PRODUCTS OF IRRATIONAL NUMBERS

James Constant

math@collissues.com

        Let r be a rational number and i,j,k be irrational numbers.

1. Prove product rj is an irrational number.

         Let rj=p/q where p/q is a rational number. Then j=p/qr is a rational number which is impossible.

2. Prove that product ij if not obviously rational or irrational is an irrational number.

        Some products ij of irrational numbers i and j are obviously rational or irrational numbers. For example, ij=a^1/2a^1/2=a where a is algebraic rational or irrational, or a is transcedental and, therefore, irrational. More generally, when either one or both i and j are irrational numbers, their products ij if not obviously rational or irrational are irrational numbers. For proof, note that an irrational number can always be written as the sum of a rational number and an irrational number and let i = r₁ + i₁ and j = r₂ + i₂ and write ij = r₁ r₂ + i₁ r₂ + r₁ i₂ + i₁ i₂ or ij =k + i₁ i₂ where k = r₁ r₂ + i₁ r₂ + r₁ i₂ is an irrational number.

        Let ij =k + i₁ i₂ = p/q where p/q is a rational number, an impossibility because k + i₁ i₂ is always an irrational number regardless whether i₁ i₂ is a rational or irrational number. However, i₁ i₂ is an irrational number for the same reason.

Copyright © 2009 James Constant

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