A technician monitoring a high-precision chemical reaction states that the temperature, TTT in ∘C^\circ C∘C, must satisfy the constraint 54.4<T<54.554.4 < T < 54.554.4<T<54.5. The technician claims that 54.554.554.5 is the largest possible temperature within this specification.
Explain the error in the technician's statement.
A research scientist argues that there is no largest value for TTT in the interval 54.4<T<54.554.4 < T < 54.554.4<T<54.5 and provides the following proof by contradiction:
Step 1: Assume there is a largest temperature in the interval 54.4<T<54.554.4 < T < 54.554.4<T<54.5 and let this largest value be LLL.
Step 2: Let k=L+54.52k = \frac{L + 54.5}{2}k=2L+54.5.
Step 3: L<k<54.5L < k < 54.5L<k<54.5, which is a contradiction.
Step 4: Therefore, there is no largest temperature in the interval 54.4<T<54.554.4 < T < 54.554.4<T<54.5.
Explain the contradiction stated in Step 3.
Using a similar method, prove that there is no smallest value of TTT in the interval 54.4<T<54.554.4 < T < 54.554.4<T<54.5.
368 exam-style questions on Edexcel A Level Maths Algebraic Methods, covering 1.1 Proof by Contradiction, 1.2 Algebraic Fractions, 1.3 Partial Fractions, 1.4 Repeated Factors, and 1.5 Algebraic Division. Each one has a worked solution and a mark scheme showing where the marks go.