What does real multiplication have to do with area?

Question

As a preliminary question, how is real multiplication defined? Sorry I’m just forgetful about middle school math, but how is multiplication of two real numbers defined? What does it mean by multiplying two real numbers? For integers, $2\times3=2+2+2$, but what is multiplication by a never-ending, never-repeating decimal expansion? How is it calculated? Why should it exist? I’m not talking about university analysis, but everyday math that’s taken for granted.

Now, why is the area of a rectangle equal to the product of two sides, which are real numbers? My guess is to place 1-by-1 tiles inside, then 0.1-by-0.1 tiles, and then 0.01-by-0.01 tiles, and so on, which is like approximating the product by increasing the precision. But is there a more direct understanding using the continuum, which is our intuition?

Here real numbers are the lengths of line segments as well as infinite decimal expansions.

Answer 1

Middle school is typically a ragbag of results not fully linked together. So "decimal representation of a real number" was added with little connection with "basic Euclidean geometry (area, length, angles etc.)". In middle school, we assume the number line $\mathbb{R}$ exists in some form, never even bother defining it.

We start with the area of rectangles given a segment (which we assign unit length). And we rearrange so that $x\times y=1\times z$ becomes $1:x::y:z$ and use constructions such as Euclid's Elements, book IV, Proposition 12 to actually construct the line segment $z$.

Then we have an alternative, defining as decimal representation. This allows you to "write down" an answer, but even addition/subtraction is problematic (there is no guarantee how far back we need to go so the answer is correct at the $n$ decimal place). But we can now write ratio of lengths of segments as decimal expansion, and go back and change half of the definition.

So in short, in middle school, the definition of product of decimal expansions of reals is: you are expected to use number line to get a representation of line segment (modulo the sign), multiply using Euclidean geometry, and convert the result back to decimal expansion. Finally we take care of the sign. Most students will just use a good enough rational approximation, multiply the two rational numbers, and round the answer to whatever precision the question asked for.

Answer 2

The multiplication of natural numbers can be defined by laying out pebbles in rectangular patterns. From multiplication of naturals, we readily extend to multiplication of integers (using $(-n)\cdot(-k)=-(n\cdot(-k))=n\cdot k$ and similar provable rules), and from that to multiplication of rational numbers (using $\frac ab\cdot \frac cd=\frac{ac}{bd}$). As every real number can be considered the limit of a Cauchy sequence of rational numbers, we can define the product of real numbers in terms of Cauchy sequences converging to the two real factors - their termwise product of rational numbers is then again a Cauchy sequence(!) and thus determines a real number, its limit; "fortunately", the limit does not depend on the Cauchy sequences originally picked ...

And that is in effect also what we do in everyday life - because in everyday life, we do not even have to deal with all real numbers, but at most with rational approximations to them (e.g., numbers with just a handful of decimals)

Answer 3

If, for you, a real number is an integer followed by a decimal point followed by a sequence of digits, then you can define the multiplication as follows: tak two real number $x$ and $y$. For you, $x=x_0+0.d_1d_2d_3\ldots$, with $x\in\mathbb Z$ and each $d_i$ is a digit. And $y=y_0+0.d_1^*d_2^*d_3^*\ldots$ If $n\in\mathbb N$, let $x_n=x_0+0.d_1d_2\ldots d_n$ and let $y_n=y_0+0.d_1^*d_2^*\ldots d_n^*$. Then $x_n$ and $y_n$ are rational numbers and I suppose that you know what is the meaning of multiplying them. It can be proved that the sequence $(x_ny_n)_{n\in\mathbb N}$ converges to a real number $z$. Then, by definition, $z$ will be $x\times y$.

Answer 4

Area and multiplication are connected as far back as Euclid, and the geometric notion of similarity - so that if lengths scale linearly, areas scale quadratically. Euclid did not have a definition of real numbers to hand, but was able to compare the lengths of line segments and also to deal with areas in the plane using dissection and congruence (congruent figures have equal areas).

Essentially Euclid's definition of multiplication comes from area, rather than the other way around, and comes from multiplying lengths identified geometrically rather than abstract real numbers.

The significance of this is that Geometry acts as a model of the real world, and the initial definitions related to mathematics modelling reality - except that there were Platonic ideas of a perfect or pure reality in the background, and Geometry was seen as the perfect thing of which the real world was an imperfect image.

When real numbers were being defined it was important to ensure that the connection with "reality" did not get broken, so no definition would have been allowed which broke the link between area and multiplication.

The Geometric method breaks down in three dimensions - it is not always possible to dissect two three-dimensional objects of equal volume bounded by (segments of) straight lines and (polygonal pieces of flat) planes into a finite number of congruent pieces.

Obviously, arithmetic separated itself from geometry (you don't need to construct four dimensional objects to multiply four numbers together); the real numbers arose from insights that not every number was rational, and not every point on a line could be identified by a finite Euclidean construction (and other considerations too). Area and volume developed into more sophisticated measure theory in which equality of volume could be rigorously defined.