Can you find a rational number that is exactly √2

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9514 1404 393

Step-by-step explanation:

1.√2 is not the square root of a perfect square integer, so is irrational. There is no rational number that is exactly √2.

The proof usually looks something like this. Let p/q represent the reduced square root. Then 2 = p²/q², or 2q² = p², which means p must be even. If p is even, then p² will have at least 2 factors of 2. This requires that q² be even, hence q must be even. This contradicts our assumption that p/q is a reduced fraction, so there cannot be any p/q that is equal to 2.

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2. Any truncation or rounding of the decimal representation of √2 will be a rational number: 1.4, 1.41, 1.414, and so on. In addition, there are some fractions that represent √2 fairly well. A couple are 99/70 and 19601/13860.

Additional fractions can be found by evaluating the continued fraction ...

  1 +1/(2 +1/(2 +1/(2+ ...)))

at some level of truncation. A few of these values are 7/5, 17/12, 41/29, 99/70, 239/169, 577/408.

The Babylonian method can also be used to find rational approximations. If you start with any rational approximation p/q, then a better approximation will be (2q² +p²)/(2pq). This formulation can be iterated as many times as you like. It very rapidly converges to very good approximations.

answer

the correct answer is

explanation

the given function is

the excluded values are obtained by solving:

we factor to obtain:

the excluded values are

the correct answer is 5 6 and 7