Prove that real multiplication distributes over addition

For college and university level advanced Mathematics
Jalal
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Joined:Mon Mar 22, 2021 10:39 am
Prove that real multiplication distributes over addition

Unread post by Jalal » Wed Apr 28, 2021 10:57 pm

The distributive property of real numbers states that $“$for all $a, b, c \in \mathbb{R}$, we've $a⋅(b + c) = a⋅b + a⋅c$ and $(b + c)⋅a = b⋅a + c⋅a”$. How to prove this axiomatic property of real numbers? Is there any rigorous proof of this? Why was this property accepted as an axiom? It doesn't seem trivial to me like other axioms of real numbers. Was the multiplication between two negative real numbers or a negative real number & a positive real number defined before setting up the distributive property of real numbers as an axiomatic property of real numbers?

Basically, I want to know the proof of the following property of $\mathbb{R}$:‌ $“(-a)⋅(-b + c) = (-a)⋅(-b) + (-a)⋅c$ and $(-b + c)⋅(-a) = (-b)⋅(-a) + c⋅(-a)$ where $-a$ and $-b$ are any negative real number and $c$ is any positive real number$”$.

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