nLab real square root function

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 2012), 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 $[0, \infty)$.

Approximate square root functions

Sometimes, the square root function cannot be defined exactly to satisfy the equation $\mathrm{sqrt}(x)^2 = x$. This is the case in some parts of numerical analysis where the focus is on computation and numerical algorithms, and the real numbers end up as rational numbers since the medium which stores the data for the computation, such as physical paper or the calculator or the computer, cannot store an infinite amount of data required to define a real number exactly.

Instead, there are various notions of approximate square root functions. These include the $\epsilon$-tolerant square root functions, which, for a given positive? rational number $\epsilon \in \mathbb{Q}_+$ representing the tolerance?, is a function $f$ from the non-negative? real numbers to the real numbers, which satisfies the following inequality for all non-negative real numbers:

$$\vert f(x)^2 - x \vert \lt \epsilon$$

There are multiple possible $\epsilon$-tolerant square root functions for each tolerance $\epsilon$.

See also

• square root, for square roots in more general mathematical structures

• real quadratic function

• real absolute value

• sign function

• Euclidean field

• real nth root function

References

• Fred Richman, Algebraic functions, calculus style. Communications in Algebra, Volume 40, Issue 7, July 2012, Pages 2671-2683 [doi:10.1080/00927872.2011.584337]