9514 1404 393
Answer:
- no, √2 is irrational
- 7/5, 99/70, 19601/13860, any truncation of the decimal value
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.
