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Mathematics
Posted: Mon Apr 26, 2021 2:01 am
by Jalal
"সকল a, b, c ∈ ℝ এর জন্য, a (b + c) = ab + ac এবং
(b + c) a = ba + ca হবে"।
বাস্তব সংখ্যার এই স্বীকার্যভিত্তিক ধর্মটি কীভাবে প্রমাণ করব? এটার কি কোনো কড়াকড়ি প্রমাণ আছে?
আমি মূলত "যখন a এবং b যেকোনো ঋণাত্নক বাস্তব সংখ্যা এবং c যেকোনো ধনাত্মক বাস্তব সংখ্যা, তখন
a (b + c) = ab + ac এবং (b + c) a = ba + ca হবে" এই ধর্মটির প্রমাণটা জানতে চাচ্ছি।
Re: Mathematics
Posted: Tue Apr 27, 2021 10:56 am
by Asif Hossain
Jalal wrote: ↑Mon Apr 26, 2021 2:01 am
"সকল a, b, c ∈ ℝ এর জন্য, a (b + c) = ab + ac এবং
(b + c) a = ba + ca হবে"।
বাস্তব সংখ্যার এই স্বীকার্যভিত্তিক ধর্মটি কীভাবে প্রমাণ করব? এটার কি কোনো কড়াকড়ি প্রমাণ আছে?
আমি মূলত "যখন a এবং b যেকোনো ঋণাত্নক বাস্তব সংখ্যা এবং c যেকোনো ধনাত্মক বাস্তব সংখ্যা, তখন
a (b + c) = ab + ac এবং (b + c) a = ba + ca হবে" এই ধর্মটির প্রমাণটা জানতে চাচ্ছি।
Isn't it a axiom itself called "Distributive Law"?
Re: Mathematics
Posted: Wed Apr 28, 2021 11:12 pm
by Jalal
Yes! it's the distributive property of real numbers. Isn't there any proof of this? Why is this property called an axiom? It doesn't seem trivial to me. Was the multiplication between two negative real numbers or a negative real number and a positive real number defined before setting up the distributive property of real numbers as an axiomatic property of real numbers?
(Though it was a testing post, I have also asked this question on "College / University Level Forum".)
Re: Mathematics
Posted: Thu Apr 29, 2021 8:44 am
by Asif Hossain
Jalal wrote: ↑Wed Apr 28, 2021 11:12 pm
Yes! it's the distributive property of real numbers. Isn't there any proof of this? Why is this property called an axiom? It doesn't seem trivial to me. Was the multiplication between two negative real numbers or a negative real number and a positive real number defined before setting up the distributive property of real numbers as an axiomatic property of real numbers?
(Though it was a testing post, I have also asked this question on "College / University Level Forum".)
Well there is a "PROOF" of it through linear algebra and matrix stuff and there is a much simpler proof by geometrically defining multiplication by area of rectangle you may try to prove it yourself geometrically it is easy enough
.
Re: Mathematics
Posted: Fri Apr 30, 2021 3:01 pm
by Jalal
How could you geometrically prove that $“(-a)⋅(-b + c) = (-a)⋅(-b) + (-a)⋅c$ and $(-b + c)⋅(-a) = (-b)⋅(-a) + c⋅(-a)$ where $-a$ and $-b$ are any negative real number and $c$ is any positive real number$”$? I can prove the distributive property of nonnegative real numbers geometrically.