Consider the following: $a$ 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 that $$\frac{1}{a} + \frac{1}{b}$$ is rational. Here is her proof: Step 1 $$\frac{1}{a} + \frac{1}{b}$$ Step 2 2 is rational and $a + b$ is non-zero and rational. Step 3 Therefore $$\frac{2}{a + b}$$ is rational. Step 4 Hence $$\frac{1}{a} + \frac{1}{b}$$ is rational. (b) Prove by contradiction that the difference of any rational number and any irrational number is irrational.

A-Level AQA Maths Pure Proof by Contradiction: Consider the following: $a$ and

Consider the following: $a$ and $b$ are two positive irrational numbers - AQA - A-Level Maths Pure - Question 7 - 2020 - Paper 2

Question 7

Consider the following: $a$ 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 ... show full transcript

Worked Solution & Example Answer:Consider the following: $a$ and $b$ are two positive irrational numbers - AQA - A-Level Maths Pure - Question 7 - 2020 - Paper 2

Step 1

Identify the error lies in step 1 without contradiction

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Answer

In Caroline's proof, the mistake lies in step 1. The expression $\frac{1}{a} + \frac{1}{b}$ cannot be concluded as rational merely from the information given that the sum of $a$ and $b$ is rational. A proper rigor in argumentation is required to link the components accurately.

Step 2

Recall correct addition $$\frac{b}{ab}$$

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Answer

When proving that $\frac{1}{a} + \frac{1}{b}$ is rational, the correct approach is to express it as: $\frac{1}{a} + \frac{1}{b} = \frac{b + a}{ab}$ Here, if $ab$ is rational and non-zero, and $a + b$ is rational, then by definition, the sum $\frac{1}{a} + \frac{1}{b}$ must also be rational.

Step 3

Prove by contradiction that the difference of any rational number and any irrational number is irrational.

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Answer

Assume that the difference between a rational number, $r$ , and an irrational number, $x$ , is rational:

$r - x = y$

where $y$ is rational. Therefore, we can rearrange this to:

$x = r - y$

If both $r$ and $y$ are rational, then $x$ , being the result of a rational number minus another rational number, must also be rational. This contradicts the assumption that $x$ is irrational.

Thus, the initial assumption must be false, proving that the difference between any rational number and any irrational number is indeed irrational.

Consider the following: $a$ and $b$ are two positive irrational numbers - AQA - A-Level Maths Pure - Question 7 - 2020 - Paper 2

Worked Solution & Example Answer:Consider the following: $a$ and $b$ are two positive irrational numbers - AQA - A-Level Maths Pure - Question 7 - 2020 - Paper 2

Identify the error lies in step 1 without contradiction

Recall correct addition $$\frac{b}{ab}$$

Prove by contradiction that the difference of any rational number and any irrational number is irrational.

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