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E

Status: Stable

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

Description

E

is the exponential constant e (base of natural logarithms), with numerical value ~= 2.71828.

NumericQ[E]

is True, and D[E, x] is 0. N[E, prec] evaluates it to any

Notes E is a mathematical constant: it has attributes Constant and Protected, precision.

Examples (4)

Every input below was run against the current Mathilda build and its output recorded.

Applications (4)

In[1]:= Log[E^3]
Out[1]= 3

In[2]:= N[E, 40]
Out[2]= 2.7182818284590452353602874713526624977572

In[3]:= Sum[1/n!, {n, 0, Infinity}]
Out[3]= E

In[4]:= Limit[(1 + 1/n)^n, n -> Infinity]
Out[4]= E

Implementation notes

Definition. E is Euler's number e = 2.71828…, the base of the natural logarithm and the unique real with D[E^x, x] == E^x. It is a constant symbol: row { "E", M_E, fill_mpfr_e } of the kConstants[] table in src/numeric.c supplies its numeric value, and the numericalize_init loop stamps it Constant | Protected (so Attributes[E] is {Constant, Protected}; the base Protected flag is also set in the default-attributes table in src/attr.c). Interned name SYM_E (src/sym_names.c), docstring in src/info.c. E has no DownValues; the identities that fold back to it — Log[E] -> 1, Log[E^3] -> 3, Exp[1] -> E, and the series / limit characterisations Sum[1/n!, {n,0,Infinity}] -> E, Limit[(1+1/n)^n, n->Infinity] -> E — are produced by the log-exp, sum and limit subsystems. D[E, x] -> 0 follows from Constant.

Representation & numeric value. E is a bare EXPR_SYMBOL that prints as E and stays exact under evaluation. N[E] returns the machine double M_E; N[E, k] runs fill_mpfr_e, which evaluates mpfr_exp(1) at the requested precision (MPFR ships no dedicated const_e) and returns an EXPR_MPFR. Machine value 2.71828; N[E, 50] = 2.71828182845904523536028747135266249775724709369996.

Usage & limits. E is the fixed point of the exponential closed forms and is recognised as numeric (NumericQ[E] is True, via is_numeric_quantity in src/core.c). Being transcendental it is never rationalised; its digits are produced on demand by N.

  • Attributes Constant, Protected. Attributes[E] = {Constant, Protected}; the symbol cannot be reassigned.
  • Propagated as an exact, unevaluated symbol; NumericQ[E] is True and D[E, x] = 0.
  • N[E] gives the machine value 2.71828; N[E, prec] gives any precision (MPFR mpfr_exp of 1), e.g. N[E, 50] = 2.71828182845904523536028747135266249775724709369996.
  • Participates in exact numeric work, e.g. Round[E^100] = 26881171418161354484126255515800135873611119.

Attributes: Constant, Protected.

References

Notes & additional examples

Notes

E is the exponential constant e, the base of the natural logarithm. It is a protected Constant, so D[E, x] is 0 and it survives evaluation symbolically until N is applied — N[E, prec] returns it to any requested precision via the MPFR backend. The constant is recognised by the rest of the system, so the classic limit and series characterisations of e both fold back to E, and Log[E^3] simplifies to its exponent.