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Jay
Mathematics
19 January, 03:45
Prove that root 3 + root 5 is an irrational no
+5
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Bryan Berry
19 January, 04:12
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Use proof by contradiction. Assume that the sum is rationial, that is
2-√ + 5-√ = ab 2+5 = ab
where aa and bb are integers with b≠0 b≠0. Now rewrite this as
5-√ = ab - 2-√. 5 = ab - 2.
Squaring both sides of this equation we obtain
5 = a2 b2 - 2 2-√ ab + 2. 5 = a2 b2 - 22 ab + 2.
Now, carefully solve for 2-√ 2 and obtain
2-√ = - 3b 2a + a 2b. 2 = - 3b 2a + a 2b.
This implies that 2-√ 2 is a rational number which is a contradiction. Thus
2-√ + 5-√ 2+5
is an irrational number.
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