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

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

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

This square root function on positive real numbers is differentiable and in fact analytic on the positive real numbers.

One can also define the usual notion of the principal square root on the non-negative real numbers, which is continuous on non-negative real numbers but not differentiable at zero.

Definition

Classically, the principal square root function is defined by case analysis:

$$\sqrt{x} \coloneqq \begin{cases} 0 & x = 0 \\ \exp(\frac{1}{2}\ln(x)) & x \gt 0 \end{cases}$$

This definition also works in constructive mathematics for any set of real numbers that satisfy analytic LPO. In the absence of the analytic LPO, the usual principal square root on non-negative real numbers can still be defined on any Cauchy complete Archimedean ordered field, but one has to continuously patch the principal square root on positive real numbers to cover zero.

This following definition of a principal square root function is François G. Dorais?‘s construction of a principal square root function in the non-negative real numbers from Birchfield, Dorais & Strickland 2022:

Definition

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 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.

In constructive mathematics, the Dirchelet indicator function is only defined on the subset of the real numbers which are decidably rational or strongly irrational. This domain is equivalent to the entire real numbers if and only if analytic LPO holds of the real numbers. As a result, the resulting square root function is in general only a partial function 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]

• Madeleine Birchfield, François G. Dorais?, Neil Strickland, Proof in constructive mathematics that the principal square root function exists in any Cauchy complete Archimedean ordered field , MathOverflow, website