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Home Real and Hyperreal Numbers Standard Parts Theorem 3: | |||||||||||||||||
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Theorem 3:
THEOREM 3 Let a and b be finite hyperreal numbers. Then
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.
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Home Real and Hyperreal Numbers Standard Parts Theorem 3: |