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Ed Quanta
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Is there any way to prove the distributive law for integers? I heard that there is yet I don't understand how being that the distributive law is an axiom and therefore what the understanding our number system is based on.
Originally posted by Ed Quanta
Is there any way to prove the distributive law for integers? I heard that there is yet I don't understand how being that the distributive law is an axiom and therefore what the understanding our number system is based on.
Originally posted by Ed Quanta
So to just prove a(b+c)= ab + ac in general would not be possible?
Originally posted by HallsofIvy
On the contrary, it is possible to prove the distributive law starting from "Peano's axioms" for the natural numbers. That is basically equivalent to "induction" and all proofs of properties of natural numbers are inductive.
Here's a link to a PDF file that contains such proofs:
http://academic.gallaudet.edu/courses/MAT/MAT000Ivew.nsf/ID/918f9bc4dda7eb1c8525688700561c74/ $file/NUMBERS.pdf
Click on "Numbers".
Originally posted by HallsofIvy
By the way, one does not have to prove that "1+ 1= 2" because that is basically how "2" is define. That "2+ 2= 4" is a theorem and has to be proven (by someone). It a simple two or three line proof, of course.
Originally posted by HallsofIvy
Hmm, Yes, 2 is defined as the sucessor of 1 and n+1 is defined as "the sucessor of n" therefore--
Gosh, I just might be forced to agree with you!
The Distributive Law for Integers states that when multiplying a number by a sum of two other numbers, the result is the same as multiplying the number by each individual number and then adding the products together. In other words, a(b+c) = ab + ac.
Proving the Distributive Law for Integers is important because it is a fundamental property of integers that is used in a wide range of mathematical operations. It allows us to simplify and manipulate expressions, making problem-solving easier and more efficient.
The most common method for proving the Distributive Law for Integers is through the use of algebraic manipulation. This involves starting with the left side of the equation and using the properties of addition and multiplication to transform it into the right side of the equation.
No, the Distributive Law for Integers holds true for all integers. It is a fundamental property of numbers and is applicable in all situations.
The Distributive Law for Integers is related to the Associative and Commutative Laws, which also involve the manipulation of expressions using addition and multiplication. These laws work together to help us solve more complex problems and demonstrate the interconnectedness of mathematical concepts.