Why does commutivity of real numbers exist for multiplication

Click For Summary
SUMMARY

The commutativity of real numbers for multiplication, expressed as a * b = b * a, is fundamentally rooted in the properties of natural numbers and the construction of real numbers from them. The historical motivation for this property arises from geometric measurements, where the area of a rectangle defined by sides a and b is congruent to that defined by sides b and a. This congruence necessitated the definition of multiplication as a commutative operation.

PREREQUISITES
  • Understanding of natural numbers and their properties
  • Basic knowledge of real number construction
  • Familiarity with geometric concepts, particularly area
  • Comprehension of mathematical definitions and axioms
NEXT STEPS
  • Research the properties of natural numbers and their role in mathematics
  • Explore the construction of real numbers from natural numbers
  • Study geometric interpretations of multiplication and area
  • Examine mathematical definitions and axioms related to commutativity
USEFUL FOR

Mathematicians, educators, students in mathematics, and anyone interested in the foundational principles of arithmetic and geometry.

roger
Messages
318
Reaction score
0
Why does commutivity of real numbers exist for multiplication ie why a*b=b*a ?




Roger
 
Last edited:
Physics news on Phys.org
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.
 
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.
 

Similar threads

Undergrad Algebraic property of real numbers
  • · Replies 30 ·
2
Replies
30
Views
3K
Undergrad Formal definition of multiplication for real and complex numbers
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Anti-dual numbers and what are their properties?
  • · Replies 1 ·
Replies
1
Views
2K
MHB Is z + z¯ and z × z¯ Real for Any Complex Number z?
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Trouble with a passage in Zariski & Samuel's Commutative algebra text
  • · Replies 5 ·
Replies
5
Views
1K
High School Elegant way to prove ##AB^{n}## commute
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Proving the Last Gauss Lemma for Real Numbers Using Induction
  • · Replies 1 ·
Replies
1
Views
2K
High School Inductive proof for multiplicative property of sdet
  • · Replies 1 ·
Replies
1
Views
2K
High School Commutators of functions of operators
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad On base-b expansion of nonnegative reals
  • · Replies 7 ·
Replies
7
Views
2K