Elementary Abstract Algebra- Examples and Applications, 2019a

2.1 THE ORIGIN OF COMPLEX NUMBERS 33
possible to state a proof very succinctly in “statement−reason” format. For
instance, here is a “statement−reason” proof of Proposition 2.1.10:
Statement
Reason
x is the hypotenuse of the right Given
triangle in Figure 2.1.1
x is rational
supposition (will be contradicted)
x 2 =2
Pythagorean Theorem
x = m/n where m, n are integers Definition of rational
m, n have no common factors Fraction can always be reduced
(m/n) 2 =2
Substitution
m 2 =2n 2
Rearrangement
m =2k where k is an integer Exercise 2.1.9 part (b)
(2k/n) 2 =2
Substitution
n 2 =2k 2
Rearrangement
n =2j where j is an integer Exercise 2.1.9 part (b)
m and n have a common factor 2 is a factor of both
supposition is false
Contradictory statements
x cannot be rational
Negation of supposition
Note that the preceding proof amounts to a proof that √ 2 is irrational,
since we know that √ 2 is the length of the hypothesis in question. Given the
results of Exercise 2.1.9, we can use a similar proof to find more irrational
numbers.
Exercise 2.1.11.
(a) Prove that the cube root of 2 is irrational. (*Hint*)
(b) Prove that the nth root of 2 is irrational, if n is a positive integer greater
than 1.
(c) Prove that 2 1/n is irrational, if n is a negative integer less than -1.
In the proof of Proposition 2.1.10, we “plugged in” or substituted one
expression for another. For example, when we discovered that m was divisible
by 2 we substituted 2j for m, which was useful for the algebra that
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