Hello Cassandra from Alexandria, IN, and others that enjoy life and humor: Your question, "When I answer my Question, How would I know what the answer is?" is quite profound as it applies to math and more. The best answer is it depends on the type of question you ask in mathematics or another subject area. Getting a second opinion from an expert in the field is another way to verify, asides checking, if the answer key agrees. Sometimes that is the best when the text book or exam answer is not right. However, for mathematics, depending on the branch of math and depending on the type of question, you can plug in your answer to see if it works, is the approach most often recommended. For graphs, you may need to make a graph to determine if your answer fits the solution set or not, so graphing it properly and knowing how to read it tells you if you have it right (with your mentor confirming things). With equations, you show that it works out properly (since the laws of logic and mathematics do not change, the supposition is foundational for math and all the ordering sciences depend on this). With inequalities, you show the solution set is part of what you want to demonstrate. With simultaneous equations, you will work with matrices possibly to see if the system is consistent and what each variable would be or what values it would be in terms of with other variables or not inconsistent. For proofs, the QED at the end requires someone else to perhaps check your work and if you are consistent logically, they will be able to say that is spot on (as the Brits would say). Now, if you are asking an even deeper question of why 1 + 1 = 2 , why 1 + 2 = 3, why 1 + n = k and so on there is a dispute on this where some mathematicians side with Kurt Friedrich Gödel was an Austrian American logician, mathematician, and philosopher and his incompleteness theorem and apply this to the extent that proof is a weaker notion than truth and that we know some things to be true without proving it. Hence, living in an orderly universe, predisposes us to believe in the existence of order and the reality around us without proof but by faith based on our senses and experiences of our own and the history of which we find ourselves. So back to your question of how do we know my answer is correct? The best answer is "You need to ask your self the question and depending on the context, what is the framework based upon the rules we are working with to confirm it or show it is complete and consistent, or incomplete but consistent or complete and inconsistent or incomplete and inconsistent." Asking an expert will get you a bigger answer than you asked... especially if they are more than a mathematician with a sense of humor. We need at some point to take some things by faith in the orderliness of the framework we are adhering to or look for a supercomplete and superconsistent basis for our faith or we continue the quest for the answer to the question. I realize you asked this question a ways back, however, the answer is more for a larger audience if your question was not this big. The largest answers go beyond our humanity to the totality of reality...
6/4/2013

Tai W.