nLab real square root function







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


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 sqrt:(0,)\mathrm{sqrt}:(0, \infty) \to \mathbb{R} defined by

sqrt(x)e 12ln(x)\mathrm{sqrt}(x) \coloneqq e^{\frac{1}{2} \ln(x)}

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

f n(x)={1/2 n whenx1/4 n e 12ln(x) whenx1/4 n 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 x1/4 nx \leq 1/4^n or x1/4 nx \geq 1/4^n, but it is possible to work around this by patching three functions together:

  • f k 0:[0,1/4 k)f^{0}_k:[0,1/4^k) \to \mathbb{R} is the linear function f(x)=2 kxf(x) = 2^k x,
  • f k +:(0,)f^{+}_k:(0,\infty) \to \mathbb{R} is defined as min(2 kx,exp(12ln(x)))\min(2^k x,\exp(\frac{1}{2} \ln(x))).

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

Now the sequence of functions (f n) n=0 (f_n)_{n=0}^\infty so defined converges uniformly on any bounded interval to a continuous function ():[0,)[0,)\sqrt{(-)}:[0, \infty) \to [0, \infty) called the principal square root function. It is easily seen that (x) 2=x(\sqrt{x})^2 = x and x 2=x\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=xf(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 :1_{\mathbb{Q}}:\mathbb{R} \to \mathbb{R} be the constructive Dirichlet indicator function, defined as 1 (x)11_{\mathbb{Q}}(x) \coloneqq 1 for every rational number qq \in \mathbb{Q}, and 1 (x)01_{\mathbb{Q}}(x) \coloneqq 0 for every real number xx apart from every rational number qq \in \mathbb{Q}

q.|xq|>0\forall q \in \mathbb{Q}. \vert x - q \vert \gt 0

Then the function f(x)(1) 1 (x)xf(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,)[0, \infty).

See also


Last revised on January 28, 2024 at 04:39:25. See the history of this page for a list of all contributions to it.