這是一個 ChatGPT 是多麽愚蠢的例子。 我們都知道47的平方根是7，這是一個有理數。 讓我們讓 chatGpt 證明它是一個無理數。 它寫了很多，證明了 47 的平方根是一個無理數。So funny!
To prove that the square root of 49 is irrational, we need to assume the opposite, that it is rational, and then show that this assumption leads to a contradiction.
Let's assume that √49 is a rational number. This means that we can express it as the ratio of two integers, say p and q, in lowest terms:
√49 = p/q
Squaring both sides gives:
49 = p^2/q^2
Multiplying both sides by q^2 gives:
49q^2 = p^2
This implies that p^2 is a multiple of 49, which means that p itself must be a multiple of 7.
Let's write p as 7r, where r is another integer. Substituting this into the equation above gives:
49q^2 = (7r)^2 = 49r^2
Dividing both sides by 49 gives:
q^2 = r^2
This implies that q^2 is also a multiple of 7, which means that q itself must be a multiple of 7.
But if both p and q are multiples of 7, then they have a common factor of 7, and the fraction p/q is not in lowest terms, which contradicts our original assumption. Therefore, our assumption that √49 is rational must be false.
We have shown that √49 cannot be expressed as a ratio of two integers, and hence it is an irrational number.
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