Why does commutivity of real numbers exist for multiplication

In summary, the commutivity of real numbers for multiplication exists because of its definition and historical motivation in geometry.
  • #1
roger
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Why does commutivity of real numbers exist for multiplication ie why a*b=b*a ?




Roger
 
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  • #2
It boils down to the natural numbers being commutative plus the way the set of real numbers is constructed (in a few steps) from N.
 
  • #3
But why is multiplication in N commutative? How's that for a prediction. And to show some predictability here's what my favourite source of maths comes up with:

http://www.dpmms.cam.ac.uk/~wtg10/commutative.html
 
  • #4
The entirely unenlightening answer is because that's how we defined them.


Historically speaking, the motivation behind the real numbers was measurement in geometry. For example, the product of two sides of a rectangle is the area of that rectangle, and the "rectangle with sides of length a and b" is congruent to the "rectangle with sides of length b and a", so commutativity was desired.
 

1. Why is commutivity important in mathematics?

Commutivity is important in mathematics because it allows us to rearrange the order of operations without changing the result. This makes it easier to solve complex equations and perform calculations.

2. What does commutivity mean for real numbers?

Commutivity for real numbers means that the order in which we multiply them does not change the result. In other words, a x b = b x a, where a and b are real numbers.

3. How does commutivity apply to multiplication?

Commutivity applies to multiplication because it allows us to multiply numbers in any order and still get the same result. For example, 3 x 4 = 4 x 3.

4. Why do real numbers have commutivity for multiplication?

The commutivity of real numbers for multiplication is a fundamental property of mathematics that comes from the distributive law. It is a result of how we define multiplication and the properties of real numbers.

5. Are there any other operations besides multiplication that have commutivity?

Yes, addition also has commutivity for real numbers. This means that the order in which we add numbers does not change the result. For example, 2 + 3 = 3 + 2.

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