What is the link between proportion and multiplication?

[Haris045]

TL;DR

What is the link between proportion and multiplication? WHY DOES and how can multiplication model proportionality?

I found this quote online:

“Multiplication is the mathematical manifestation of the fundamental physical phenomenon of proportionality (as addition is to combination).”

Question 1: How are multiplication and proportion linked? How can and WHY DOES multiplication model proportion? (My understanding of proportion is the equivalence of ratios)

Also saw this "Multiplication is scaling a number up proportionally up or down"

Question 2: What does this mean to scale something up proportionally , proportionally to WHAT? . e.g. if I had 7.2 * 3.2 , how do I think of that as proportionally scaling? What am I scaling proportionally to what?

Edit:

These questions were formed when I was doing re-search on the true nature of multiplication, some things I have read about this topic:

https://www.maa.org/external_archive/devlin/devlin_06_08.html and https://www.maa.org/external_archive/devlin/devlin_01_11.html

I have also tried to read " Development of Multiplicative Reasoning in the Learning of Mathematics" it talked about multiplicative conceptual fields but it got quite confusing and dragged on and I didn't really understand the book.

Thanks in advance for the help!

 

[Drakkith]

Question 2: What does this mean to scale something up proportionally , proportionally to WHAT? . e.g. if I had 7.2 * 3.2 , how do I think of that as proportionally scaling? What am I scaling proportionally to what?

You are scaling either number proportionally to its original 'size' or 'quantity'.
It's easier with whole numbers, so let's say we are multiplying 7x3.
Think of the number 7 as a line that's 7 units long. Multiplying by 3 means that we increase the length of the original line until it is 3x longer. And, since multiplication of two real numbers is commutative, meaning that the order you place them in doesn't matter, you can consider 3 to be scaled up by 7x. You can think about it as if it is the same line that's just been stretched out. The line is now larger than it was originally, meaning its proportions have increased.

In this case the only proportion it has is length since it's a line, but the same concept applies to things like 2d and 3d shapes where a multiplication by a single number scales all proportions up or down such that the new shape retains the same ratios in its proportions as the old. That is, a square whose side lengths are multiplied by 2x still has a 1-to-1 ratio of its new side lengths. A rectangle with sides 1x2 in length that are multiplied by 2x now has sides of 2x4, which have the same 1-to-2 ratio as their original lengths.

 

[russ_watters]

Did you find any of the replies you got to a similar question a week and a half ago useful?

 

[Dale]

Summary: What is the link between proportion and multiplication? WHY DOES and how can multiplication model proportionality?

How are multiplication and proportion linked?

The statement “A is proportional to B” means $A=k \times B$