Abstract
Division undoes a multiplication. And what stands in the way of the undoing is not of one kind. This paper counts five obstructions and shows in numbers what happens under each. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, one $12\div3=4$ is three different fours. $12\,\mathrm{m}\div3$ is a length, $12\,\mathrm{m}\div3\,\mathrm{m}$ is a bare count, and $12\,\mathrm{m}\div3\,\mathrm{s}$ is a speed — what separates them is the units (Section 2).
Second, in the integers division almost never works. Over $200000$ random pairs the quotient landed back in the integers $1404$ times, that is $\mathbf{0.7020\%}$ (Section 3).
Third, the divisor may have no inverse. In $\mathbb{Z}/12$, $6x=1$ has no solution at all while $6x=0$ has $\mathbf{6}$ — not merely absent but not unique either (Section 4).
Fourth, when the multiplication does not commute there are two quotients. Over $19977$ random matrix pairs the left and right quotients agreed $0$ times (Section 5).
Fifth, this is the core. In $S_3$ every subgroup has a whole-number index, and yet the quotient is a group only when the subgroup is normal — the numbers always divide and the objects do not (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — Lagrange's theorem, normal subgroups and quotient groups, the Riemann sphere, IEEE 754, wheel algebras, left and right quotients in a non-commutative setting, and the partitive/quotitive distinction are all standard. No completeness is claimed — five obstructions appear and are not said to exhaust the matter. No correct definition of division by zero is chosen — which convention is better is not argued. This is not a textbook on algebra — neither localisation nor the general theory of quotient rings is treated. No position is taken on the teaching of arithmetic — the partitive and quotitive senses are cited as a distinction that already has a name. Relation to earlier papers: Paper 16 counted six roots of the equals sign, Paper 227 two kinds of inequality, Paper 228 eight plus signs — this is the fourth. Paper 228 closed by finding that only the failure to commute admits no coordinate — Section 5 here is that same obstruction appearing a second time, on the division side. Paper 181 showed that multiplication does not decide the second system — division undoes that multiplication and so inherits the ambiguity whole. What is added is separating three fours by their units, measuring how often integers divide, exhibiting both the absence and the non-uniqueness of an inverse in $\mathbb{Z}/12$, counting the agreements of the left and right quotients over $19977$ pairs, and tabulating index against quotient group for every subgroup of $S_3$.
1. Notation and Position
Division undoes a multiplication: $a\div b$ asks for the thing that, multiplied by $b$, gives $a$. What this paper asks is what stands in the way of that undoing.
Position: This is the fourth in the sorting of symbols, after Paper 16 (the equals sign), Paper 227 (inequality) and Paper 228 (the plus). The obstruction that Paper 228 found irreducible appears here a second time.
2. One Four, Three Different Things
Check 2.1:
| What is asked | Expression | Answer | Is the numerator's unit kept? |
|---|---|---|---|
| Share $12$ m among $3$ | $12\,\mathrm{m}\div3$ | $4$ m | yes |
| How many $3$ m in $12$ m | $12\,\mathrm{m}\div3\,\mathrm{m}$ | $4$ | no – a bare count |
| $12$ m in $3$ s | $12\,\mathrm{m}\div3\,\mathrm{s}$ | $4$ m/s | no – a new unit appears |
Observation 2.2 The number is $4$ in all three and the thing is different in all three. One is a length, one is a dimensionless count, one is a speed — what separates them is not the calculation but the units. Arithmetic teaching already names these the partitive and the quotitive.
3. The Answer Leaves the Set
Check 3.1: draw $200000$ pairs of integers from $1$ to $999$ and count how often the quotient is again an integer.
| Measured | Value |
|---|---|
| Pairs tried | $200000$ |
| Quotient an integer | $1404$ |
| Share | $\mathbf{0.7020\%}$ |
Observation 3.2 Inside the integers, division is simply undefined most of the time. The cure is not to adjust the operation but to enlarge the set — that is what passing from $\mathbb{Z}$ to $\mathbb{Q}$ is.
4. The Divisor Has No Inverse
Check 4.1: what $1\div0$ is depends on where one stands.
| Standing inside | What $1\div0$ is |
|---|---|
| The real numbers | undefined: no such number exists |
| The Riemann sphere | one point, at infinity |
| IEEE 754 | a signed infinity, and $0/0$ is not a number |
| A wheel | a defined element, with weaker laws |
| A limit | it depends on the path |
Measured: $1/0=\infty$, $-1/0=-\infty$, $0/0=\mathrm{NaN}$; and $x/x^2$ is $+10^{6}$ at $x=10^{-6}$ and $-10^{6}$ at $x=-10^{-6}$.
Check 4.2: dividing by $6$ in $\mathbb{Z}/12$.
| Question | Solutions |
|---|---|
| $6x=1 \pmod{12}$ | none at all |
| $6x=0 \pmod{12}$ | $\mathbf{0,2,4,6,8,10}$ (six of them) |
The share of invertible residues is $1.0000$ modulo $7$, $0.3636$ modulo $12$ and $0.2759$ modulo $30$.
Observation 4.3 The inverse is not merely absent but not unique either. Six solutions to $6x=0$ means that dividing $0$ by $6$ has six answers — a third kind of failure.
5. Without Commutativity There Are Two Quotients
Check 5.1: let $A$ have rows $(1,2)$ and $(3,4)$, and $B$ rows $(0,1)$ and $(1,1)$.
| Which quotient | Value |
|---|---|
| $B^{-1}A$ | rows $(2,2)$ and $(1,2)$ |
| $AB^{-1}$ | rows $(1,1)$ and $(1,3)$ |
| Largest difference | $\mathbf{1.000000}$ |
Over $19977$ random matrix pairs, the left and right quotients agreed $\mathbf{0}$ times.
Observation 5.2 There is no single object called $A$ divided by $B$. There is one from the left and one from the right, and they are different — the obstruction Paper 228 found irreducible is here a second time.
6. The Core: The Numbers Divide and the Objects Do Not
Check 6.1: every subgroup of the six-element group $S_3$.
| Subgroup $H$ | $|H|$ | Index $|S_3|/|H|$ | Normal? | Is $S_3/H$ a group? |
|---|---|---|---|---|
| The identity alone | $1$ | $6$ | yes | yes |
| $A_3$, the rotations | $3$ | $2$ | yes | yes |
| $\{e,(0\,1)\}$ | $2$ | $\mathbf{3}$ | no | no |
| $\{e,(0\,2)\}$ | $2$ | $\mathbf{3}$ | no | no |
| $S_3$ itself | $6$ | $1$ | yes | yes |
Observation 6.2 The index $|G|/|H|$ is a whole number for every subgroup (Lagrange). And the quotient is a group only when $H$ is normal — at index $3$ the numbers divide cleanly while the cosets have no multiplication.
Observation 6.3 Here nothing is being inverted at all. Dividing $S_3$ by $A_3$ does not undo a multiplication; it collapses an object — under one word, one gets either a number or a thing.
7. The Census: Five Obstructions
Check 7.1:
| What stops the undoing | Example | What one gets |
|---|---|---|
| Nothing (it is a field) | $12\div3$ in the rationals | a number |
| The answer leaves the set | $3\div2$ in the integers | nothing, until you enlarge |
| The divisor has no inverse | $1\div0$, or $1\div6$ in $\mathbb{Z}/12$ | up to five answers |
| The product does not commute | $A\div B$ for matrices | two answers, left and right |
| Nothing is being inverted | $S_3\div A_3$ | an object, not a number |
Observation 7.2 The five slashes share only the symbol. The first is ordinary division, the second is undefined, the third has several answers, the fourth has two, and the fifth is a different operation altogether.
8. The Separator and the Fence
Observation 8.1 (the separator) What separates the senses of the slash is what stops the multiplication from being undone. It is not the symbol.
Observation 8.2 (the fence) The fence around this divides is inside which set, and what is being undone. Outside it the sentence is not even false but empty — the same shape as these can be added in Paper 228 and this quantity is bounded in Paper 227.
Observation 8.3 In Paper 228, seven of eight were rescued by a change of coordinate. Here not one kind is — the obstructions to division do not vanish under relabelling. One must enlarge the set, or accept several answers, or distinguish left from right, or admit that it was a different operation.
Observation 8.4 The sorting of symbols now stands at four — six for the equals sign, two for inequality, eight for the plus, five for the slash. And it is not finished.
9. The Census and the Close
What this paper measured:
- Three fours (Check 2.1): a length, a count, a speed.
- Integers almost never divide (Check 3.1): $0.7020\%$.
- Five answers for division by zero (Check 4.1): $1/0=\infty$, $0/0=\mathrm{NaN}$, path-dependent.
- No inverse, and not unique either (Check 4.2): six solutions to $6x=0$.
- Left and right quotients (Check 5.1): $0$ agreements in $19977$.
- Numbers divide, objects do not (Check 6.1): index $3$ with no quotient group.
- Five obstructions (Check 7.1).
References
- Lagrange, J.-L. (1771). Réflexions sur la résolution algébrique des équations. Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin.
- Riemann, B. (1851). Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Göttingen.
- Ore, O. (1931). Linear equations in non-commutative fields. Annals of Mathematics, 32(3), 463–477.
- Fischbein, E., Deri, M., Nello, M. S., Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3–17.
- Greer, B. (1992). Multiplication and division as models of situations. In: Grouws, D. A. (ed.), Handbook of Research on Mathematics Teaching and Learning. New York: Macmillan.
- Carlström, J. (2004). Wheels: on division by zero. Mathematical Structures in Computer Science, 14(1), 143–184.
- IEEE (2019). IEEE Standard for Floating-Point Arithmetic. IEEE Std 754-2019.
- Yamagishi, Y. (2026). The Equals Sign Has Distinct Roots Too [Paper 16]. Zenodo. 10.5281/zenodo.21600471
- Yamagishi, Y. (2026). The Multiplication Does Not Decide the Second System [Paper 181]. Zenodo. 10.5281/zenodo.22104155
- Yamagishi, Y. (2026). The Four Means Are One Family, and There Is Only One Equality Fence [Paper 194]. Zenodo. 10.5281/zenodo.22106809
- Yamagishi, Y. (2026). The Sun Peaks in the Green, in the Orange and in the Infrared [Paper 222]. Zenodo. 10.5281/zenodo.22120199
- Yamagishi, Y. (2026). The Same $\le$ Points at a Wall You Can Touch and at One You Cannot [Paper 227]. Zenodo. 10.5281/zenodo.22123129
- Yamagishi, Y. (2026). There Are Eight Plus Signs, and Seven of Them Become Real Addition in the Right Coordinate [Paper 228]. Zenodo. 10.5281/zenodo.22126254
Reference
Yamagishi, Y. (2026). There Are Five Divisions, and Each One Fails for a Different Reason [Paper 229]. YOI: https://www.yamagishiyuuki.com/yoi/229
DOI: 10.5281/zenodo.22126445
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 229. This paper is a preprint and has not been peer reviewed.