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This is an elementary math question. How to interpret multiplication of positive integers geometrically when the number of factors exceed three?

length * width (area of a rectangle)

length * width * height It's visible that the ordering doesn't matter in the calculation of 3-dimensional volume

But what about 4 factors? How to see that?

It's harder to see the associative property.

estrella
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  • Associativity for more than three "factors" is done here for example. – Dietrich Burde Sep 29 '23 at 17:43
  • Frequently, density works well for a 4th dimension. Then multiplying the three x, y, z, dimensions by density gives total mass of an object. Another simplification trick to visualize is to only visualize 3 dimensions at a time and hold the 4th constant. How much sense does either approach make to you? – nickalh Sep 29 '23 at 19:36
  • @nickalh Holding the fourth constant separately doesn't illustrate associativity nicely like those two examples. Imagining a lot of cubes in 3-dimensions all connected together illustrates this property. – estrella Sep 30 '23 at 12:04
  • 6x7x10x10 can be thought of as 10 layers of 6 rows of 7. Hold this in your mind as one cube. Multiplying this by ten 'extends' this cube by 10, all side by side. Now one can see that each long row of 10x 42 is going down ten times 10x(10x(6x7)) or 10 x (10x42). Since each column is 60, you can think you have 7 of those in one cube and multiply that by ten for ten cubes (6x10)x7)x10 or you can highlight each column of 60 from each of the 10 cubes once and see that multiplying it by 7 makes 70 of those or (6x10)x10)x7, both give you 70 x 60. – estrella Sep 30 '23 at 15:57
  • One can see that each cube is 6x70, and since this 70 is extending 10 times side by side, this equals 6x700, or alternatively we have 7x 600. All of this shows that 6x700=420x10=70x60=100x42 – estrella Sep 30 '23 at 15:58

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You can do geometry in more than three dimensions, using algebraic tools to describe structures.

To specify a point in the plane you use two numbers, traditionally named "$x$" and "$y$". Then you can describe a rectangle algebraically and calculate its area as the product of difference between corner values of $x$ (the "length") and the corresponding difference for $y$ (the "width").

For three dimensions you use one more coordinate, and call that the "height".

It's clear how to generalize this to any number of dimensions, although we don't have names for those new dimensions. You can't "see" them in the way we see three dimensional objects like boxes.

See What's new in higher dimensions? .

(Associativity is not a problem in these calculations - it's just the ordinary associativity of multiplication of numbers.)

Ethan Bolker
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  • My question was about the ordinary multiplication and it's associativity. How to see that these multiplications are associative when there are more than three factors? – estrella Sep 29 '23 at 17:31
  • @estrella That's an entirely different question, nothing to do with the geometric interpretation your question asks about. How you prove the associativity of ordinary multiplication depends on where you start. If you define multiplication of integers as repeated addition then you prove associativity by induction on the number of factors and proceed through the rationals to the reals, extending the definition of multiplication and the proof. Or you start with associativity of three factors as an axiom and prove it for more factors by induction. – Ethan Bolker Sep 29 '23 at 17:36
  • I have edited my question for clarity. I wanted to know whether there was some simple visual way to see that multiplication is symmetric (e.g: area of a rectangle, volume of a rectangular prism) when there are more factors involved. – estrella Sep 29 '23 at 17:44
  • The edit suggests that you are indeed hoping to "see" a four dimensional box. Many mathematicians share that (probably vain) hope. – Ethan Bolker Sep 29 '23 at 19:25