( of the real number system )
The elements a in R such that a is in will of course be called positive,
those such that -a is in negative.
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Note also that both (a is greater than b) and (b is less than a) mean that is in .
And of course and mean "greater than or equal to" and "less than or equal to" respectively.
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It took me a long time to appreciate the facts presented in the following pages. I think it is worthwhile to spend quite a lot of time on them, and I am working toward being able to recall and prove them from memory. I'm doing this by occasionally rereading, or by testing my ability to recall the facts and to deduce the proofs.
O1 : ( Trichotomy )
O2 : ( Transitivity )
O3 : (Larger Added to Larger)
O4 : (Larger Positives Multiplied)
O5 : (Sign Rules in Addition and Multiplication)
O6 : (Squares Are Greater than Zero)
O7 : (Multiplicative Inversion Does Not Change Sign)
O8 : (Larger Numbers Have Smaller Multiplicative Inverses)
O9 : (Rules of Arithmetic)
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We also have the following:
O10. Negation of an Inequality :
O11. Multiplication of an Inequality by a Positive Real Number :
O12. Multiplication of an Inequality by a Negative Real Number :
O13. a less than b implies (a+b)/2 is between a and b :