Abstract
Subtraction undoes an addition. And one thing wearing that symbol undoes no addition at all. This paper counts seven minus signs and sorts them by three questions. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, among the counting numbers it fails about half the time. Over $200000$ pairs, $a-b$ stayed a counting number $\mathbf{50.0575\%}$ of the time — the negative numbers were invented for this (Section 2).
Second, this is the core. Set difference does not undo union. $(A\cup B)\setminus B$ came back to $A$ in $\mathbf{44.0115\%}$ of $200000$ pairs, and only when the two are disjoint (counterexamples: $\mathbf{0}$) (Section 3).
Third, a point minus a point changes type. Weights summing to $1$ give a point, to $0$ give a vector, and anything else gives nothing at all — $P+Q$ is not unusual; it is undefined (Section 4).
Fourth, infinity minus infinity depends on the road. Four expressions of the same shape returned $0$, $-1.667\times10^{-7}$, $+3.333\times10^{-7}$, and a divergence (Section 5).
Fifth, subtracting a divergence depends on the scheme. Three ways of removing the same divergence left $0$, $-0.6931471806$ and $+0.6931471806$ behind — differing by $\ln2$ (Section 6).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — affine spaces and barycentric coordinates, set difference, indeterminate forms, the scheme dependence of renormalisation and the unary inverse are all standard. No completeness is claimed — seven appear and are not said to exhaust the matter. No theory of renormalisation is offered — the removal of the divergence is not treated, only that the finite part left behind is not settled. No correct scheme is chosen. This is not a textbook on algebra — the general theory of groups, rings and affine spaces is not treated. Relation to earlier papers: This is the fifth in the sorting of symbols, after Paper 16 (six for the equals sign), 227 (two for inequality), 228 (eight for the plus) and 229 (five for the slash). Paper 228 closed with before asking whether things can be added, ask in which variable — the answer here is its mirror image: before subtracting, ask what is supposed to come back. Paper 170 showed that cancelling the differentials gives $-1$, not $+1$ — the same family, a sign hidden inside a notation. Paper 226 treated the case where the sign is the orientation itself. What is added is measuring how often the counting numbers fail, measuring how often set difference undoes union and confirming the condition with no counterexample, sorting expressions into point, vector and meaningless by the sum of the weights, comparing four indeterminate forms, and exhibiting the $\ln2$ that separates three schemes.
1. Notation and Position
Subtraction undoes an addition: $a-b$ asks for the thing that, added to $b$, gives $a$. Three questions are asked of each case. (i) Does it really undo an addition? (ii) Is what comes out the same type as what went in? (iii) Is the answer unique?
Position: This is the fifth in the sorting of symbols. The closing line of Paper 228 turned around becomes the closing line here.
2. Among the Counting Numbers It Fails About Half the Time
Check 2.1: draw $200000$ pairs of counting numbers from $0$ to $999$.
| Measured | Value |
|---|---|
| Pairs tried | $200000$ |
| $a-b$ stayed a counting number | $100115$ |
| Share | $\mathbf{50.0575\%}$ |
Observation 2.2 The cure is not to adjust the operation but to enlarge the set — exactly the shape of division in Paper 229. The negative numbers were made for this and for nothing else.
3. The Core: Set Difference Does Not Undo Union
Check 3.1: let $A=\{1,2\}$ and $B=\{2,3\}$.
| Expression | Value |
|---|---|
| $A\cup B$ | $\{1,2,3\}$ |
| $(A\cup B)\setminus B$ | $\mathbf{\{1\}}$ |
| $A$ | $\{1,2\}$ |
The element $2$ belonged to $A$, and removing $B$ threw it out with the rest.
Check 3.2: over $200000$ random pairs of small sets.
| Measured | Value |
|---|---|
| $(A\cup B)\setminus B=A$ occurred | $88023$ times |
| Share | $\mathbf{44.0115\%}$ |
| Disagreements with the rule “exactly when disjoint” | $\mathbf{0}$ (over $20000$ pairs) |
Observation 3.3 Set difference is not an operation that undoes union. It undoes it only when $A$ and $B$ are disjoint — the very sentence subtraction undoes addition fails here. And the symbol is the same.
4. A Point Minus a Point Changes Type
Check 4.1: with $P=(3,1)$, $Q=(1,4)$ and $R=(-2,2)$, the gap between $(P-Q)+(Q-R)$ and $P-R$ was $\mathbf{0}$. Differences of points chain like arrows.
Check 4.2 (the check that failed): The first version treated every result as a point. Moving the origin shifts a point's coordinates and leaves a vector's alone, so the two need different tests. Moving the origin by $(10,-7)$:
| Expression | Weights sum | As a point? | As a vector? | Verdict |
|---|---|---|---|---|
| $P-Q$ | $0$ | no | yes | a vector |
| $(P+Q)/2$ | $1$ | yes | no | a point |
| $2P-Q$ | $1$ | yes | no | a point |
| $3P-2Q$ | $1$ | yes | no | a point |
| $P+Q$ | $2$ | no | no | meaningless |
| $2P+Q$ | $3$ | no | no | meaningless |
Observation 4.3 Weights summing to $1$ give a point, to $0$ a vector, and anything else moves with the origin and means nothing. Here the minus does what the plus cannot — it takes two points and returns an object of another type. $P+Q$ is not unusual but undefined. Two points may be averaged and may not be added.
5. Infinity Minus Infinity Depends on the Road
Check 5.1: the difference of two quantities that both blow up, as $x\to0$.
| Expression | $x=10^{-2}$ | $x=10^{-4}$ | $x=10^{-6}$ |
|---|---|---|---|
| $1/x-1/x$ | $0$ | $0$ | $0$ |
| $1/x-1/\sin x$ | $-0.00166669$ | $-1.66667\times10^{-5}$ | $-1.66707\times10^{-7}$ |
| $1/x-1/\tan x$ | $+0.00333336$ | $+3.33333\times10^{-5}$ | $+3.33297\times10^{-7}$ |
| $1/x^2-1/x$ | $+9900$ | $+9.999\times10^{7}$ | $\mathbf{+9.99999\times10^{11}}$ |
Observation 5.2 All four look like infinity minus infinity, and the answers are zero, zero from below, zero from above, and a divergence. The symbol does not distinguish them.
6. Subtracting a Divergence Depends on the Scheme
Check 6.1: cut off a divergent integral at $L$ and remove the divergence three ways.
| Scheme | Finite part left behind |
|---|---|
| Subtract $\ln L$ | $+0.0000000000$ |
| Subtract $\ln(2L)$ | $\mathbf{-0.6931471806}$ |
| Subtract $\ln(L/2)$ | $\mathbf{+0.6931471806}$ |
Here $\ln2=0.6931471806$.
Observation 6.2 All three remove the same divergence. And what is left over differs by $\ln2$ — and no computation inside a scheme can say which is right. The same shape as Paper 227, where the symbol did not settle whether a bound is reached.
7. The Unary Minus Is Not Subtraction
Observation 7.1
| Expression | Arguments | What it does |
|---|---|---|
| $a-b$ | two | an operation: undo an addition |
| $-a$ | one | a name: the element that gives $0$ |
| $a^{-1}$ | one | the same idea, written for a product |
| $A\setminus B$ | two | an operation, but it undoes nothing |
Observation 7.2 $-a+a=0$ is a definition and not a computation. One symbol holds a binary job and a unary job — and saying minus does not say which.
8. The Census and the Separator
Check 8.1:
| Which minus | Undoes an addition? | Output type | Unique? |
|---|---|---|---|
| $a-b$ in the integers | yes | same | yes |
| $a-b$ in the counting numbers | only when $a\ge b$ | same | yes, when defined |
| Point minus point | yes | a vector | yes |
| Set difference | no | same | yes |
| The unary $-a$ | not an operation | same | yes |
| Infinity minus infinity | no such addition | anything | no (the path) |
| Renormalisation | removes a divergence | same | no (the scheme) |
Observation 8.2 (the separator) What separates the senses of minus is three questions — whether it truly undoes an addition, whether the output has the same type, and whether the answer is unique. The symbol answers none of them.
Observation 8.3 (the fence) The fence around this can be subtracted is having settled what is supposed to come back. Outside it the sentence is not even false but empty — the same shape as this divides in Paper 229 and these can be added in Paper 228.
Observation 8.4 Paper 228 closed with before asking whether things can be added, ask in which variable. The answer here is its mirror — before subtracting, ask what is supposed to come back.
9. The Census and the Close
What this paper measured:
- The counting numbers fail about half the time (Check 2.1): $50.0575\%$.
- Set difference does not undo union (Check 3.2): $44.0115\%$, only when disjoint, $0$ counterexamples.
- The first check was wrong (Check 4.2): points and vectors need different tests.
- Three classes by the sum of the weights (Check 4.2): $1$, $0$, and neither.
- Four answers for infinity minus infinity (Check 5.1).
- The schemes differ by $\ln2$ (Check 6.1): $\pm0.6931471806$.
- The unary $-a$ is not an operation (Observation 7.1): it is a name.
References
- Grassmann, H. (1844). Die lineale Ausdehnungslehre. Leipzig: Wigand.
- Möbius, A. F. (1827). Der barycentrische Calcul. Leipzig: Barth.
- Cantor, G. (1895). Beiträge zur Begründung der transfiniten Mengenlehre. Mathematische Annalen, 46(4), 481–512.
- Bogoliubov, N. N., Parasiuk, O. S. (1957). Über die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder. Acta Mathematica, 97, 227–266.
- Hepp, K. (1966). Proof of the Bogoliubov–Parasiuk theorem on renormalization. Communications in Mathematical Physics, 2(1), 301–326.
- 't Hooft, G. (1973). Dimensional regularization and the renormalization group. Nuclear Physics B, 61, 455–468.
- Berger, M. (1987). Geometry I. Berlin: Springer.
- Yamagishi, Y. (2026). The Equals Sign Has Distinct Roots Too [Paper 16]. Zenodo. 10.5281/zenodo.21600471
- Yamagishi, Y. (2026). Cancelling the Differentials Gives $-1$, Not $+1$ [Paper 170]. Zenodo. 10.5281/zenodo.22093731
- Yamagishi, Y. (2026). A Surface That Comes Back Inside Out Still Has an Area [Paper 226]. Zenodo. 10.5281/zenodo.22121247
- 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
- Yamagishi, Y. (2026). There Are Five Divisions, and Each One Fails for a Different Reason [Paper 229]. Zenodo. 10.5281/zenodo.22126445
Reference
Yamagishi, Y. (2026). There Are Seven Minus Signs, and One of Them Undoes No Addition [Paper 230]. YOI: https://www.yamagishiyuuki.com/yoi/230
DOI: 10.5281/zenodo.22126779
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 230. This paper is a preprint and has not been peer reviewed.