Theorem 1. Cancellation Law for Addition.
If a + b = b + c , then b = c
Theorem 2. Possibility of Subteation .
Given a and b , there exactly one x s.t. a + x = b
Theorem 3. b - a = b + ( -a )
Theorem 4. - (-a) = a
Theorem 5. a( b - c ) = ab - ac
Theorem 6. 0*a = a*0 = 0
Theorem 7. Cancellation of Mutiplication .
If ab = ac and a ≠ 0 , then b = c
Theorem 8. Possibility of Division
Given a and b with a ≠ 0 , there is exactly one x s.t. ax = b .
This x is denoted by b/a is called the quotient of b and a .
In particular , 1/a is also written a^(-1) and is called the
reciprocal of a .
Theorem 9. If a ≠ 0 , then b/a = b * a^(-1)
Theorem 10. If a ≠ 0 , then [a^(-1)]^(-1) = a
Theorem 11. If ab = 0 , then a = 0 or b = 0 .
Theorem 12. (-a)b = - (ab) and (-a)(-b) = ab
Theorem 13. (a/b) + (c/d) = (ad + bc)/(bd) if b ≠ 0 and d ≠ 0
Theorem 15. (a/b)(c/d) = (ad)/(bc) if b ≠ 0 , c ≠ 0 and d ≠ 0
Theorem 16. Trichotomy Law . For arbitrary real numbers a and b ,
exactly one of the three relations a < b , b < a , a = b holds .
Theorem 17. Transitive Law. If a < b and b < c , then a < c .
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