4. Prove that: 2√3-4 is an irrational number, using the fact that √3 is an irrational number.
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Step-by-step explanation:
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To prove that 2√3 - 4 is an irrational number,
we can assume the opposite, that it is a rational number. Let's assume that it can be written as a ratio of two integers, a and b, such that a and b have no common factors other than 1 (i.e. they are in lowest terms). That is:
2√3 - 4 = a/b
Squaring both sides of the equation, we get:
(2√3 - 4)² = (a/b)²
4(3) + 16 - 8√3 = a²/b²
12 + 16 - 8√3 = a²/b²
28 = a²/b²
Now, let's substitute the fact that √3 is an irrational number. Since √3 is not equal to any fraction of integers,
we can assume that a²/b² = 3:
28 = 3
This is a contradiction, because 28 is not equal to 3.
So, our assumption that 2√3 - 4 is a rational number is false.
Hence, we have proved that 2√3 - 4 is an irrational number.
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