One times one equals two

One times one equals two

1 × 1 = 2

The table of Pythagoras laid out on Napier's rods, from Popular Science Monthly, 1884. The rod headed 1, read against the index 1, gives nought above the diagonal and one below.^[1]

Claimed by	Terrence Howard, actor
Name given to it	Terryology
First widely reported	2015
Supporting argument	The square root of two should be one
Standing
Peer-reviewed publication	None in the record
Refutation requires	One line
What it would cost	The identity element, and what rests on it
Conserved?	No. Nothing is, at this point

One times one equals two is a claim about elementary arithmetic advanced publicly since about 2015 by the American actor Terrence Howard, who calls the system built on it Terryology. The claim is that the product of one with itself is two rather than one, and that conventional mathematics is in error on the point.

It is false, and the demonstration takes one line. That is not why it is here. The claim is worth an article because the objection underneath it is a real one, asked by people learning arithmetic and not often answered well; because the argument offered in support of it fails in an unusually clean way; and because mathematics does in fact change its rules, has done so repeatedly and profitably, and has a settled procedure for it that this proposal does not follow.

The claim, as made

The position is of long standing and has been stated consistently since a profile in Rolling Stone in September 2015.^[2] The objection, as reported, runs roughly thus: if one times one is one, then multiplying a number by one accomplishes nothing, and a quantity that accomplishes nothing is of doubtful value.

A supporting argument is drawn from square roots. The square root of four is two; so, on the pattern, the square root of two ought to be one rather than the conventional 1.414…

The claim was published as a proof on social media in 2017, and returned to wide notice in May 2024 after an appearance on a podcast, at which further propositions were advanced concerning gravity and the status of zero.^[3] The astrophysicist Neil deGrasse Tyson, responding in June 2024, described the material as containing assumptions and statements that were "under-informed, misinformed, or simply false".^[4] No peer-reviewed publication of the system appears in the record.

Why one times one is one

The point that matters is that $1 \times 1 = 1$ is not a measurement, an observation, or a convention adopted for convenience. It is a consequence of what the symbol 1 is defined to mean.

In any system with a multiplication, an identity element is an element that leaves every other unchanged from either side: $a \times e = e \times a = a$ for every $a$. The number 1 is defined to be that element for ordinary multiplication. Setting $a = 1$ gives

immediately, and not as a separate fact to be checked. So if $1 \times 1 = 2$, then by the same substitution $1 = 2$, and from there every number is equal to every other, since one may multiply that equality by anything. The system does not produce a different arithmetic. It produces one number.

This is the whole of the refutation, and it is worth being clear about what it does and does not show. It does not show that no system can have an element whose square is twice itself. It shows that such an element cannot also be the multiplicative identity, and therefore cannot be what everybody else means by 1.

The argument from square roots

The supporting argument deserves separate treatment, because it fails in a way that is instructive rather than merely wrong.

The observation is that $\sqrt{4} = 2$, and the inference is that $\sqrt{2}$ should therefore be 1. But $\sqrt{4}$ is 2 for one reason only: because $2 \times 2 = 4$. The square root of a number is the thing that, multiplied by itself, gives it. So $\sqrt{2} = 1$ holds if and only if $1 \times 1 = 2$.

The argument therefore assumes precisely what it was offered to establish. It is not a second line of evidence for the claim but the claim again, restated in the notation of roots, and it cannot support the thing it is derived from. The circularity is easy to miss because the two statements look different on the page, which is a fair description of how a good deal of pseudomathematics works and is not peculiar to this instance.

What an identity is for

Underneath the claim there is a real complaint, and it is the part of this article most worth reading. The objection is that multiplying by one does nothing, and that an operation which does nothing is suspicious.

The answer is that doing nothing is the job. Every useful algebraic structure needs an element that changes nothing, because without one there is nothing for an inverse to return to. To say that $x$ has a reciprocal is to say that $x$ times it gives the identity. Solving an equation means manipulating it until one side is reduced to the identity and the other holds the answer. Remove that element and the machinery does not become simpler; it stops. The number 1 is not an idle number that happens to sit in the middle of the sequence. It is the fixed point the whole operation is defined against, and its being ineffective is exactly the property that makes everything else effective.

The complaint, in other words, is aimed at a load-bearing wall on the grounds that it is not doing anything visible.

Rules that have been changed, and what they cost

Against the suggestion that mathematics is defensive about its axioms, the record is the other way. It abandons them frequently, and the abandonments are among its best work. What it insists on is that the price be stated and the result given a new name.

• Commutativity. For centuries multiplication was assumed to satisfy $ab = ba$. On 16 October 1843, walking along the Royal Canal in Dublin, William Rowan Hamilton gave it up, and cut the defining relation of the quaternions into the stone of Broome Bridge. In quaternions $ij = k$ but $ji = -k$. The system is in daily use for rotations in computer graphics and spacecraft attitude control.
• Associativity. The octonions go further and abandon $(ab)c = a(bc)$. They remain a respectable object of study.
• The size of the numbers. In modular arithmetic, worked every time a clock passes noon, $12 + 1 = 1$. Nothing is wrong with the clock.
• What addition means. In Boolean algebra $1 + 1 = 1$. In the tropical semiring, "multiplication" is ordinary addition and "addition" is taking the smaller of two numbers, with the result that the multiplicative identity is 0.

In every case the same procedure is followed. The altered rule is named, the consequences are worked out, the structures that no longer apply are dropped, and the outcome is published as a new system rather than as a correction to the old one. Quaternion multiplication is not a claim that ordinary multiplication was wrong.

Two further habits of the discipline belong beside the claim, because between them they cover the ground it wants. A rule may simply hold in one place and fail in another, and be stated that way from the outset: the hairy ball theorem is true of spheres of even dimension and false of the odd ones, and says so in its own statement rather than being defended. And a change may be argued for properly and then declined on the costing: tau is a serious case for replacing π with the constant the subject actually uses, and its own advocates grant that the change would now cost more than it saved.

That is the respect in which the present proposal differs, and it is a difference of method rather than of daring. It changes one rule and retains the rest, and the rest are what make the changed rule mean anything.

See also

• The doctrinal value of π – a constant very nearly settled by a legislature, and the nearest thing here to a precedent
• Whole-radian geometry – a discredited system that did at least rebuild its own units to match
• Pi – a number that cannot be tidied, and the several attempts
• Grimble's razor – on paying for a premise and then declining the bill
• The hairy ball theorem – a rule that genuinely does change with the dimension, and says so
• Tau – a change to a convention argued for properly, costed, and then declined

References

1. ^ Popular Science Monthly vol. 26 (1884-85). The device is a set of Napier's rods, on which each cell carries the tens of a product above the diagonal and the units below, so that a multiplication is read off by laying the rods side by side. It is included because it settles the article's question mechanically: the answer was cut into a stick before anyone thought to dispute it.
2. ^ Erik Hedegaard, "Terrence Howard's Dangerous Mind", Rolling Stone, 14 September 2015. This article paraphrases rather than quotes, and the reason is worth stating: the profile is behind a paywall this encyclopedia could not get through, and the renderings of the remark that circulate in secondary sources do not agree with one another word for word. Putting quotation marks round a living man's sentence on that basis would be a worse fault than declining to.
3. ^ The proof was posted to social media in 2017; the wider circulation dates from an appearance on The Joe Rogan Experience in May 2024, at which claims were also made regarding gravity and the number zero. This article takes no view on those and confines itself to the arithmetic, which is checkable.
4. ^ Neil deGrasse Tyson, in a video response of June 2024. The full phrase is that the material contained "assumptions and statements that are under-informed, misinformed, or simply false", which is a description of the work and not of the man, and is the register this article has tried to keep.

External links

• W. R. Hamilton, "On quaternions; or on a new system of imaginaries in algebra" (1844) – the paper in which commutativity was given up and the price stated, Philosophical Magazine, October 1844
• Erik Hedegaard, "Terrence Howard's Dangerous Mind", Rolling Stone, 14 September 2015 – the profile in which the claim was first widely reported
• Neil deGrasse Tyson's response, June 2024 – reported in The Root
• Table of Pythagoras on slats, Popular Science Monthly vol. 26 (1884) – the engraving reproduced above