# Evaluate numerically an extremely small number

How to evaluate numerically $e^{-4 \cdotp 10^{35}}$ in the form $0,a_1a_2...\times 10^{-n}$

N[E^(-4*10^35)]

General::unfl: Underflow occurred in computation. >>
Underflow[]


doesn't work

• How many digits would you be satisfied with? – Michael E2 Feb 25 '16 at 0:11

n = 35;
{pow, logdec} = {#[[1]][[;; #[[2]]]], #[[1]][[#[[2]] + 1 ;;]]} &@
RealDigits[N[Log[10, E^(-4 10^n)], 200]];


So there are

FromDigits@pow

(* 173717792761300731060451567566642032 *)


zeroes,followed by

dec = N[1/10^(FromDigits[logdec] 10^-Length[logdec]), 50]

(* 0.12084848148616706326389685430970719629910021783715 *)


... or may be I'm off by one :)

Checking with smaller numbers:

n=4;
{pow, logdec} = {#[[1]][[;; #[[2]]]], #[[1]][[#[[2]] + 1 ;;]]} &@
RealDigits[N[Log[10, E^(-4 10^n)], 200]];
FromDigits@pow
(* 17371 *)
dec = N[1/10^(FromDigits[logdec] 10^-Length[logdec]), 50]
0.16623553671520518223181112083319039297273582477854

E^(-4 10^n) // N
(* 1.66235536715*10^-17372 *)

• Somewhat simpler: exp = -4*10^35; tenExp = Floor[exp/Log[10]]; rem = exp - tenExp Log[10]; N[Exp[rem], 50]*Inactivate@Power[10, tenExp] Also, rem can be written as Mod[exp, Log[10]] but I wanted to make sure (and make it obvious) that the integer and fractional part add up right. – The Vee Mar 19 '16 at 23:59
$MinMachineNumber $\text{2.2250738585072014$\grave{ }$*${}^{\wedge}$-308}$and as Karsten 7 points out: Log[$MinNumber]


$-3.121657384082590881601471993929\times 10^{15}$

Use Exp[x] == 10^(x/Log[10])

and Exp[a b] = Exp[a] + Exp[b] to find:

$0.0183156\ 10^{-10^{35}}$