Hello,
The addition property of real numbers says if a=b,c=d then a+c=b+d.can someone tell me why is it true? is there an algebraic proof for this or should we accept this as being true based on inductive reasoning?i really tried doing a google search for a proof but couldn't find any.
(i know i had asked a similer question a couple of days ago but i feel i might get a much more reasonable answer to this one..may be the way i had put accross the question wasn't sensible)
Informal reasoning: in a=b, the 'a' and 'b' are merely placeholders that act as pointers to the same underlying value, so they can be used interchangeably. It's no different than referring to "seven" and "the integer directly following six" - I could use either reference any place I could use the other with no effect on the overall statement. Ditto for c=d. All that is happening with a+c=b+d is that two references to values are being changed to different references with <i>the exact same underlying values</i>, so the result is necessarily the same.<p>See also <a href="http://en.wikipedia.org/wiki/Peano_axioms" rel="nofollow">http://en.wikipedia.org/wiki/Peano_axioms</a> if you prefer the math and logic jargon. High school geometry taught me to dislike dealing with formal proofs, but I think that should be about the right area to look.
There are a bunch of ways to prove this. Let's start with addition, subtraction, equality being commutative, that a + 0 = a, and that a - a = 0<p>Suppose that d != a - b + c. Since a = b, a - b = 0. This implies d != c. This is a contradiction, so our supposition is inaccurate. d = a - b + c<p>Now, suppose a + c != b + d. Plug in what we just learned. a + c != b + a - b + c. The b's cancel, leaving another contradiction. Thus, supposition inaccurate, so a + c = b + d. QED
Tangentially related question: how are real numbers formally defined?<p>I remember that integers are usually defined in terms of successors: Succ 1 = 2. But this doesn't help for real numbers because they can't really be enumerated?
a = a (equality is reflexive under Peano arithmetic)<p>=> a + c = a + c (addition is commutative under Peano arithmetic)<p>=> a + c = b + c (a == b)<p>=> a + c = b + d (c == d)<p>qed