called a real “square root function”.

For example, let $1_{\mathbb{Q}}:\mathbb{R} \to \mathbb{R}$ be the constructive Dirichlet indicator function, defined as $1_{\mathbb{Q}}(x) \coloneqq 1$ for every rational number $q \in \mathbb{Q}$, and $1_{\mathbb{Q}}(x) \coloneqq 0$ for every real number $x$ apart from every rational number $q \in \mathbb{Q}$

$\forall q \in \mathbb{Q}. \vert x - q \vert \gt 0$

Then the function $f(x) \coloneqq (-1)^{1_{\mathbb{Q}}(x)} \sqrt{x}$ is a real square root function, even though it is nowhere continuous, and not defined on the entire half-open interval $[0, \infty)$.