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Theorem 3:

THEOREM 3

Let a and b be finite hyperreal numbers. Then

(i) st(-a) = -st(a)
(ii) st(a + b) = st(a) + st(b)
(iii) st(a - b) = st(a) - st(b)
(iv) st(ab) = st(a) · st(b)
(v) If st(b) ≠ 0, then st(a/b) = st(a)/st(b)
(vi) st(an) = (st(a))n
(vii) If a ≥ 0, then st(01_real_and_hyperreal_numbers-167.gif) =01_real_and_hyperreal_numbers-168.gif
(viii) If a ≤ b, then st(a) ≤ st(b)

This theorem gives formulas for the standard parts of the simplest expressions.

All of the rules in Theorem 3 follow from our three principles for hyperreal numbers. As an illustration, let us prove the formula (iv) for st(ab). Let r be the standard part of a and s the standard part of b, so that

a = r + ε, b = s + δ,

where ε and δ are infinitesimal. Then

ab = (r + ε)(s + δ) = rs + rδ + sε + sδ ≈ rs.

Therefore

st(ab) = rs = st(a) · st(b).

Often the symbols Δx, Δy, etc. are used for infinitesimals. In the following examples we use the rules in Theorem 3 as a starting point for computing standard parts of more complicated expressions.


Last Update: 2010-11-25