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## The Mathematical Constants

This section lists the mathematical constants, their use(s) (and sometimes rationale for their inclusion).

Table 4.1. Mathematical Constants

name | formula | Value (6 decimals) | Uses and Rationale
--- | --- | --- | ---
Rational fractions |  |  |
half | 1/2 | 0.5 |
third | 1/3 | 0.333333 |
two_thirds | 2/3 | 0.66667 |
three_quarters | 3/4 | 0.75 |
two and related |  |  |
root_two | √2 | 1.41421 | Equivalent to POSIX constant M_SQRT2
root_three | √3 | 1.73205 |
half_root_two | √2 /2 | 0.707106 |
ln_two | ln(2) | 0.693147 | Equivalent to POSIX constant M_LN2
ln_ten | ln(10) | 2.30258 | Equivalent to POSIX constant M_LN10
ln_ln_two | ln(ln(2)) | -0.366512 | Gumbel distribution median
root_ln_four | √ln(4) | 1.177410 |
one_div_root_two | 1/√2 | 0.707106 | Equivalent to POSIX constant M_SQRT1_2
π and related |  |  |
pi | π | 3.14159 | Ubiquitous. Archimedes constant π. Equivalent to POSIX constant M_PI
half_pi | π/2 | 1.570796 | Equivalent to POSIX constant M_PI2
third_pi | π/3 | 1.04719 |
quarter_pi | π/4 | 0.78539816 | Equivalent to POSIX constant M_PI_4
sixth_pi | π/6 | 0.523598 |
two_pi | 2π | 6.28318 | Many uses, most simply, circumference of a circle
tau | τ | 6.28318 | @https://en.wikipedia.org/wiki/Turn_(angle)#Tau_proposals Many uses, most simply, circumference of a circle. Equal to two_pi.
two_thirds_pi | 2/3 π | 2.09439 | volume of a hemi-sphere = 4/3 π r³
three_quarters_pi | 3/4 π | 2.35619 | = 3/4 π
four_thirds_pi | 4/3 π | 4.18879 | volume of a sphere = 4/3 π r³
one_div_two_pi | 1/(2π) | 1.59155 | Widely used
root_pi | √π | 1.77245 | Widely used
root_half_pi | √ π/2 | 1.25331 | Widely used
root_two_pi | √ π*2 | 2.50662 | Widely used
one_div_pi | 1/π | 0.31830988 | Equivalent to POSIX constant M_1_PI
two_div_pi | 2/π | 0.63661977 | Equivalent to POSIX constant M_2_PI
one_div_root_pi | 1/√π | 0.564189 |
two_div_root_pi | 2/√π | 1.128379 | Equivalent to POSIX constant M_2_SQRTPI
one_div_root_two_pi | 1/√(2π) | 0.398942 |
root_one_div_pi | √(1/π | 0.564189 |
pi_minus_three | π-3 | 0.141593 |
four_minus_pi | 4 -π | 0.858407 |
pi_pow_e | πe | 22.4591 |
pi_sqr | π2 | 9.86960 |
pi_sqr_div_six | π2/6 | 1.64493 |
pi_cubed | π3 | 31.00627 |
cbrt_pi | √3 π | 1.46459 |
one_div_cbrt_pi | 1/√3 π | 0.682784 |
Euler's e and related |  |  |
e | e | 2.71828 | Euler's constant e, equivalent to POSIX constant M_E
exp_minus_half | e -1/2 | 0.606530 |
e_pow_pi | e π | 23.14069 |
root_e | √ e | 1.64872 |
log10_e | log10(e) | 0.434294 | Equivalent to POSIX constant M_LOG10E
one_div_log10_e | 1/log10(e) | 2.30258 |
log2_e | log2(e) | 1.442695 | This is the same as 1/ln(2) and is equivalent to POSIX constant M_LOG2E
Trigonometric |  |  |
degree | radians = π / 180 | 0.017453 |
radian | degrees = 180 / π | 57.2957 |
sin_one | sin(1) | 0.841470 |
cos_one | cos(1) | 0.54030 |
sinh_one | sinh(1) | 1.17520 |
cosh_one | cosh(1) | 1.54308 |
Phi | Phidias golden ratio | Phidias golden ratio |
phi | (1 + √5) /2 | 1.61803 | finance
ln_phi | ln(φ) | 0.48121 |
one_div_ln_phi | 1/ln(φ) | 2.07808 |
Euler's Gamma |  |  |
euler | euler | 0.577215 | Euler-Mascheroni gamma constant
one_div_euler | 1/euler | 1.73245 |
euler_sqr | euler2 | 0.333177 |
Misc |  |  |
zeta_two | ζ(2) | 1.64493 | Riemann zeta function
zeta_three | ζ(3) | 1.20205 | Riemann zeta function
catalan | K | 0.915965 | Catalan (or Glaisher) combinatorial constant
glaisher | A | 1.28242 | Decimal expansion of Glaisher-Kinkelin constant
khinchin | k | 2.685452 | Decimal expansion of Khinchin constant
extreme_value_skewness | 12√6 ζ(3)/ π3 | 1.139547 | Extreme value distribution
rayleigh_skewness | 2√π(π-3)/(4 - π)3/2 | 0.631110 | Rayleigh distribution skewness
rayleigh_kurtosis_excess | -(6π2-24π+16)/(4-π)2 | 0.245089 | Rayleigh distribution kurtosis excess
rayleigh_kurtosis | 3+(6π2-24π+16)/(4-π)2 | 3.245089 | Rayleigh distribution kurtosis
first_feigenbaum |  | 4.6692016 | First Feigenbaum constant
plastic | Real solution of x3 = x + 1 | 1.324717957 | Plastic constant
gauss | Reciprocal of agm(1, √2) | 0.8346268 | Gauss's constant
dottie | Solution of cos(x) = x | 0.739085 | Dottie's number
reciprocal_fibonacci | Sum of reciprocals of Fibonacci numbers | 3.359885666 | Reciprocal Fibonacci constant
laplace_limit |  | .6627434193 | Laplace Limit

| Note
Integer values are not included in this list of math constants, however interesting, because they can be so easily and exactly constructed, even for UDT, for example: `static_cast<cpp_float>(42)`.

| Tip
If you know the approximate value of the constant, you can search for the value to find Boost.Math chosen name in this table.

| Tip
Bernoulli numbers are available at Bernoulli numbers.

| Tip
Factorials are available at factorial.