Abstract
The same symbol for a square root is applied to numbers, to operators, to time, to matrices and to functions. What this paper shows is that the five behave differently, with different counts of solutions and different answers on uniqueness. No new theorem or law is claimed.
First, for numbers the count depends on where they are housed——things squaring to $-1$ number $\mathbf{0}$ in the reals, $\mathbf{2}$ in the complexes, and every unit imaginary quaternion, with a worst error of $\mathbf{2.22\times10^{-16}}$ over $200$ trials (Section 2).
Second, for operators the anticommutation relations decide——the Dirac operator squares back to the scalar $p^{2}=\mathbf{-3.3700000000}$, with a worst error including off-diagonal entries of $\mathbf{4.44\times10^{-16}}$ (Section 3).
Third, for time it is something measured——a random walk reaches a root-mean-square of $\mathbf{63.301193}$ at $t=4000$, and the exponent, measured rather than assumed, is $\mathbf{0.502469}$ (Section 4).
Fourth, and this is the core. For matrices it is not unique——Cholesky and symmetric square roots both reproduce the same covariance to within $10^{-12}$, and yet stand $\mathbf{0.562852}$ apart in Frobenius distance (Section 5).
Fifth, for functions the placement decides whether it is a norm——$L^{2}$ scales $\|2f\|$ by exactly $\mathbf{2.000000}$, while $\sqrt{\int|f|}$ scales by only $\mathbf{1.414214}$ (Section 6).
Sixth, the separator is what the root is taken of——number, operator, time, matrix or function: different questions can be asked of each (Section 7).
Keywords: square root, quaternions, Dirac matrices, matrix square root, random walk.
Scope (what is not claimed)
No new theorem or law is claimed——quaternions, Dirac matrices, random walks, matrix square roots and $L^{p}$ spaces are all standard. It is not claimed that five exhausts the list——five were counted, and no claim is made that there are no others. This is not the reverse of Paper 157 or Paper 294——Paper 157 took a census of squares and Paper 294 of square roots of scale. The axis here is what the root is taken of, neither the field nor the scale. Operator theory is not entered——existence and uniqueness of square roots of unbounded operators are not treated. No interpretation of quantum mechanics is offered——the Dirac operator is used as a specimen of algebra. Algorithms for matrix square roots are not compared——which of Cholesky and the symmetric root is faster or more stable is not the subject. No general theory of $L^{p}$ is given——only $p=2$ and its neighbourhood are examined. Relation to earlier papers: Paper 157 took a census of squares——this runs the other way, but the axis is what the root acts on, so it is not that table reversed. Paper 294 showed that separation improves only as the square root of length——that is one case of the third kind here. Paper 236 showed that one and the same “$=0$” returns five different kinds of collection——Section 2 is a special case, with the solutions of $x^{2}=-1$ changing with the ambient set. Paper 375 showed that units, not data, choose the first principal component——the whitening used there is a covariance square root, whose non-uniqueness is measured here. Paper 216 treated approaching without coming closer——the distance there is $L^{2}$, the fifth kind here. Paper 300 showed that sharing a root is decidable——the five are separate roots under the criterion of what a quantity carries. What is added is setting the five side by side and measuring counts and uniqueness for each, confirming over $200$ quaternions, checking the Dirac square to $\mathbf{4.44\times10^{-16}}$, measuring two matrix square roots $\mathbf{0.562852}$ apart, and placing the separator on what the root is taken of.
1. Notation and the Question
Setting: the radical sign returns something that squares to what is under it. What is being squared, however, is not written.
Observation 1.1 if squaring means different things, so does the root. Multiplication of numbers, composition of operators, matrix products, integrals of functions——one symbol inverting four or more different operations.
Observation 1.2 this is not Paper 157 reversed. That paper took a census of where the number two appears——this one sorts by what the radical acts on, neither by field nor by scale.
Question: how do counts and uniqueness differ across the five?
2. Numbers: the Count Depends on the Ambient Set
Computation 2.1: counting the things that square to $-1$ in three settings.
| ambient set | count |
|---|---|
| reals | $\mathbf{0}$ |
| complexes | $\mathbf{2}$ ($i$ and $-i$) |
| quaternions | every unit imaginary quaternion (a two-sphere) |
Observation 2.2 the quaternion case was checked directly. Two hundred random unit imaginary quaternions squared, worst error $\mathbf{2.22\times10^{-16}}$——not merely the distinguished $i$, $j$ and $k$.
Observation 2.3 without unit length it fails. An imaginary quaternion of length two squares to $-4$——the far side of the boundary, confirming the check does not say “any quaternion at all”.
Observation 2.4 a special case of Paper 236. There, one “$=0$” returned five kinds of collection——here the solutions of $x^{2}+1=0$ run from the empty set to two points to a two-sphere.
3. Operators: Anticommutation Decides
Computation 3.1: assembling the gamma matrices in the Dirac representation and computing $(\gamma\cdot p)^{2}$ with $p=(1.3,-0.7,0.4,2.1)$.
| quantity | value |
|---|---|
| diagonal of $(\gamma\cdot p)^{2}$ | $\mathbf{-3.3700000000}$ |
| $p^{2}=E^{2}-|\mathbf{p}|^{2}$ | $\mathbf{-3.3700000000}$ |
| worst error including off-diagonal | $\mathbf{4.44\times10^{-16}}$ |
Observation 3.2 squaring a matrix returns a scalar. Every off-diagonal entry vanishes at the $10^{-16}$ level——not a coincidence but a consequence of the anticommutation relations.
Observation 3.3 here the root is built rather than taken. Asking for a square root of the Laplacian produced matrices——the search for a solution forced an enlargement of where things live, the same shape as the quaternions of Section 2.
Observation 3.4 uniqueness behaves differently from numbers. The representation of $\gamma$ is not unique, and the Dirac and Weyl representations satisfy the same relations——asking about equivalence is more natural than counting.
4. Time: Something Measured
Computation 4.1: running $4000$ one-dimensional random walks for $4000$ steps and measuring the root-mean-square displacement.
| $t$ | r.m.s. | ratio to $\sqrt{t}$ |
|---|---|---|
| $100$ | $\mathbf{9.918064}$ | $0.991806$ |
| $400$ | $\mathbf{20.030277}$ | $1.001514$ |
| $1600$ | $\mathbf{40.387189}$ | $1.009680$ |
| $4000$ | $\mathbf{63.301193}$ | $1.000880$ |
Observation 4.2 the exponent was measured, not assumed. A logarithmic slope of $\mathbf{0.502469}$——$\sqrt{t}$ was not put in to get $0.5$ out. It is neither $1$ nor $0$.
Observation 4.3 this root does not come from algebra. It comes from variances adding under independent summation——Sections 2 and 3 were algebra; this is probability.
Observation 4.4 Paper 294 treated this kind. Separation improving only as the square root of length——that paper took a census of this root across three scales; here it is one of five.
5. Matrices: Not Unique
Computation 5.1: taking a covariance with $\Sigma_{11}=4$, $\Sigma_{12}=1.5$, $\Sigma_{22}=2$ and constructing two square roots.
| root | entries | error reproducing $\Sigma$ |
|---|---|---|
| Cholesky | $[[2.000000,0],[0.750000,1.198958]]$ | $\mathbf{2.22\times10^{-16}}$ |
| symmetric | $[[1.947200,0.456524],[0.456524,1.338501]]$ | $\mathbf{0.00}$ |
| distance between them (Frobenius) | $\mathbf{0.562852}$ | — |
Observation 5.2 this is the core. Both reproduce $\Sigma$ to within $10^{-12}$ and stand $\mathbf{0.562852}$ apart——the phrase “the square root of $\Sigma$” does not name one object.
Observation 5.3 the choice changes the result. Where coordinates are read after whitening, the chosen root fixes the coordinates——the same shape as Paper 375, where units chose the first principal component; here the choice of root chooses.
Observation 5.4 this is not the “two” of numbers either. Sign alone would give two, but any orthogonal factor on the right gives another——a continuum, not a pair.
6. Functions: Placement Decides Whether It Is a Norm
Computation 6.1: scaling $f$ by a constant and measuring two quantities.
| $c$ | $L^{2}$: $\|cf\|/\|f\|$ | $\sqrt{\int|cf|}/\sqrt{\int|f|}$ |
|---|---|---|
| $2$ | $\mathbf{2.000000}$ | $\mathbf{1.414214}$ |
| $9$ | $\mathbf{9.000000}$ | $\mathbf{3.000000}$ |
Observation 6.2 $L^{2}$ is homogeneous and the other is not. At $c=9$, $\mathbf{9.000000}$ against $\mathbf{3.000000}$——whether the radical sits outside or inside the integral decides whether the norm axioms hold.
Observation 6.3 before writing, the prediction was that the triangle inequality would fail. It did not. Since $\sqrt{a+b}\le\sqrt{a}+\sqrt{b}$ always holds, $\sqrt{\int|f|}$ is subadditive too——only homogeneity breaks. Without recording the prediction, this error would not appear anywhere.
Observation 6.4 the distance of Paper 216 is this kind. The “closeness” in approaching without coming closer is $L^{2}$——its radical is placed where it is for the sake of homogeneity.
7. The Separator
Observation 7.1 the separator is what the root is taken of.
| of what | count of solutions | uniqueness |
|---|---|---|
| number | $0$, $2$, or a two-sphere | depends on the ambient set |
| operator | as many as representations | equivalence is the question |
| time | a measured quantity | exponent $\mathbf{0.502469}$ |
| matrix | a continuum | none (distance $\mathbf{0.562852}$) |
| function | part of the definition | fixed by homogeneity |
Observation 7.2 they part under the criterion of Paper 300. A number’s root carries an ambient set; a matrix’s root carries a choice——write “square root” without its belongings and it is undetermined which is meant.
Observation 7.3 no claim that five exhausts the list. Five were counted——to call a list exhaustive one must first fix how to exhaust it.
Observation 7.4 what is not said. Neither operator theory nor the general theory of $L^{p}$ is entered——the Dirac operator served as a specimen of algebra, and no interpretation of quantum mechanics is offered.
8. Roll Call and Closing
- $\mathbf{0}$ in the reals (Computation 2.1): $\mathbf{2}$ in the complexes.
- every unit imaginary quaternion (Observation 2.2): $200$ tested, error $\mathbf{2.22\times10^{-16}}$.
- length two gives $-4$ (Observation 2.3): the far side.
- $(\gamma\cdot p)^{2}=\mathbf{-3.3700000000}$ (Computation 3.1): error $\mathbf{4.44\times10^{-16}}$.
- a matrix squares to a scalar (Observation 3.2): from anticommutation.
- diffusion exponent $\mathbf{0.502469}$ (Observation 4.2): measured.
- roots $\mathbf{0.562852}$ apart (Computation 5.1): the core.
- both reproduce $\Sigma$ (Computation 5.1): not unique.
- $L^{2}$ scales by $\mathbf{9.000000}$ (Computation 6.1): homogeneous.
- the other by $\mathbf{3.000000}$ (Observation 6.2): not homogeneous.
- the triangle inequality held (Observation 6.3): the prediction was wrong.
Closing: The same symbol for a square root is applied to numbers, operators, time, matrices and functions. For numbers the count depends on the ambient set——things squaring to $-1$ number $\mathbf{0}$ in the reals, $\mathbf{2}$ in the complexes and every unit imaginary quaternion, checked over $200$ random ones with a worst error of $\mathbf{2.22\times10^{-16}}$. Unit length is required, since an imaginary quaternion of length two squares to $-4$. This is a special case of Paper 236, with the solutions of $x^{2}+1=0$ running from the empty set to two points to a two-sphere. For operators the anticommutation relations decide——$(\gamma\cdot p)^{2}$ has diagonal $\mathbf{-3.3700000000}$, equal to $p^{2}$, with a worst error including off-diagonal entries of $\mathbf{4.44\times10^{-16}}$. A matrix squaring to a scalar is a consequence of the algebra, and here the root is built rather than taken. The representation is not unique, so equivalence is the natural question rather than a count. For time it is measured——the random walk reaches $\mathbf{63.301193}$ at $t=4000$ and the logarithmic slope gives $\mathbf{0.502469}$, obtained without assuming $\sqrt{t}$. It comes not from algebra but from variances adding, and Paper 294 treated this kind. For matrices it is not unique——Cholesky and symmetric roots of one covariance both reproduce it to within $10^{-12}$ and stand $\mathbf{0.562852}$ apart in Frobenius distance. Where coordinates are read after whitening, the chosen root fixes them, the same shape as Paper 375. Nor is it the “two” of signs: any orthogonal factor on the right gives another, so there is a continuum. For functions the placement decides——$L^{2}$ scales by $\mathbf{2.000000}$ and $\mathbf{9.000000}$ while $\sqrt{\int|f|}$ scales by $\mathbf{1.414214}$ and $\mathbf{3.000000}$. The prediction before writing was that the triangle inequality would fail; it did not, because $\sqrt{a+b}\le\sqrt{a}+\sqrt{b}$ always holds, and only homogeneity breaks. The distance of Paper 216 is this $L^{2}$, whose radical sits where it does for homogeneity. The separator is what the root is taken of——an ambient set, a representation, a measurement, a choice, or an axiom. Under the criterion of Paper 300 the belongings differ, so writing “square root” without them leaves it undetermined. Paper 157 took a census of the number two; sorting by what the radical acts on does not reverse that table. To be honest——no claim is made that five exhausts the list. Operator theory was not entered, and existence and uniqueness for unbounded operators were not treated. The Dirac operator served as a specimen of algebra with no interpretation of quantum mechanics offered, algorithms for matrix square roots were not compared, and no general theory of $L^{p}$ was given. What can be said is that one symbol inverts five different operations, and that counts and uniqueness differ in all five, and no further.
References
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- Einstein, A. (1905). Uber die von der molekularkinetischen Theorie der Warme geforderte Bewegung von in ruhenden Flussigkeiten suspendierten Teilchen. Annalen der Physik.
- Hamilton, W. R. (1847). On Quaternions (read 11 November 1844). Proceedings of the Royal Irish Academy.
- Higham, N. J. (2008). Functions of Matrices: Theory and Computation. SIAM.
- Yamagishi, Y. (2026). A Census of the Twos [Paper 157]. Zenodo. 10.5281/zenodo.22091931
- Yamagishi, Y. (2026). It Approaches and Has Not Come One Step Closer [Paper 216]. Zenodo. 10.5281/zenodo.22119579
- Yamagishi, Y. (2026). One and the Same “= 0” Returns Five Different Kinds of Collection [Paper 236]. Zenodo. 10.5281/zenodo.22129043
- Yamagishi, Y. (2026). Separation Improves Only as the Square Root of the Length [Paper 294]. Zenodo. 10.5281/zenodo.22171603
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). What Chooses the First Principal Component Is Not the Data but the Units [Paper 375]. Zenodo. 10.5281/zenodo.22698703
Reference
Yamagishi, Y. (2026). One and the Same Square Root Is Taken of Five Different Things [Paper 531]. YOI: https://www.yamagishiyuuki.com/yoi/531
DOI: 10.5281/zenodo.23014103
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 531. This paper is a preprint and has not been peer reviewed.