o Prove that for each real number x, (x + v2) is irrational or is irrational.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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8a
Note: You may even assume that we have proven that 5 is an irra-
tional number. (We have not proven this.)
(c) Is the real number 2+ /5 a rational number or an irrational number?
Justify your
conclusion.
8. (a) Prove that for each real number x, (x + v2) is irrational or (-x + v2)
is irrational.
(b) Generalize the proposition in Part (a) for any irrational number (instead
of just 2) and then prove the new proposition.
9. Is the following statement true or false?
For all positive real numbers x and y, x+ y < VI+ ỹ.
10. Is the following proposition true or false? Justify your conclusion.
1
For each real number x, x (1 – x) < -.
* 11. (a) Is the base 2 logarithm of 32, log2(32), a rational number or an irra-
tional number? Justify your conclusion.
Transcribed Image Text:Note: You may even assume that we have proven that 5 is an irra- tional number. (We have not proven this.) (c) Is the real number 2+ /5 a rational number or an irrational number? Justify your conclusion. 8. (a) Prove that for each real number x, (x + v2) is irrational or (-x + v2) is irrational. (b) Generalize the proposition in Part (a) for any irrational number (instead of just 2) and then prove the new proposition. 9. Is the following statement true or false? For all positive real numbers x and y, x+ y < VI+ ỹ. 10. Is the following proposition true or false? Justify your conclusion. 1 For each real number x, x (1 – x) < -. * 11. (a) Is the base 2 logarithm of 32, log2(32), a rational number or an irra- tional number? Justify your conclusion.
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