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a and b are two positive irrational numbers - AQA - A-Level Maths Mechanics - Question 7 - 2020 - Paper 2
Question 7
a and b are two positive irrational numbers. The sum of a and b is rational. The product of a and b is rational. Caroline is trying to prove $\frac{1}{a} + \frac{... show full transcript
Worked Solution & Example Answer:a and b are two positive irrational numbers - AQA - A-Level Maths Mechanics - Question 7 - 2020 - Paper 2
Step 1
Prove by contradiction that the difference of any rational number and any irrational number is irrational.
Answer
To prove this by contradiction, we start by assuming that the difference is rational.
Let [ x = r - a ] where ( r ) is a rational number and ( a ) is an irrational number. This implies:
[ r = x + a ]
Since ( x ) is assumed to be rational and ( a ) is irrational, the sum of a rational number and an irrational number yields an irrational number. Therefore, we have:
- Let ( x ) be a rational number, then: [ x + a = r ]
This leads to a contradiction, as it implies that an irrational number can be expressed as a rational number. Thus, our initial assumption must be incorrect, confirming that the difference of any rational number and any irrational number is indeed irrational.