Pi¶

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description¶

Pi
    is pi, with numerical value ~= 3.14159.
Pi is a mathematical constant: it has attributes Constant and Protected,
NumericQ[Pi] is True, and D[Pi, x] is 0. N[Pi, prec] evaluates it to any
precision.

Examples¶

No verified examples yet for this function.

Implementation notes¶

Attributes Constant, Protected. Attributes[Pi] = {Constant, Protected};

Attributes: Constant, Protected.

Implementation status¶

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References¶

Source: src/info.c
Specification: docs/spec/builtins/mathematical-constants.md

Notes & additional examples¶

Worked examples¶

The symbol Pi stays exact until you ask for a numeric value, and N[Pi, prec] delivers it to any requested precision:

In[1]:= N[Pi]
Out[1]= 3.14159

In[2]:= N[Pi, 40]
Out[2]= 3.1415926535897932384626433832795028841971

Exact special values of the elementary functions are reduced symbolically in terms of Pi:

In[1]:= Sin[Pi/6]
Out[1]= 1/2

In[2]:= Cos[Pi/3] + Sin[Pi/4]^2
Out[2]= 1

Pi arises naturally as the closed form of inverse trigonometric values — and Cos[Pi/5] evaluates to the golden-ratio surd, a non-obvious exact constructible number:

In[1]:= ArcTan[1]
Out[1]= 1/4 Pi

In[2]:= Cos[Pi/5]
Out[2]= 1/4 (1 + Sqrt[5])

As a recognised constant, Pi participates in exact closed forms that you can then numericalise to high precision — e.g. the Basel value Pi^2/6 to 40 digits:

In[1]:= N[Pi^2/6, 40]
Out[1]= 1.6449340668482264364724151666460251892188

It is treated as constant by calculus, so its derivative is zero:

In[1]:= D[Pi, x]
Out[1]= 0

Notes¶

Pi is the mathematical constant π. It carries the Constant and Protected attributes, NumericQ[Pi] is True, and D[Pi, x] is 0. It remains an exact symbol through symbolic computation — driving the special-value reductions of the trigonometric and inverse-trigonometric functions — and is evaluated to arbitrary precision only on demand via N[Pi, prec], which uses the MPFR numeric core.