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I want to show that this magic equation is true for all kinds numbers (rational, irrational, imaginary, natural, etc..) which is true as far as I've tested.$$\displaystyle\large\frac{2x+10}{2}-x=5$$How would I exactly do that?

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It is not magic. – Pedro Tamaroff Sep 7 '12 at 21:20
It is trivially correct ;) – Seyhmus Güngören Sep 7 '12 at 21:24
Actually, you might say that your "equation" is actually more like an "identity" ... – Old John Sep 7 '12 at 21:33
Why so many downvotes? – t.b. Sep 7 '12 at 21:37
David - I just took a look at your previous questions. If you are seriously worried about this one, I am not sure I understand how you have the mathematical understanding to ask the others. This one is really basic - so perhaps you could try to tell us why it is an issue for you? – Mark Bennet Sep 7 '12 at 21:40
up vote 7 down vote accepted

First, we have

$$\frac{2x + 10}{2} = \frac{2(x + 5)}{2} = x + 5$$

so the overall equation simplifies to

$$x + 5 - x = 5$$

which is obviously true for all $x$.

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You divide through your fraction by 2 to get $$\frac{2x+10}{2}-x= x+5-x = 5$$

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$$\frac{2x + 10}{2} - x = \frac{2x}{2} + \frac{10}{2} - x = x + 5 - x = 5$$

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The proof is the same ("universal") in all the number systems that you mention simply because they all obey the laws that one employs to prove the equality (associative, commutative, distributive laws, etc). In addition to these ring axioms, the proof also uses the fact that one can uniquely divide by $\,2,\,$ which is not true in every ring. For example, in modular arithmetic $$\rm mod\ 10\!:\ \ x\equiv1\ \Rightarrow\ \frac{2\,x+10}2-x\, \equiv\, \frac{2}2 - 1\,\equiv\, 0\,\not\equiv\, 5.$$

The problem is that $\rm\:mod\ 10\!:\ 2\, y \equiv 2\:$ has two solutions, $\rm\ y\equiv 1,6,\:$ hence the "fraction" $\rm\, y\equiv 2/2\,$ does not denote a unique value. Similar difficulties occur when attempting to generally apply the quadratic formula (which involves division by $2$). Analogous difficulties occur generally for fractions whose denominator is not coprime to the modulus. These matters are clarified when one studies abstract algebra (look up zero-divisors and integral domains).

In summary the "magic" is simply abstraction. By choosing ring axioms that abstract out the common algebraic structure of these familiar number systems, we are able to give a single proof that works universally for all number systems (rings). Thus we need prove only once familiar identities like the Binomial Theorem, Difference of Squares factorization, etc, since the proof can be specialized as need be for any particular ring (this specialization process will become even clearer algebraically when one learns about the universal properties of polynomial rings).

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