Approximations to e include a zeroless pandigital approximation that is correct to 18457734525360901453873570 decimal digits and is given by
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(1)
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which was found by R. Sabey in 2004 (Friedman 2004). An improved zeroless pandigital approximation
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(2)
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which is correct to 8368428989068425943817590916445001887164 decimal digits, was found by D. Bamberger (pers. comm., Mar. 13, 2024; Friedman 2004 updated page). A pandigital approximation including 0
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(3)
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which is correct to 5447761679516886279045570843725804037563002422 decimal digits, was subsequently found by Reddit user Fastfaxr (2024).
M. Verheyen (pers. comm., Nov. 8, 2026) found a zeroless pandigital approximation allowing nested square roots,
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(4)
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Here 
, so the expression
equals 
with 
. The absolute
error is less than 
, and its negative base-10
logarithm is about 
.
Castellanos (1988ab) gives several curious approximations to 
,
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(5)
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(6)
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(7)
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(8)
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(9)
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(10)
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which are good to 6, 7, 9, 10, 12, and 15 digits respectively.
E. Weisstein (using Wolfram|Alpha, Oct. 29, 2026) obtained
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(11)
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which is good to 10 decimal digits, and the root 
of
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(12)
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which approximates 
to 7 decimal digits.
E. Pegg Jr. (pers. comm., Mar. 2, 2002), found
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(13)
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which is good to 7 digits.
A compact approximation attributed to G. Morley by Pegg (2005) is
![e approx (1+5RadicalBox[6, 5])/3,](/images/equations/eApproximations/NumberedEquation8.svg) |
(14)
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whose right side is smaller than 
by approximately 
.
J. Lafont (pers. comm., MAy 16, 2008) found
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(15)
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where 
is a harmonic number, which is good to 7 digits.
N. Davidson (pers. comm., Sept. 7, 2004) found
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(16)
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which is good to 6 digits.
D. Barron noticed the curious approximation
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(17)
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where 
is Catalan's constant and 
is the Euler-Mascheroni
constant, which however, is only good to 3 digits.
