nLab
carrying

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Idea

In the context of arithmetic, carrying is part of the operation of representing addition of natural numbers by digits with respect to a base.

In terms of commutative algebra

Given the rig of natural numbers \mathbb{N}, there exists a free commutative \mathbb{N}-algebra [b]\mathbb{N}[b] on one generator bb called the base. Since multiplication in a commutative algebra is power-associative, there exists a right \mathbb{N}-action on [b]\mathbb{N}[b] () ():[b]×[b](-)^{(-)}:\mathbb{N}[b]\times\mathbb{N}\to\mathbb{N}[b] called the power, and every element in [b]\mathbb{N}[b] could be written as a polynomial

p= n=0 ka(n)b np = \sum_{n=0}^{k} a(n) b^n

When the algebra is quotiented out by the relation b10b \sim 10, the resulting quotient algebra is isomorphic to the original rig of natural numbers \mathbb{N}. This means that every natural number could be expressed a polynomial with base ten,

p= n=0 ka(n)10 np = \sum_{n=0}^{k} a(n) 10^n

There is a canonical such polynomial, one where all natural numbers in the sequence a(n)<10a(n) \lt 10 in the polynomial. Carrying arises from adding two canonical polynomials, when the sum a 1(n)+a 2(n)10a_1(n) + a_2(n) \geq 10 and the polynomial is no longer canonical; in order to make the polynomial canonical again, one would have to take the sum a 1(n)+a 2(n)a_1(n) + a_2(n) modulo 10 and add 1 to the sum a 1(n+1)+a 2(n+1)a_1(n+1) + a_2(n+1) in the next power of ten. This means there ought to be another representation of the digits in terms of integers modulo 10.

In terms of cohomology

Write /10\mathbb{Z}/10 for the abelian group of addition of integers modulo 10. In the following we identify the elements as

/10={0,1,2,,9}, \mathbb{Z}/{10} = \{0,1,2, \cdots, 9\} \,,

as usual.

Being an abelian group, every delooping n-groupoid B n(/10)\mathbf{B}^n (\mathbb{Z}/{10}) exists.

Carrying is a 2-cocycle in the group cohomology, hence a morphism of infinity-groupoids

c:B(/10)B 2(/10). c : \mathbf{B} (\mathbb{Z}/{10}) \to \mathbf{B}^2 (\mathbb{Z}/{10}) \,.

It sends

a = b a+bmod10 id c(a,b) id id , \array{ && \bullet \\ & {}^{\mathllap{a}}\nearrow &\Downarrow^=& \searrow^{\mathrlap{b}} \\ \bullet &&\stackrel{a+b mod 10}{\to}&& } \;\;\; \mapsto \;\;\; \array{ && \bullet \\ & {}^{\mathllap{id}}\nearrow &\Downarrow^{c(a,b)}& \searrow^{\mathrlap{id}} \\ \bullet &&\stackrel{id}{\to}&& \bullet } \,,

where

c(a,b)={1 a+b10 0 a+b<10. c(a,b) = \left\{ \array{ 1 & a + b \geq 10 \\ 0 & a + b \lt 10 \,. } \right.

The central extension classified by this 2-cocycle, hence the homotopy fiber of this morphism is /100\mathbb{Z}/{100}

B(/100) * B(/10) c B 2(/10). \array{ \mathbf{B} (\mathbb{Z}/{100}) &\to& * \\ \downarrow && \downarrow \\ \mathbf{B} (\mathbb{Z}/{10}) &\stackrel{\mathbf{c}}{\to}& \mathbf{B}^2 (\mathbb{Z}/{10}) } \,.

That now carries a 2-cocycle

B(/100)B 2(/10), \mathbf{B} (\mathbb{Z}/{100}) \to \mathbf{B}^2 (\mathbb{Z}/{10}) \,,

and so on.

B(/1000) c B 2(/10) B(/100) c B 2(/10) B(/10) c B 2(/10) \array{ \vdots \\ \downarrow \\ \mathbf{B} (\mathbb{Z}/{1000}) &\stackrel{c}{\to}& \mathbf{B}^2 (\mathbb{Z}/{10}) \\ \downarrow \\ \mathbf{B} (\mathbb{Z}/{100}) &\stackrel{c}{\to}& \mathbf{B}^2 (\mathbb{Z}/{10}) \\ \downarrow \\ \mathbf{B} (\mathbb{Z}/{10}) &\stackrel{c}{\to}& \mathbf{B}^2 (\mathbb{Z}/{10}) }

This tower can be viewed as a sort of “Postnikov tower” of \mathbb{Z} (although it is of course not a Postnikov tower in the usual sense). Note that it is not “convergent”: the limit of the tower is the ring of 1010-adic integers 10\mathbb{Z}_{10}. This makes perfect sense in terms of carrying: the 1010-adic integers can be identified with “decimal numbers” that can be “infinite to the left”, with addition and multiplication defined using the usual carrying rules “on off to infinity”.

References

  • Dan Isaksen, A cohomological viewpoint on elementary school arithmetic, The American Mathematical Monthly, Vol. 109, No. 9. (Nov., 2002), pp. 796-805. (jstor)

Last revised on May 29, 2021 at 00:16:42. See the history of this page for a list of all contributions to it.