THE FIELD AXIOMS

field

  1. If x and y are real numbers, then… x+y is a unique real number. [CLOSURE PROPERTY OF ADDITION]

  2. If x and y are real numbers, then… xy is a unique real number. [CLOSURE PROPERTY OF MULTIPLICATION]

  3. If x and y are real numbers, then… x+y = y+x. [COMMUTATIVE PROPERTY OF ADDITION]

  4. If x and y are real numbers, then… xy = yx. [COMMUTATIVE PROPERTY OF MULTIPLICATION]

  5. If x, y and z are real numbers, then… (x+y)+z = x+(y+z). [ASSOCIATIVE PROPERTY OF ADDITION]

  6. If x, y and z are real numbers, then… (xy)z = x(yz). [ASSOCIATIVE PROPERTY OF MULTIPLICATION]

  7. Multiplication distributes over addition. If x, y and z are real numbers, then… x(y+z) = xy + yz. [DISTRIBUTIVE PROPERTY]

  8. The IDENTITY ELEMENT FOR ADDITION is 0, i.e., for any real number x, … x+0 = x. [ADDITION PROPERTY OF 0]

  9. The IDENTITY ELEMENT FOR MULTIPLICATION is 1, i.e., for any real number x, … x.1 or 1x = x. [MULTIPLICATION PROPERTY OF 1]

  10. A unique ADDITIVE INVERSE exists for every real number, i.e., for every x the additive inverse is -x such that… x + (-x) = 0.

  11. A unique MULTIPLICATIVE INVERSE exists for every real number, i.e., for every non-zero x the multiplicative inverse is 1/x such that… x . (1/x) = 1.

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