An exact definition of multiplication

Question

I am looking into repeated operations, and it seems really hard to precisely define multiplication.

Of course, for integer $b$ and real number $a$, we use the grade school definition we all know:

$$ab = \underbrace{a + a + a + \cdots + a}_{b\text{ times}}$$

but what about for real numbers $a$ and $b$?

For exponentiating (for integers: repeated multiplication), we have a precise formula to define it, which is easy to derive:

$$a^x = \sum_{n=0}^{\infty} \frac{x^n \left(\ln(a)\right)^n}{n!}$$

which is nice because we only have integer powers in the sum, which we already know how to define:

$$x^n = \underbrace{x \times x\times x \times \cdots \times x}_{n\text{ times}}$$

But this just raises the question of how we define $x \times x$ precisely.

• Is there an analogous formula to this for multiplication?

• How does the calculator compute multiplication of reals?

Note: According to sources, just approximating multiplication for real numbers uses calculus or numerical methods. I cannot grasp why we need these advanced concepts to precisely define this fundamental operation, especially when comparing it to the simple formula for exponentiation. But I still don’t have a formula yet.

Answer 1

The way we define these fundamental operations depends on how we define (or construct) the real numbers. When we do that, we define what we mean with "adding" and "multiplying" two numbers, and it's done in a certain way that we construct a field, with several other properties. See for example Construction of real numbers.

However, one way you can think we define the multiplication of two arbitrary numbers $a$ and $b$ is by taking sequences $\left\lbrace a_n\right\rbrace$ and $\left\lbrace b_n\right\rbrace$ of rational numbers that approximate $a$ and $b$, respectively (that is, whose limits are $a$ and $b$), and we can define the product $ab$ as $\displaystyle\lim_{n\rightarrow \infty} (a_n b_n) $, which makes sense since the product of two rational numbers can be defined "intuitively".

Of course that it's needed to prove the product is well defined (basically, that you get the same result no matter what sequences you choose, as long as their limits are $a$ and $b$), but that's a way you can think how the product can be defined.

P.S. Note that in order for that to make sense, we must first define what we mean with "real numbers" $a$ and $b$, so what I wrote here is rather a property of the real numbers than a definition of multiplication. I suggest reading about Dedekind cuts and Cauchy sequences.

Answer 2

To answer your question for an analogous explicit formula -- if $b$ is some real number written as $$x.x_1x_2x_3\ldots$$ (with $x$ being the integer part and each $x_i$ representing a digit in the decimal expansion), then $a\times b$ can be written as follows: $$ax + \sum_{n=1}^\infty \frac{ax_i}{10^i}.$$

To answer your question about calculators -- as mentioned by others, calculators don't actually work with real numbers, since they can't store infinitely many decimal places, or perform infinitely many sub-calculations. They "calculate" $\sqrt2 + \sqrt3$ just like you would -- write down ten or so digits of each and call it a day.

This is the precalculus-level answer (with real numbers understood loosely as "(potentially infinite) decimal"), but the other answers and comments are referring to an important issue, which I'd summarize as follows -- it's difficult to precisely define multiplication of real numbers because it's difficult to precisely define real numbers.

I'm glossing over this when I say "write $b$ as $x.x_1x_2x_3\ldots$". The usual definitions of the reals are Dedekind cuts and Cauchy sequences, but it's possible to make the "naive" idea of decimals work -- this is formalized at an approachable level in Appendix B of Terence Tao's Analysis I.

Answer 3

You can define multiplication as a binary operation of a particular algebra.

For example the definition of a ring. From A.H. Lightstone Symbolic logic and the real number system

DEFINITION 3.6.1. An algebraic system, say $(R,+,\cdot,0)$, where $+$ and $\cdot$ are binary operations on $R$ and $0\in R$ is said to be a ring iff

• (i) $(R,+,0)$ is an abelian group,
• (ii) $(R,\cdot)$ is a semi group,
• (iii) $\forall x \forall y \forall z [x \cdot (y+z) = x \cdot y + x \cdot z]$,
• (iv) $\forall x \forall y \forall z [(y+z) \cdot x = y \cdot x + z \cdot x]$.

And the binary operation $\cdot$ in the definition of a ring represents multiplication.

For multiplication of real numbers the definition is

DEFINITION 5.2.4. $(a_n) \cdot (b_n) = ND(a_n \cdot b_n)$ whenever $(a_n)$ and $(b_n)$ are real numbers.

And the $N$ and $D$ are unary operations on infinite sequences $(a_n)$ and $(b_n)$ of decimal rationals that represent a real number. Here $D$ rounds down certain digits and $N$ rounds of a block of infinite 9's, to turn the sequence $(a_n \cdot b_n)$, which may not be a real number, into a real number.

Answer 4

There's a very elegant way to define multiplication with a bit of machinery. I learnt this (the Tate/Eudoxos Reals) from Steve Schanuel in 1995.

On the wikipedia page https://en.wikipedia.org/wiki/Construction_of_the_real_numbers linked by Ivan above, there's a reference to Construction from integers (Eudoxus reals), here's another description with some more details: https://mattbaker.blog/2021/12/15/the-eudoxus-reals/.

In this construction, multiplication of reals is composition of functions.

The trick is to identify each real with the function (n -> r * n) on the integers, which (as the linked pages show) with the right quotient allows you to find an isomorph of the reals as a suitable quotient of suitable functions Z -> Z.

Anyway, just a little gem that I think belonged in this thread.