Prove that the sum of a rational number and an irrational number is always irrational. - AQA - A-Level Maths Mechanics - Question 9 - 2019 - Paper 1
Question 9
Prove that the sum of a rational number and an irrational number is always irrational.
Worked Solution & Example Answer:Prove that the sum of a rational number and an irrational number is always irrational. - AQA - A-Level Maths Mechanics - Question 9 - 2019 - Paper 1
Step 1
Assume the sum is rational
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Answer
Let us assume that the sum of a rational number and an irrational number is rational. Suppose that we have a rational number ( a ) and an irrational number ( n ). Therefore, we assume:
c=a+n
where ( c ) is rational.
Step 2
Express the irrational number
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Answer
Rearranging the equation gives us:
n = c - a$$
Since both \( c \) and \( a \) are rational (by definition), their difference \( c - a \) must also be rational.
Step 3
Contradiction
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Answer
This leads to a contradiction because we assumed that ( n ) is irrational. Thus, the statement that the sum of a rational number and an irrational number is rational must be false.
Step 4
Conclusion
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Answer
Hence, it follows that the sum of a rational number and an irrational number must always be irrational.
