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22
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7
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333
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106
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355
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113
. These numbers are among the best-known and most widely used historical approximations of the constant. Each approximation generated in this way is a best rational approximation; that is, each is closer to π than any other fraction with the same or a smaller denominator.[30] Because π is known to be transcendental, it is by definition not algebraic and so cannot be a quadratic irrational. Therefore, π cannot have a periodic continued fraction. Although the simple continued fraction for π (shown above) also does not exhibit any other obvious pattern,[31][32] mathematicians have discovered several generalized continued fractions that do, such as
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Ladies and gentlemen, Nate!