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Homework Statement
I referenced this theorem from the following webpage:
http://mathonline.wikidot.com/theorems-on-the-properties-of-the-real-numbers
Homework Equations
The Attempt at a Solution
The proof makes perfect sense, but why must it be so roundabout? A real number axiom is a + 0 = 0. Can I not just state this, making b = 0 obvious, and be done with the proof? The actual proof does reference other identities, but it seems roundabout in a way. The same can be said for theorem 2 (if a x b = a, can I not simply reference the real number property a x 1 = a ( making b = 1 obvious ) and be done with it? )To state it more concisely, what would be wrong with the following line of thinking?
Suppose a + b = a
According to Axiom A3, a + 0 = a.
Clearly, since the right side of the equation shows that a is the answer, the only number that a on the left side could have been added to was 0.
Therefore, b must be 0
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