Axiom 1. Commutative Laws. x + y = y + x , xy = yx
Axiom 2. Associative Laws. ( x + y ) + z = x + ( y + z )
Axiom 3. Distribute Law. x ( y + z ) = xy + xz
Axiom 4. Existence of Identity Elements .
x + 0 = x , x * 1 = x
Axiom 5. Existence of Negatives .
For every real x there is a real y s.t. x + y = 0
Axiom 6. Existence of Reciprocals .
For every real x ≠ 0 there is a real y s.t. xy = 1
Axiom 7. If x , y are in R+ , so are x + y and xy .
Axiom 8. For every x ≠ 0 , either x belongs to R+ or -x belongs to R+
but not both .
Axiom 9. 0 is not belongs to R+
Axiom 10. Every nonempty set S of real numbers which is bounded above has a
supremum , that is , there is a rwal B s.t. B = sup S
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以上是實數的 Field Axioms 是微積分的基礎 很簡單
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