Algebra 1 Chapter 1 Part 13

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""We must be careful to avoid ending up with too many postulates," the king said. "We will want to PROVE most of our behavior rules, rather than simply assume them to be true." "Are we just making these postulates up?" the professor asked philosophically. "Or do they exist already, in which case we are just discovering them?" "Who cares?" Recordis said. "I think that gremlin was wrong! Algebra will be as easy as pie--none of these properties are hard at all! The only hard part is getting used to the idea of using letters to stand for numbers, and I think I am even beginning to get used to that idea. I remember that at first I didn't like spinach, but once I became accustomed to it I thought it was great." Next, we developed our first result that we proved, rather than assumed. A proved result is called a THEOREM. We set out to prove the addition properties for odd and even numbers. We realized that any even number could be written in the form 2 x n, where n is some natural number. An odd number can be written in the form 2 x n +1. For example, the even number 12 is equal to 2 x 6. The od number 15 is equal to 2 x 7 + 1. We tried adding together one even number called 2 x n and another even number called 2 x m, calling the resulting sum s. s = (2 x m) +(2 x n) Using the distributive property, x =2 x (m +n) From the closure property, m + n must be a natural number, so s can be written in the form s = 2 x (some Natural number) Therefore, s must be even. Next we tried adding together two odd numbers, called 2 x m + 1 and 2 x n + 1 (again calling the result s). s = (2 x m _ 1) + (2 x n + 1) s = 2 x m +2 x n +2 s =2 x (m + n + 1) Since m + n + 1 must be a natural number, it follows that s must be even. We had one more combination to do: the sum of an odd number (which we called 2 x m + 1) and an even number (which we called 2 x n): s = (2 x n) + (2 x m =1) s = 2 x m + n) + 1 Since 2 x (m + n must be even, it follows that 2 x (m + n + 1 must be odd. "We did it!" the professor exclaimed in amazement. "We can prove general behavior rules by using symbols to stand for letters! I wasn't even sure that it could be done ! "Now we don't have to worry about the ferry boat tipping over, no matter what the actual number of cars happen to be." We were all pleased that we had been able to prove our very first theorems in algebra...It was time to return to Capital City."