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Blast into Math! Analatic nummer theora: ants, ghosts and giants
7. For every x ∈ R with x =0, there exists x −1 ∈ R such that
x ∗ x −1 =1. (There aremultiplicativeinversesin R just likein Q.)
Assumptions (8-12) below mean that we can put the real numbers in order from smaller (left)
to larger (right), and the order is consistent with the binary operations + and , just like in
∗
the rational numbers. This means that we can put the real numbers in the number line.
8. For all x ∈ R ,
x ≤ x.
9. For all x and y ∈ R , either x ≤ y or y ≤ x , and if both x ≤ y and y ≤ x then
x = y .
10. For all x , y and z ∈ R , if x ≤ y , then x + z ≤ y + z . If x ≤ y and y ≤ z then
x ≤ z
11. For all x , y and z ∈ R , if 0 ≤ x and 0 ≤ y , then 0 ≤ xy .
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