Symbol: ConstPi — π \pi π — The constant pi (3.14...)
The real number giving the ratio of a circle's circumference to its diameter.
Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("b5d706"),
SymbolDefinition(ConstPi, ConstPi, "The constant pi (3.14...)"),
Description("The real number giving the ratio of a circle's circumference to its diameter.")) π ∈ [ 3.1415926535897932384626433832795028841971693993751 ± 5.83 ⋅ 1 0 − 51 ] \pi \in \left[3.1415926535897932384626433832795028841971693993751 \pm 5.83 \cdot 10^{-51}\right] π ∈ [ 3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 ± 5 . 8 3 ⋅ 1 0 − 5 1 ] TeX:
\pi \in \left[3.1415926535897932384626433832795028841971693993751 \pm 5.83 \cdot 10^{-51}\right] Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("6505a9"),
Formula(Element(ConstPi, RealBall(Decimal("3.1415926535897932384626433832795028841971693993751"), Decimal("5.83e-51"))))) π ∉ Q \pi \notin \mathbb{Q} π ∈ / Q TeX:
\pi \notin \mathbb{Q} Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...) QQ Q \mathbb{Q} Q Rational numbers
Source code for this entry:
Entry(ID("0c838a"),
Formula(NotElement(ConstPi, QQ))) π = 4 atan ( 1 ) \pi = 4 \operatorname{atan}\!\left(1\right) π = 4 a t a n ( 1 ) TeX:
\pi = 4 \operatorname{atan}\!\left(1\right) Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...) Atan atan ( z ) \operatorname{atan}\!\left(z\right) a t a n ( z ) Inverse tangent
Source code for this entry:
Entry(ID("0c9939"),
Formula(Equal(ConstPi, Mul(4, Atan(1))))) π = 16 acot ( 5 ) − 4 acot ( 239 ) \pi = 16 \operatorname{acot}\!\left(5\right) - 4 \operatorname{acot}\!\left(239\right) π = 1 6 a c o t ( 5 ) − 4 a c o t ( 2 3 9 ) TeX:
\pi = 16 \operatorname{acot}\!\left(5\right) - 4 \operatorname{acot}\!\left(239\right) Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("f8d280"),
Formula(Equal(ConstPi, Sub(Mul(16, Acot(5)), Mul(4, Acot(239)))))) π = − i log ( − 1 ) \pi = -i \log\!\left(-1\right) π = − i log ( − 1 ) TeX:
\pi = -i \log\!\left(-1\right) Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...) ConstI i i i Imaginary unit Log log ( z ) \log\!\left(z\right) log ( z ) Natural logarithm
Source code for this entry:
Entry(ID("590136"),
Formula(Equal(ConstPi, Neg(Mul(ConstI, Log(-1)))))) π = 2 ∫ − 1 1 1 − x 2 d x \pi = 2 \int_{-1}^{1} \sqrt{1 - {x}^{2}} \, dx π = 2 ∫ − 1 1 1 − x 2 d x TeX:
\pi = 2 \int_{-1}^{1} \sqrt{1 - {x}^{2}} \, dx Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...) Sqrt z \sqrt{z} z Principal square root Pow a b {a}^{b} a b Power
Source code for this entry:
Entry(ID("464961"),
Formula(Equal(ConstPi, Mul(2, Integral(Sqrt(Sub(1, Pow(x, 2))), Tuple(x, -1, 1)))))) π = ∫ − ∞ ∞ 1 x 2 + 1 d x \pi = \int_{-\infty}^{\infty} \frac{1}{{x}^{2} + 1} \, dx π = ∫ − ∞ ∞ x 2 + 1 1 d x TeX:
\pi = \int_{-\infty}^{\infty} \frac{1}{{x}^{2} + 1} \, dx Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...) Pow a b {a}^{b} a b Power Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("04cd99"),
Formula(Equal(ConstPi, Integral(Div(1, Add(Pow(x, 2), 1)), Tuple(x, Neg(Infinity), Infinity))))) π = ( ∫ − ∞ ∞ e − x 2 d x ) 2 \pi = {\left(\int_{-\infty}^{\infty} {e}^{-{x}^{2}} \, dx\right)}^{2} π = ( ∫ − ∞ ∞ e − x 2 d x ) 2 TeX:
\pi = {\left(\int_{-\infty}^{\infty} {e}^{-{x}^{2}} \, dx\right)}^{2} Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...) Pow a b {a}^{b} a b Power Exp e z {e}^{z} e z Exponential function Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("dae4a7"),
Formula(Equal(ConstPi, Pow(Integral(Exp(Neg(Pow(x, 2))), Tuple(x, Neg(Infinity), Infinity)), 2)))) π = 4 ∑ k = 0 ∞ ( − 1 ) k 2 k + 1 \pi = 4 \sum_{k=0}^{\infty} \frac{{\left(-1\right)}^{k}}{2 k + 1} π = 4 k = 0 ∑ ∞ 2 k + 1 ( − 1 ) k TeX:
\pi = 4 \sum_{k=0}^{\infty} \frac{{\left(-1\right)}^{k}}{2 k + 1} Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...) Pow a b {a}^{b} a b Power Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("f617c0"),
Formula(Equal(ConstPi, Mul(4, Sum(Div(Pow(-1, k), Add(Mul(2, k), 1)), Tuple(k, 0, Infinity)))))) π = ∑ k = 0 ∞ 1 16 k ( 4 8 k + 1 − 2 8 k + 4 − 1 8 k + 5 − 1 8 k + 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) π = k = 0 ∑ ∞ 1 6 k 1 ( 8 k + 1 4 − 8 k + 4 2 − 8 k + 5 1 − 8 k + 6 1 ) References:
D. H. Bailey and P. B. Borwein and S. Plouffe (1997). On the rapid computation of various polylogarithmic constants. Mathematics of Computation. vol 66, no 218, p. 903–913. DOI:10.1090/S0025-5718-97-00856-9 TeX:
\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) Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...) Pow a b {a}^{b} a b Power Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("fddfe6"),
Formula(Equal(ConstPi, Sum(Mul(Div(1, Pow(16, k)), Sub(Sub(Sub(Div(4, Add(Mul(8, k), 1)), Div(2, Add(Mul(8, k), 4))), Div(1, Add(Mul(8, k), 5))), Div(1, Add(Mul(8, k), 6)))), Tuple(k, 0, Infinity)))),
References("D. H. Bailey and P. B. Borwein and S. Plouffe (1997). On the rapid computation of various polylogarithmic constants. Mathematics of Computation. vol 66, no 218, p. 903–913. DOI:10.1090/S0025-5718-97-00856-9")) π = 2 ∏ k = 1 ∞ 4 k 2 4 k 2 − 1 \pi = 2 \prod_{k=1}^{\infty} \frac{4 {k}^{2}}{4 {k}^{2} - 1} π = 2 k = 1 ∏ ∞ 4 k 2 − 1 4 k 2 TeX:
\pi = 2 \prod_{k=1}^{\infty} \frac{4 {k}^{2}}{4 {k}^{2} - 1} Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...) Pow a b {a}^{b} a b Power Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("69fe63"),
Formula(Equal(ConstPi, Mul(2, Product(Div(Mul(4, Pow(k, 2)), Sub(Mul(4, Pow(k, 2)), 1)), Tuple(k, 1, Infinity)))))) π = lim k → ∞ 16 k k ( 2 k k ) 2 \pi = \lim_{k \to \infty} \frac{{16}^{k}}{k {{2 k \choose k}}^{2}} π = k → ∞ lim k ( k 2 k ) 2 1 6 k TeX:
\pi = \lim_{k \to \infty} \frac{{16}^{k}}{k {{2 k \choose k}}^{2}} Definitions:
Fungrim symbol Notation Short description ConstPi π \pi π The constant pi (3.14...) SequenceLimit lim n → a f ( n ) \lim_{n \to a} f\!\left(n\right) lim n → a f ( n ) Limiting value of sequence Pow a b {a}^{b} a b Power Binomial ( n k ) {n \choose k} ( k n ) Binomial coefficient Infinity ∞ \infty ∞ Positive infinity
Source code for this entry:
Entry(ID("e1e106"),
Formula(Equal(ConstPi, SequenceLimit(Div(Pow(16, k), Mul(k, Pow(Binomial(Mul(2, k), k), 2))), k, Infinity)))) ∣ π − 22 7 ∣ < 0.00127 \left|\pi - \frac{22}{7}\right| \lt 0.00127 ∣ ∣ ∣ ∣ π − 7 2 2 ∣ ∣ ∣ ∣ < 0 . 0 0 1 2 7 TeX:
\left|\pi - \frac{22}{7}\right| \lt 0.00127 Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value ConstPi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("2516c2"),
Formula(Less(Abs(Sub(ConstPi, Div(22, 7))), Decimal("0.00127")))) ∣ π − 355 113 ∣ < 2.67 ⋅ 1 0 − 7 \left|\pi - \frac{355}{113}\right| \lt 2.67 \cdot 10^{-7} ∣ ∣ ∣ ∣ π − 1 1 3 3 5 5 ∣ ∣ ∣ ∣ < 2 . 6 7 ⋅ 1 0 − 7 TeX:
\left|\pi - \frac{355}{113}\right| \lt 2.67 \cdot 10^{-7} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value ConstPi π \pi π The constant pi (3.14...)
Source code for this entry:
Entry(ID("1e3a25"),
Formula(Less(Abs(Sub(ConstPi, Div(355, 113))), Decimal("2.67e-7")))) ∣ π − log ( 640320 3 + 744 ) 163 ∣ < 2.24 ⋅ 1 0 − 31 \left|\pi - \frac{\log\!\left({640320}^{3} + 744\right)}{\sqrt{163}}\right| \lt 2.24 \cdot 10^{-31} ∣ ∣ ∣ ∣ ∣ π − 1 6 3 log ( 6 4 0 3 2 0 3 + 7 4 4 ) ∣ ∣ ∣ ∣ ∣ < 2 . 2 4 ⋅ 1 0 − 3 1 TeX:
\left|\pi - \frac{\log\!\left({640320}^{3} + 744\right)}{\sqrt{163}}\right| \lt 2.24 \cdot 10^{-31} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value ConstPi π \pi π The constant pi (3.14...) Log log ( z ) \log\!\left(z\right) log ( z ) Natural logarithm Pow a b {a}^{b} a b Power Sqrt z \sqrt{z} z Principal square root
Source code for this entry:
Entry(ID("fdc3a3"),
Formula(Less(Abs(Sub(ConstPi, Div(Log(Add(Pow(640320, 3), 744)), Sqrt(163)))), Decimal("2.24e-31")))) ∣ 1 π − ( 12 ∑ k = 0 N − 1 ( − 1 ) k ( 6 k ) ! ( 13591409 + 545140134 k ) ( 3 k ) ! ( k ! ) 3 640320 3 k + 3 / 2 ) ∣ < 1 151931373056000 N \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| \lt \frac{1}{{151931373056000}^{N}} ∣ ∣ ∣ ∣ ∣ π 1 − ( 1 2 k = 0 ∑ N − 1 ( 3 k ) ! ( k ! ) 3 6 4 0 3 2 0 3 k + 3 / 2 ( − 1 ) k ( 6 k ) ! ( 1 3 5 9 1 4 0 9 + 5 4 5 1 4 0 1 3 4 k ) ) ∣ ∣ ∣ ∣ ∣ < 1 5 1 9 3 1 3 7 3 0 5 6 0 0 0 N 1 Assumptions: N ∈ Z ≥ 0 N \in \mathbb{Z}_{\ge 0} N ∈ Z ≥ 0 TeX:
\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| \lt \frac{1}{{151931373056000}^{N}}
N \in \mathbb{Z}_{\ge 0} Definitions:
Fungrim symbol Notation Short description Abs ∣ z ∣ \left|z\right| ∣ z ∣ Absolute value ConstPi π \pi π The constant pi (3.14...) Pow a b {a}^{b} a b Power Factorial n ! n ! n ! Factorial ZZGreaterEqual Z ≥ n \mathbb{Z}_{\ge n} Z ≥ n Integers greater than or equal to n
Source code for this entry:
Entry(ID("4c0698"),
Formula(Less(Abs(Sub(Div(1, ConstPi), Parentheses(Mul(12, Sum(Div(Mul(Mul(Pow(-1, k), Factorial(Mul(6, k))), Add(13591409, Mul(545140134, k))), Mul(Mul(Factorial(Mul(3, k)), Pow(Factorial(k), 3)), Pow(640320, Add(Mul(3, k), Div(3, 2))))), Tuple(k, 0, Sub(N, 1))))))), Div(1, Pow(151931373056000, N)))),
Variables(N),
Assumptions(Element(N, ZZGreaterEqual(0))))