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Pi

Table of contents: Definitions - Numerical value - Elementary function representations - Integral representations - Series representations - Product representations - Limit representations - Approximations

Definitions

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Symbol: Pi π\pi The constant pi (3.14...)

Numerical value

6505a9
π[3.1415926535897932384626433832795028841971693993751±5.831051]\pi \in \left[3.1415926535897932384626433832795028841971693993751 \pm 5.83 \cdot 10^{-51}\right]
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Table of simple expressions involving π\pi to 50 digits
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πQ\pi \notin \mathbb{Q}
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πQ\pi \notin \overline{\mathbb{Q}}
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π=n=1A000796 ⁣(n)101n\pi = \sum_{n=1}^{\infty} \text{A000796}\!\left(n\right) {10}^{1 - n}

Elementary function representations

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π=4atan(1)\pi = 4 \operatorname{atan}(1)
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π=2acos(0)\pi = 2 \operatorname{acos}(0)
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π=2asin(1)\pi = 2 \operatorname{asin}(1)
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π=zero*x[3,4]sin(x)\pi = \mathop{\operatorname{zero*}\,}\limits_{x \in \left[3, 4\right]} \sin(x)
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π=16acot(5)4acot(239)\pi = 16 \operatorname{acot}(5) - 4 \operatorname{acot}(239)
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π=ilog(1)\pi = -i \log(-1)

Integral representations

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π=2111x2dx\pi = 2 \int_{-1}^{1} \sqrt{1 - {x}^{2}} \, dx
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π=1x2+1dx\pi = \int_{-\infty}^{\infty} \frac{1}{{x}^{2} + 1} \, dx
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π=(ex2dx)2\pi = {\left(\int_{-\infty}^{\infty} {e}^{-{x}^{2}} \, dx\right)}^{2}

Series representations

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π=4k=0(1)k2k+1\pi = 4 \sum_{k=0}^{\infty} \frac{{\left(-1\right)}^{k}}{2 k + 1}
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π=k=0116k(48k+128k+418k+518k+6)\pi = \sum_{k=0}^{\infty} \frac{1}{{16}^{k}} \left(\frac{4}{8 k + 1} - \frac{2}{8 k + 4} - \frac{1}{8 k + 5} - \frac{1}{8 k + 6}\right)

Product representations

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π=2k=14k24k21\pi = 2 \prod_{k=1}^{\infty} \frac{4 {k}^{2}}{4 {k}^{2} - 1}

Limit representations

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π=limk16kk(2kk)2\pi = \lim_{k \to \infty} \frac{{16}^{k}}{k {{2 k \choose k}}^{2}}

Approximations

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π227<0.00127\left|\pi - \frac{22}{7}\right| < 0.00127
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π355113<2.67107\left|\pi - \frac{355}{113}\right| < 2.67 \cdot 10^{-7}
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πlog ⁣(6403203+744)163<2.241031\left|\pi - \frac{\log\!\left({640320}^{3} + 744\right)}{\sqrt{163}}\right| < 2.24 \cdot 10^{-31}
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1π(12k=0N1(1)k(6k)!(13591409+545140134k)(3k)!(k!)36403203k+3/2)<1151931373056000N\left|\frac{1}{\pi} - \left(12 \sum_{k=0}^{N - 1} \frac{{\left(-1\right)}^{k} \left(6 k\right)! \left(13591409 + 545140134 k\right)}{\left(3 k\right)! {\left(k !\right)}^{3} {640320}^{3 k + 3 / 2}}\right)\right| < \frac{1}{{151931373056000}^{N}}

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2019-11-11 15:50:15.016492 UTC