

Ask HN:properties of real numbers - pencil

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)
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gdl
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 _the exact same underlying values_ , so the result is
necessarily the same.

See also <http://en.wikipedia.org/wiki/Peano_axioms> 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.

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pencil
i know even i used to hate formal proofs!!!i assumed this might have a formal
proof by not knowing that this property is based on informal reasoning. (but i
personally like formal proofs!!!!!)

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patio11
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

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

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

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sleepdev
Tangentially related question: how are real numbers formally defined?

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?

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pencil
well..this gives the definition of real numbers
<http://en.wikipedia.org/wiki/Real_numbers..but> doesn't mention even a single
bit about the truthfullness of the properties of realnumbers.(to be honest
i'am not in a position to come out with a rational explanation!!!!!)

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mfukar
a = a (equality is reflexive under Peano arithmetic)

=> a + c = a + c (addition is commutative under Peano arithmetic)

=> a + c = b + c (a == b)

=> a + c = b + d (c == d)

qed

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KoZeN
_if a=b,c=d then a+c=b+d_

Also, d= a-b+c

Probably doesn't help but thats about the limit of my capabilities!

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pencil
we are in the same page !!!!

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TomK32
a = b c = d

a + c = b + d a + c = a + c # replaced b with a and d with c

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pencil
oh ya..that looks like a formal proof!!!!!!!!!

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vladoh
Ohhhhh NOOO... my brain evaporated because of the stupidity of this thread...

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pencil
no this isn't stupid.

