132 Irrational and Real Numbers
Table 9-2. An S/R proof that 21/2 is not a rational number.
Statements Reasons
Assume 21/2 is rational We begin with this statement and will prove it false
The quantity 21/2 can be represented
as a ratio of two integers, p and q,
in lowest termsDefinition of rational number2 1/2 = p /q This is a mathematical statement of the claim made above
(21/2)^2 = (p /q)^2 Square both sides of the equation in the previous line
2 = p^2 /q^2 Use arithmetic to manipulate the equation in the previous line
2 q^2 = p^2 Multiply each side of the equation in the previous line by q^2
q^2 = p^2 /2 Divide each side of the equation in the previous line by 2
We know q is an integer, so q^2 is
an integerInteger-squared ruleWe know p is an integer, so p^2 is an
integerInteger-squared rulep^2 /2 is an integer We know this because q^2 = p^2 /2, and q^2 is an integer
p^2 is an even integer This follows from the definition of even integer
p is an even integer According to the odd-integer-squared rule, if p were odd,
then p^2 would be odd; but p^2 is even
p /2 is an integer This follows from the definition of even integer
Callp /2 = t This will make things a little simpler
p = 2t Multiply each side of the equation in the previous line by 2
2 q^2 = (2t)^2 Substitute 2t for p in the equation 2q^2 = p^2 from earlier in
this proof
2 q^2 = 4t^2 Simplify the right-hand side of the equation in the previous line
q^2 /2 = t^2 Divide each side of the equation in the previous line by 4
We know t is an integer, so t^2 is an
integerInteger-squared ruleq^2 /2 is an integer This follows from the previous two lines
q^2 is an even integer This follows from the definition of even integer
q is an even integer According to the odd-integer-squared rule, if q were odd,
then q^2 would be odd; but q^2 is even
q /2 is an integer This follows from the definition of even integer
The quotient p /q is a ratio of
integers in lowest termsPart of the assumption we made at the beginning of this proofp /2 is an integer, and q /2 is an integer We have proven both of these facts
(p /2)/(q /2) is a ratio of integers This follows from the statement immediately above this line
The ratio p /q is not given in
lowest terms(p /2)/(q /2) is in lower terms than p /q, but the two
expressions represent the same quantity
We have produced a logical absurdity The preceding line contradicts our original assumption
about p /q
2 1/2 is not rational Reductio ad absurdum
Q.E.D. Mission accomplished