Mathematics

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Jalal
Posts:8
Joined:Mon Mar 22, 2021 10:39 am
Mathematics

Unread post by Jalal » 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।

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

Re: Mathematics

Unread post by Asif Hossain » Tue Apr 27, 2021 10:56 am

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"?
Hmm..Hammer...Treat everything as nail

Jalal
Posts:8
Joined:Mon Mar 22, 2021 10:39 am

Re: Mathematics

Unread post by Jalal » 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".)

Asif Hossain
Posts:194
Joined:Sat Jan 02, 2021 9:28 pm

Re: Mathematics

Unread post by Asif Hossain » Thu Apr 29, 2021 8:44 am

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 :D .
Hmm..Hammer...Treat everything as nail

Jalal
Posts:8
Joined:Mon Mar 22, 2021 10:39 am

Re: Mathematics

Unread post by Jalal » Fri Apr 30, 2021 3:01 pm

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.

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