# Problem of the Month (April 2014)

This month we explore approximations of the constants π and e using logarithms. We allow expressions to use the four arithmetic operations (× / + –) and the ln function. We measure the complexity of an expression as the number of integer digits used plus the number of logarithms used. Among all expressions using only one digit and logarithms with a given complexity, which one best approximates these constants? What if any digits can be used?

What if we are allowed to use square roots instead of logarithms? What if we are allowed to use decimal points instead? What if the logarithms, square roots, or decimals were not counted in the complexity?

Solutions were sent by Hakan Summakoğlu and Joe DeVincentis.

Here are the best known results.

Best Known Approximations Using 1's and ln's
ComplexityExpression for πValueAuthorExpression for eValueAuthor
11π – 2.14 1e – 1.72
21 + 1π – 1.14 1 + 1e – .718
31 + 1 + 1π – .142 1 + 1 + 1e + .282
41×1 + 1 + 1π – .142 ln[11 + 1]e – .233
5ln[ (1 + 1) × 11]π – .0506 1 + 1 + ln[1 + 1]e – .0251
6ln[ (1 + 1) × 11 + 1]π – .00610 ln[111] – 1 – 1e – .00875
71 + 1 + 1 / ln[ ln[11] ] π + .00180 ln[111 + 1] – 1 – 1e + .000217
81 + ln[11 – ln[11 + 1] ] π + .000248 ln[111 + 1] – 1 – 1×1e + .000217
9ln[111 / (1+1) ln[11] ]π + .000199 (1 + 1) / (1 – ln[1 + 1] ) ln[11]e – .000155
10ln[11 × (1 + ln[111 + 1] ) ] – 1π + .00000896 1 + ln[ ln[ (111 – 1) ln[11] ] ]e + .0000234
111 + 11 × 11 / (111 / (1+1) + 1) π + .000000267 ln[ 11 / ln[1 + 1] – 1 / (ln[11] – 1) ]e + .00000145
121 × 1 + 11 × 11 / (111 / (1+1) + 1) π + .000000267JDln[11×ln[ln[ln(1+1)×111]]–1]e + .0000000275JD

Best Known Approximations Using 2's and ln's
ComplexityExpression for πValueAuthorExpression for eValueAuthor
12π – 1.14 2e – .718
22 + 2π + .858 2 + 2e + 1.28
3ln[22]π – .0506 2 + ln[2]e – .0251
4ln[22 + 2]π + .0365 (2 + 2) ln[2] e + .0543
5ln[22 + 2/2]π – .00610 ln[22 ln[2] ]e + .00625
6ln[22 + 2 – 2/2]π – .00610 2 + 2 / (2 × 2 ln[2] )e + .00307
7ln[22 + ln[ ln[22] ] ]π – .000527 (2 + ln[2] ) / (ln[2 + ln[2] ] )e + .000118
8(2 – 2/22) × (2 + ln[2] ) – 2π – .000130 2 + 2 / (ln[2 × 22] – 2/2)e + .0000600
9ln[22 + ln[2 + ln[ ln[22] ] ] ]π – .00000589 2 + (ln[22 – 2] – 2) / (2 ln[2] )e – .0000128
10(2 + (2 + 2 – ln[2] / 2) ln[2] ) ln[2]π + .00000144 (2+ln[2])×(2/2–ln[ln[2+ln[2]]])e + .000000364JD

Best Known Approximations Using 3's and ln's
ComplexityExpression for πValueAuthorExpression for eValueAuthor
13π – .142 3e + .282
2ln[3]π – 2.04 ln[3]e – 1.62
33 + 3 – 3π – .142 3 / ln[3]e + .0124
43 + ln[ ln[3] ]π – .0475 3 – (3 / (3×3) )e – .0516
5(3 + 3) / (3 – ln[3] )π + .0140 3 – 3 ln[ ln[3] ]e – .000425
63 + 3 / (3 × (3+3) + 3)π + .00126 3 × (3/3 – 3 ln[ ln[3] ] )e – .000425
73 + 3/3 – 3 / ln[33]π + .000408 3 / ln[3] ln[3 / ln[3] ]e + .0000282
83 + (3 – 3 / ln[3] ) / (3 – ln[3] )π + .0000314 (3 + 3) / ln[3×3 + 3/33]e + .00000214
9ln[3] × ln[ (3+3) × (3 – 3/33) ]π – .000000844 (3 – 3 ln[ ln[3 / ln[3] ] ] ) / ln[3]e – .0000000430
10ln[3–(33–3)/3/(3–ln[33])]π – .000000538JD3×(3/3–ln[ln[3/ln[3]]])/ln[3]e – .0000000430JD

Best Known Approximations Using 4's and ln's
ComplexityExpression for πValueAuthorExpression for eValueAuthor
14π + .858 4e + 1.28
2ln[4]π – 1.76 ln[4]e – 1.33
34 – 4/4π – .142 ln[4 × 4]e – .0543
4(4 + 4 + 4) / 4π – .142 44 / (4 × 4)e + .0317
5ln[4×4 + 4 + 4]π + .0365 ln[4 ln[44] ]e – .00116
6ln[4 × (4+4) / ln[4] ]π – .00249 ln[4] + ln[ ln[44] ]e – .00116
7ln[4 + 4 + 4 ln[44] ]π – .000170 (4 + 4) / (4 – 4/ln[44] )e + .0000600
84 – ln[4 × (4+4) – 4/4] / 4π – .0000894 4 / ln[ (4 × 4) / (4 – ln[ ln[4] ] ) ]e + .0000691
9ln[4 × 4 ln[4] + ln[4 – ln[4] ] ]π + .0000340 ln[ (4 + (4 + 4/4) ln[4] ) ln[4] ]e – .00000162
10ln[(4+4–ln[4])/ln[ln[ln[44]]]]π + .0000000426JD4/ln[4+(4+ln[4])/(4ln[44])]e – .000000308JD

Best Known Approximations Using 5's and ln's
ComplexityExpression for πValueAuthorExpression for eValueAuthor
15π + 1.86 5e + 2.28
2ln[5]π – 1.53 ln[5]e – 1.11
35 / ln[5]π – .0349 5 / ln[5]e + .388
4ln[5] + ln[5]π + .0773 ln[5 + 5 + 5]e – .0102
5ln[25 – ln[5] ]π + .0107 ln[5×5 – 5 – 5]e – .0102
65 / ln[5 – 5/55 ]π + .000910 ln[5 × 5] – 5 / (5+5)e + .000594
75 / (ln[5] – 5 / (5 × 55) )π + .000579 (5 + 5) / (5 – (5 + ln[5] ) / 5)e + .000504
85 – ln[ ln[555 + 55] ]π + .00000860 5 / ln[5] – ln[5 / (5 – ln[5] ) ]e – .0000493
9ln[5×5 – 5 / (5×5 – 5) – ln[5] ]π – .00000564 5 + (5 – (5 + 5) / (ln[5] – 5/5) ) / 5e + .00000570
10ln[55]–5/(5–ln[ln[5+5]/5])π + .000000450JDln[5+5ln[5+5–5ln[ln[5]]]]e – .000000204JD

Best Known Approximations 6's and ln's
ComplexityExpression for πValueAuthorExpression for eValueAuthor
16π + 2.86 6e + 3.28
2ln[6]π – 1.35 ln[6]e – .927
36 / ln[6]π + .207 ln[6 + 6]e – .233
4ln[6] × ln[6]π + .0688 6 – 6 / ln[6]e – .0669
56 – ln[6 + 6 + 6]π – .0320 ln[6 ln[6 + 6] ]e – .0163
66 – (6 + 6) / ln[66] π – .00579 (6 + 6 / ln[ ln[6] ] ) / 6e – .00360
76 / (6/6 + ln[ ln[6 + 6] ] ) π – .000618 ln[ (6 + 6) / (ln[6] – 6/6) ]e + .000122
86 / ln[6] – ln[6 + 6] / (6 + 6) π – .00000445 ln[6 ln[6 + 6 + 6/(6+6) ] ]e + .00000723
9ln[(6 – ln[6 + 6] ) × (6 + ln[ ln[6] ] ) ] π – .00000591 6 / ln[ (666 – 66) / 66]e + .00000214
10ln[6×6]–(ln[6+6]+6/(6×6))/6π – .00000260JD(6+6–(6+6)×(ln[ln[66]/6]))/6 e + .000000450JD

Best Known Approximations Using 7's and ln's
ComplexityExpression for πValueAuthorExpression for eValueAuthor
17π + 3.86 7e + 4.28
2ln[7]π – 1.20 ln[7]e – .772
37 / ln[7]π + .456 ln[7 + 7]e – .0792
47 – ln[7 × 7]π – .0334 7 – ln[77]e – .0621
57 – ln[7] – ln[7] π – .0334 ln[7 + 7 + 7/7]e – .0102
6(7 + 7 + 7 + 7/7) / 7 π + .00126 (7 + 7 + 7 – ln[7] ) / 7e + .00373
7ln[7 ln[ (7+7) ln[7] ] ] π – .000310 (7+7) ln[ ln[7 + 7] ] / 7e – .000528
87 – ln[7×7 – 7 / ln[77] ] π + .0000276 7 × (ln[ (7 + 7 + 7) × 77] – 7)e + .0000132
9ln[7 + 7 + (7+7) / ln[7] + ln[7] ] π – .00000889 7 / (7 – ln[77 + 7 – 7/(7+7) ] )e + .00000287
107–ln[7×7–7/(7–7/ln[7+7])]π – .00000164JDln[7]–(ln[7–ln[7+7/7]]–7)/7e + .00000287JD

Best Known Approximations Using 8's and ln's
ComplexityExpression for πValueAuthorExpression for eValueAuthor
18π + 4.86 8e + 5.28
2ln[8]π – 1.06 ln[8]e – .639
3ln[8 + 8]π – .369 ln[8 + 8]e + .0543
4ln[8 + 8 + 8] π + .0364 ln[8 ln[8] ]e + .0933
58 – ln[8 × (8+8) ] π + .00638 ln[8 + 8 – 8/8]e – .0102
6ln[8 × 8 / ln[8 + 8] ] π – .00249 8 / ln[88/8 + 8]e – .00130
7ln[88] / ln[ ln[8 × 8] ] π – .000145 ln[88/8 + ln[8 × 8] ]e + .000305
8ln[8 × (88 – ln[8] ) ] / ln[8] π + .0000512 ln[8 + ln[ (8+8)×(88–8) ] ]e + .0000233
98 – (8 + 8) / ln [(8×8 – 8) / ln[8] ] π – .0000118 (8 – ln[8] ) / (ln[8 + 8 + 8] – 8/8)e – .00000261
10ln[8+8+8–8/(ln[8]×ln[88])]π + .00000214JDln[8]+8/(8+8+8/8–ln[88])e + .00000146JD

Best Known Approximations Using 9's and ln's
ComplexityExpression for πValueAuthorExpression for eValueAuthor
19π + 5.86 9e + 6.28
2ln[9]π – .944 ln[9]e – .521
3ln[9 + 9]π – .251 ln[9 + 9]e + .172
49 / ln[9 + 9]π – .0278 ln[9 ln[9] ]e + .266
59 / ln[9] – 9/9 π – .0455 9 / ln[9 + 9 + 9]e + .0124
6ln[999]/ ln[9] π + .00181 ln[9 + 9 – ln[9 + 9] ]e – .00295
79 – 9 ln[ ln[9 – ln[9] ] ] π + .00000182 ln[9] + ln[99 + 9] / 9e – .000820
89 ×(9/9 ln[ ln[9 – ln[9] ] ] ) π + .00000182 ln[9] / ln[ (9 + 9 + ln[9] ) / 9]e – .0000231
99 – 9 ln[ ln[9 – ln[9] ] ] × 9/9 π + .00000182 ln[9 + 9 – ln[9 + 9 – ln[ ln[9] ] ] ]e + .00000555
109×(9–9) + 9 – 9 ln[ ln[9 – ln[9] ] ]π + .00000182JD(9+99)/ln[9+9/99]/(9+9)e + .00000214JD

Best Known Approximations Using Any Digits and ln's.
ComplexityExpression for πValueAuthorExpression for eValueAuthor
13π – .142 3e + .282
23 + 0π – .142 8 / 3e – .0516
322 / 7π + .00126 19 / 7e – .00400
43 + 9 / 64π – .000968 87 / 32e – .000468
53 × (5 ln[5] – 7)π – .0000240 4 – (7 ln[3] / 6)e – .00000383
63 + 7 / (45 ln[3] )π + .000000115JD2 ln[894] / 5e + .000000482HS

Best Known Approximations Using 1's and Unlimited ln's
DigitsExpression for πValueAuthorExpression for eValueAuthor
11π – 2.14 1e – 1.72
2ln[11]π – .744 ln[11]e – .320
31 + 1 + 1π – .142 ln[11 + 1]e – .233
4ln[11] + ln[1 + 1]π – .0506 ln[1] – 1 / ln[ ln[2] ]e + .0101
51 + 1 + 1 / ln[ ln[11] ]π + .00180 ln[11] × (1 – ln[ ln[ ln[11] ] ] )e + .000928
61+ln[11–ln[1+11]]π + .000248HS1+1–ln[ln[1+ln[1+ln[ln[11]]]]]e – .00000601HS
71–ln[ln[ln[1+1]–ln[ln[ln[1+1+1]]]]–1]π + .0000101JD(ln[ln[(ln[1]–1)/(ln[ln[1+1]])]]–1)/ln[ln[1+1]]e – .0000000233JD
8ln[1]–ln[1–(ln[1+1])]/ln[1–ln[1+ln[ln[1+1]]]]π – .00000102JD1×(ln[ln[(ln[1]–1)/(ln[ln[1+1]])]]–1)/ln[ln[1+1]]e – .0000000233JD

Best Known Approximations Using 2's and Unlimited ln's
DigitsExpression for πValueAuthorExpression for eValueAuthor
12π – 1.14 2e – .718
2ln[22]π – .0506 2 + ln[2]e – .0251
32 + ln[ ln[22] ]π – .0131 ln[22] + ln [ ln[2] ]e + .00625
4ln[22+ln[ln[22]]]π – .000527HS2/ln[ln[2]]/ln[ln[ln[2]]×ln[ln[2]]]e + .0000188 JD
5ln[22+ln[2+ln[ln[22]]]]π – .00000589HS(ln[2]×ln[2])–(2×ln[ln[ln[2+2]]])e – .000000404HS
52+ln[2/ln[ln[ln[222]]]–ln[2]]π + .000000524JD2+ln[2]–(ln[2]+2)×ln[ln[ln[2]+2]]e + .000000364JD

Best Known Approximations Using 3's and Unlimited ln's
DigitsExpression for πValueAuthorExpression for eValueAuthor
13π – .142 3e + .281
23 + ln[ ln[3] ]π – .0475 3 / ln[3]e + .0124
3ln[ ln[3] + ln[3] ] – ln[ ln[ ln[3] ] ]π + .00955 3 – 3 ln[ ln[3] ]e – .000425
4ln[3/(ln[ln[3]–ln[ln[3]]/ln[ln[ln[3]]]])]π + .000163HS(3/ln[3/ln[3]])/ln[3]e + .0000282HS
5ln[3]–ln[ln[ln[3/ln[ln[3]]]]]/ln[ln[ln[ln[3]/ln[ln[3]]]]]π + .00000203HS(3–3×ln[ln[3/ln[3]]])/ln[3]e – .0000000430HS
6ln[3]+ln[ln[ln[ln[3–ln[ln[ln[3]]]]]]+ln[ln[3×3]]/ln[ln[3]]] π + .00000000534JD(3–3ln[ln[3]])/ln[3–3ln[ln[3]]]e + .0000000333JD

Best Known Approximations Using 4's and Unlimited ln's
DigitsExpression for πValueAuthorExpression for eValueAuthor
14π + .858 4e + 1.28
24 / ln[4]π – .256 ln[4] + ln[4]e + .0543
3ln[4] / ln [ ln[4] ] + ln[ ln[ ln[4] ] ]π – .0163 ln[4] + ln[ ln[44] ]e – .00116
44+ln[4–ln[4]]/ln[ln[ln[4]]]π – .000255HSln[ln[ln[ln[4]]]/ln[ln[ln[(ln[ln[4]]+4/ln[ln[4]])]]]]e – .00000745HS
5ln[(4+4–ln[4])/ln[ln[ln[44]]]]π + .00000000426HS4–(4–ln[4])×ln[ln[4–ln[ln[ln[4]]]]]e + .0000000300HS
64ln[4/ln[4]–(ln[ln[4]]–ln[ln[ln[4]]]×ln[ln[4]])]π + .000000105JDln[4ln[44]–ln[ln[ln[4×4]]]/ln[ln[ln[4]]]]e + .000000188JD

Best Known Approximations Using 5's and Unlimited ln's
DigitsExpression for πValueAuthorExpression for eValueAuthor
1ln[5]π – 1.53 ln[5]e – 1.11
25 / ln[5]π – .0349 ln[5] × ln[5]e – .128
35 + ln[ ln[ ln[ ln[5] + ln[5] ] ] ]π + .00164 ln[5 / ln[ ln[ ln[55] ] ] ]e + .00604
4ln[5×5+ln[ln[ln[ln[5×5]]]]]]π + .000110HSln[ln[ln[5]]]/ln[ln[ln[5×ln[5+ln[ln[5]]]]]]e – .000289HS
5ln[ln[ln[5–ln[5]]]]×ln[ln[ln[ln[ln[ln[5]]/5]/ln[ln[ln[5]]]]]]π – .00000104HSln[ln[5×55]]–ln[ln[ln[5+ln[ln[ln[5]]]]]]e – .00000475HS
6(ln[ln[5]–(ln[5]/ln[ln[ln[5]]])])×(ln[ln[ln[5]]]+5/ln[5])π + .0000000218JDln[ln[5]]–ln[5]/ln[ln[ln[5–(ln[ln[ln[5]+ln[5ln[ln[5]]]]])]]]e – .000000122JD

Best Known Approximations Using 6's and Unlimited ln's
DigitsExpression for πValueAuthorExpression for eValueAuthor
1ln[6]π – 1.35 ln[6]e – .927
2ln[6] × ln[6]π + .0688 ln[6 + 6]e – .233
36 ln[ ln[6 – ln[ ln[6] ] ] ]π + .00502 ln[6]–ln[ln[ln[6]]]/ln[ln[6]]e – .00192HS
4ln[6]×ln[6–ln[ln[6×ln[ln[6]]]]]π + .000275HSln[ln[6]+(6+ln[6])/ln[ln[6]]]e – .000139HS
56+ln[ln[ln[6]]]–ln[6×ln[6]–ln[ln[6]]]π – .00000351HSln[ln[66]/ln[ln[ln[6+6×6]]]]e + .00000339HS
66/ln[6–(ln[ln[ln[ln[6]]–(ln[ln[ln[ln[ln[ln[6]]+66]]]])]])]π – .0000000779JDln[ln[ln[ln[ln[ln[6]]]×ln[ln[ln[6]]]]×ln[ln[ln[6]]]/6]/ln[ln[ln[ln[6]]+ln[6]]]] e + .000000000530JD

Best Known Approximations Using 7's and Unlimited ln's
DigitsExpression for πValueAuthorExpression for eValueAuthor
1ln[7]π – 1.20 ln[7]e – .772
2ln[7] / ln[ ln[7] ]π – .219 ln[7 + 7]e – .0792
3ln[ ln[7] / 7] / ln[ ln[ ln[7] ] ]π + .00481 7 ln[ ln[7] ] – ln[7]e – .00408
4ln[7]×ln[ln[7]]+ln[7–ln[ln[7]]]π – .000168HSln[ln[ln[ln[7+ln[ln[ln[7]]]]]]/ln[ln[ln[7+7]]]]e – .0000841HS
5ln[7]×ln[7–ln[7]/ln[ln[7]]]–ln[ln[ln[7]]]π – .00000125HSln[7+ln[7]]–ln[ln[ln[7]]]×ln[ln[7]]×ln[7]e – .00000367JD
6ln[ln[7]]/ln[ln[ln[ln[ln[ln[7–ln[7ln[7]]]]]/ln[ln[ln[7+7]]]]]]π – .0000000307JDln[7]+ln[ln[7+ln[ln[7]]–ln[ln[ln[ln[7×(7+ln[7])]]]]]]e + .00000000709JD

Best Known Approximations Using 8's and Unlimited ln's
DigitsExpression for πValueAuthorExpression for eValueAuthor
1ln[8]π – 1.06 ln[8]e – .639
2ln[8] / ln[ ln[8] ]π – .301 ln[8 + 8]e + .0543
3ln[(ln[ln[8]]–8)/ln[ln[ln[8]]]]π + .00714HSln[ ln[8] ] – ln[8 – ln[ ln[8] ] ]e – .00271
4ln[ln[8–ln[ln[ln[8]]]]/(ln[ln[8]]/8)]π + .00000948HSln[8]/ln[ln[8+ln[ln[8–ln[8]]] ]]e + .0000385HS
5ln[8ln[8ln[8]]–ln[ln[ln[8]]]×ln[8]] π + .00000370HSln[8+ln[ln[ln[ln[8×8]]+ln[ln[ln[8]]]]]/ln[ln[ln[8]]]]e – .00000365HS
6ln[8–(8+8)/ln[ln[ln[ln[8]+ln[ln[ln[ln[8]]]+8]]]]] π + .00000000155JDln[8ln[ln[ln[ln[8]]]+8+ln[ln[ln[8–8/ln[8]]]]]]e – .0000000234JD

Best Known Approximations Using 9's and Unlimited ln's
DigitsExpression for πValueAuthorExpression for eValueAuthor
1ln[9]π – .944 ln[9]e – .521
2ln[9] + ln[ ln[9] ]π – .157 ln[9] / ln[ ln[9] ]e + .0729
3ln[9 / ln[ ln[ ln[9] + ln[9] ] ] ]π – .00857 ln[9]–ln[ln[ln[9]]]×ln[9]e + .00511JD
49–9×ln[ln[9–ln[9]]]π + .00000182HSln[ln[ln[9]]] + ln[9+ln[ln[ln[9]]]] + ln[ln[9]]e – .0000879HS
5ln[ln[9]/ln[ln[ln[9]+ln[ln[ln[9+ln[ln[9]]]]]/ln[ln[ln[9]]]]]]π + .00000106HSln[9]/ln[ln[9]+ln[ln[ln[9+9]]]×ln[ln[9]]]e – .00000135HS
6ln[9ln[9–ln[9]–ln[ln[ln[ln[ln[ln[ln[ln[9]]–ln[ln[ln[9]]]]]/ln[ln[ln[9]]]]]]]]]π + .0000000478JDln[9–ln[ln[ln[9]/ln[ln[ln[ln[ln[9]]]+ln[9]×ln[9]]]]]]/ln[ln[9]]e – .0000000466JD

Best Known Approximations Using Any Digits and Unlimited ln's
DigitsExpression for πValueAuthorExpression for eValueAuthor
13π – .142JDln[9]e – .282JD
2ln[23]π – .00610HSln[5/ln[ln[4]]]e + .0101JD
3ln[ln[2]–7/ln[ln[ln[8]]]]π – .00000287JDln[ln[ln[8]+ln[8]]]-ln[ln[ln[3]]]e + .0000148JD
4ln[(8–ln[4])/ln[ln[ln[44]]]]π + .00000000426HS1–ln[ln[ln[ln[7]+ln[4–ln[ln[3]]]]]]e – .0000000122JD

Best Known Approximations Using 1's and √'s
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 1 π – 2.142HS 1 e – 1.718HS
2 1+1 π – 1.142HS 1+1 e – .718HS
3 1+1+1 π – .142HS 1+1+1 e + .282HS
4 √[11–1] π + .0207HS 1+1+√1 e + .282HS
5 √[11–√1] π + .0207HS 1+√[1+1+1] e + .0138HS
6 √[11–√√1] π + .0207HS 1+1+√[1/(1+1)] e – .0112HS
7 √[1+1+1]+√[1+1] π + .00467HS (11–1–1)/√11 e – .00468HS
8 √[1+1]×(11–1)–11 π + .000543HS √[11–√[1+1+11]] e + .000992HS
9 1+1+1–√[1+1]/(1–11) π – .000171HS (1+11)/(1+1+1+√[1+1]) e + .000209HS
10 1+1+√√[1+1–1/√11] π + .0000118JD 1+1+11/(√11+1+11) e – .000108JD
11 1+11×11/(1+111/(1+1)) π + .000000267JD √√[1+1]×(1+√111)-11 e – .000000309JD
12 1×1+11×11/(1+111/(1+1)) π + .000000267JD 1×√√[1+1]×(1+√111)-11 e – .000000309JD

Best Known Approximations Using 2's and √'s
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 2 π – 1.142HS 2 e – .0718HS
2 2+2 π + .858HS 2+2 e + 1.282HS
3 1+2/2 π – .142HS 2√2 e + .11HS
4 2+√√2 π + .0476HS 2+√2/2 e – .0112HS
5 √[(22–2)/2] π + .0207HS 2+√[2/(2+2)] e – .0112HS
6 2+√[2–√2/2] π – .00454HS 2√√[2+√2] e + .000364HS
7 2/√[(2–√√2)/2] π – .000433HS (2+2)/√√√22 e –. 000234HS
8 2+√√[2–√[2/22]] π + .0000118HS 2+(√[2+√2/22])/2 e +. 0000987HS
9 2–2/(2×22–2)+√√2 π – .00000459HS √[(22/(2+2))/(2–√2)–2] e + .00000574HS
10 2+√[2–√√[√2/(2+2×2)]] π – .00000271JD √2+√√√[2+2/22] e – .00000280JD
11 (2+22×(22/(2+222/2)))/2 π + .000000267JD (2+√2)/2+√√[(2+2/22)/2] e – .000000129JD

Best Known Approximations Using 3's and √'s
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 3 π – .142HS 3 e + .282HS
2 √3 π – 1.41HS √3 e – .986HS
3 √[3×3] π – .142HS √[3+3] e – .269HS
4 3+3/33 π – .0507HS 3/3+√3 e +. 0138HS
5 √√[3×33] π + .0127HS 3–(√3/(3+3)) e – .00696HS
6 3–((√3–3)/(3×3)) π – .000709HS √[((3+3)×√3)–3] e + .000598HS
7 3+3/(3+√333) π – .000405HS (3+(√3/3))/√√3 e – .0000834HS
8 √3+√√[3√√3] π + .0000728HS √[33–√[3/33]]–3 e – .0000226HS
9 √[3×3+3/(√[3+3]+3/3)] π + .0000142HS 3–(3×3)/33–3/333 e – .0000181HS
10 3-3/(3–√[3×((3+3)×33–3)]) π + .00000512JD 3–√[(3+3/3/3)/(3×3+33)] e + .0000000866JD

Best Known Approximations Using 4's and √'s
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 4 π + .858HS 4 e + 1.282HS
2 √4 π – 1.142HS √4 e – .718HS
3 4 – 4/4 π – .142HS √[4+4] e + .11HS
4 4–√[4/4] π – .142HS 44/(4×4) e +. 0317HS
5 √[(44–4)/4] π + .0207HS 4–((4+4/4)/4) e + .0317HS
6 44/(4×4–√4) π – .00126HS 4–√[√44/4] e – .00604HS
7 4–√[4/(4+√√4)] π – .00113HS (44–√4/4)/(4×4) e + .000468HS
8 4–(√√44/(4–4/4) π – .0000958HS √[(44–4)/(4+√√4)] e – .000201HS
9 4/(4/(4×4)–4/(4/44–4)) π – .0000401HS 4–√[4/(4–√√[4+√4])] e + .0000132HS
10 √[4√[4+44/√[444]]] π + .0000102JD (√[4+4×44]–4)/√[4×4–4] e + .000000979JD
11 4/(4–44/(4×4+(4+√4)/44)) π + .000000267JD √[√√4+4+√(4+4/(4–44))] e – .000000142JD

Best Known Approximations Using 5's and √'s
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 5 π + 1.858HS 5 e + 2.282HS
2 √5 π – .906HS √5 e – .482HS
3 √[5+5] π + .0207HS 5–√5 e + .0457HS
4 5/5+√5 π + .0945HS √√55 e + .00499HS
5 √[(55–5)/5] π + .0207HS 5/(5–√[5+5]) e + .00248HS
6 5–5/(√[5+√5]) π – .000333HS 5–√[5+5/(5×5)] e + .00137HS
7 5–√[5×5/(5+√5)] π – .000333HS √√5+√√√5 e – .0000885HS
8 5/√√[√55–5/5] π + .00000495HS 5/√√[(55+√5)/5] e + .0000036HS
9 5/√√[√55–√[5/5]] π + .00000495HS 5√[5/√[5×(55+√5)]] e + .0000036HS
10 5–√√[5√[5+5/(5+√5)]] π + .00000000606JD √[55–5]–5+(√5+5/5)/5 e – .000000421JD

Best Known Approximations Using 6's and √'s
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 6 π + 2.858HS 6 e + 3.282HS
2 √6 π – .692HS √6 e – .269HS
3 √[6+6] π + .323HS 6/√6 e – .269HS
4 (6+6+6)/6 π – .142HS √[6+6/6] e –. 0725HS
5 6–√√66 π + .00814HS 6√6–6–6 e – .0213HS
6 6–√6/6–√6 π – .000669HS √[6+(6+√6)/6] e + .00353HS
7 (6×6–6)/(6+6–√6) π – .000399HS √[(66–6)/√66] e – .000656HS
8 6/6+√[6–√[(6+6)/6]] π – .000148HS 6–(√[666–6]–6)/6 e – .000026HS
9 √[6√[(6+6)/(6–√√6)]] π – .00000196HS √[6+√[66+√[6+6]]/6] e + .0000054HS
9 √[6√[6+6]/√[6–√√6]] π – .00000196JD 6–√√[66×(6–√[6+6+6])] e + .00000118JD

Best Known Approximations Using 7's and √'s
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 7 π + 3.858HS 7 e + 4.282HS
2 √7 π – .496HS √7 e – .0725HS
3 7/√7 π – .496HS 7/√7 e – .0725HS
4 √[7+√7] π – .0358HS √√[7×7] e – .0725HS
5 7/(7+7)+√7 π + .00416HS √[7+√7/7] e – .00204HS
6 (77+77)/7×7 π + .00126HS 7/((7+7)×7)+√7 e – .0011HS
7 √√[7+7+7]+7/7 π – .000898HS √[7+7/(77/7+7)] e – .0000308HS
8 √[7+7/7+7/√[7+7]] π + .000195HS √[7+7/((77+7×7)/7)] e – .0000308HS
9 √7+√[√√√77/7] π + .0000159HS √7+((7×7)/77)/√77 e – .0000102HS
10 √[7–7/(7+7)+√[7+7–√7]] π + .000000351JD 7/(7/7/7+√√[7×7–7–7]) e – .000000352JD

Best Known Approximations Using 8's and √'s
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 8 π + 4.858HS 8 e + 5.282HS
2 √8 π – .313HS √8 e + .11HS
3 8/√8 π – .313HS 8/√8 e + .11HS
4 √√88 π – .0788HS √[8–8/8] e – .0725HS
5 √√[88+8] π – .0114HS √8–8/(8×8) e – .0149HS
6 √[√8–8/8+8] π – .00656HS √8–8/((8×8)+8) e – .000966HS
7 √√[88+√88] π – .000228HS √[(88–8)/(8+√8)] e – .000201HS
8 √[(8–√√8)×√8–8] π + .000157HS 88/(8+√√8/8)–8 e +. 0000614HS
9 8/(√8–√8/8)–8/88 π – .0000136HS 8/√[8+√[(8–8/8)/(8+8)]] e +. 00000319HS
10 √[88/√[88–8–8/(8+8)]] π – .00000231JD √[8×8/(8+√[(8–8/8)/(8+8)])] e +. 000000400JD

Best Known Approximations Using 9's and √'s
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 9 π + 5.858HS 9 e + 6.282HS
2 √9 π – .142HS √9 e + .282HS
3 9/√9 π – .142HS 9/√9 e + .282HS
4 √√99 π + .0127HS √[9–9/9] e +. 11HS
5 √[(99–9)/9] π + .0207HS 9√[9/99] e – .00468HS
6 √[9–(9/9–9)/9] π + .00307HS 9√9/√99 e – .00468HS
7 √[9+(9/√[9+99])] π – .00057HS √[√[9+99]–√9] e + .00598HS
8 √[9+9/(9+(9+√9)/9)] π + .000217HS √[9–(9+99/(9+9))/9] e – .0000308HS
9 9√[9+9]/(9+√√99) π – .0000186HS √[9–9/9–99/(9×(9+9))] e – .0000308HS
10 √[9+9/(9+√√√[99/9])] π + .000000400JD √[9+9–√9]–√[9+99]/9 e + .000000979JD

Best Known Approximations Using Any Digits and √'s.
ComplexityExpression for πValueAuthorExpression for eValueAuthor
13π – .142HS3e + .282HS
2√9π – .142HS8 / 3e – .0516HS
322 / 7π + .00126HS19 / 7e – .00400HS
4√51 – 4π – .000164HS87 / 32e – .000468HS
5(9 + √45) / 5π + .0000481HS3 – √[5 / 63]e + .0000000866HS
69 / (2 + 9 / 52) – 1π + .000000267HS9 – 6 – √[5 / 63]e + .0000000866HS
7√√[97 + 9 / 22]π – .00000000101HS2 + √[7 / (8 + √31)]e – .00000000945HS

Best Known Approximations Using 1's and Decimal Points
ComplexityExpression for πValueAuthorExpression for eValueAuthor
1 1 π – 2.142HS 1 e – 1.718HS
2 1+1 π – 1.142HS 1+1 e – .718HS
3 1+1+1 π – .142HS 1+1+1 e + .282HS
4 1×(1+1+1) π – .142HS 1×(1+1+1) e + .282HS
5 1+1+1.1 π – .0416HS 1–.1+1+1 e + .182HS
6 1+1–1/(.1–1) π – .0305HS (1+1+1)/1.1 e + .00899HS
7 1+1+1/(1–.11) π – .0180HS (1+1+1)×(1/1.1) e + .00899HS
8 11/(11/(1+1)–1–1) π + .00126HS (1+1)/1.1+1–.1 e – .000100HS
9 11×(1/(11/(1+1)–1–1)) π + .00126HS ((1+1)/.1)/11+1–.1 e – .000100HS
10 (1–.1+1+1)×(1/(1+11)+1) π + .0000740HS (1+1+1)/1.1–1/111 e – .0000181HS
11 1+11×11/(1+111/(1+1)) π + .000000267JD (1+1)/(1.1)+1/(1.111) e – .0000100JD
12 1×1+11×11/(1+111/(1+1)) π + .000000267JD (1+1+1/(1–.1))/1.1–.11 e + .00000100JD
13 1×1×1+11×11/(1+111/(1+1)) π + .000000267JD 1+(1+1)/1.1–(1–.1)×.111 e – .0000000103JD

Best Known Approximations Using 2's and Decimal Points
ComplexityExpression for πValueAuthorExpression for eValueAuthor
1 2 π – 1.142HS 2 e – .718HS
2 2+2 π + .858HS 2+2 e + 1.282HS
3 2+2/2 π – .142HS 2+2/2 e + .282HS
4 (2+2+2)/2 π – .142HS 2+2/(2+2) e – .218HS
5 2–2/(.2–2) π – .0305HS 22/(2×(2+2)) e + .0317HS
6 2+2/(2–.22) π – .0180HS (2+2+2)/2.2 e + .00899HS
7 2–2/(2/(2×(2+2))–2) π + .00126HS 2/(2+2)+2.22 e + .00172HS
8 22×(2/(22–2×(2+2))) π + .00126HS 2+2+2/(2×.22–2) e – .000333HS
9 2+(2/(2+22)+2.2)/2 π + .0000740HS 2–(.2–2)/2–(2+2)/22 e – .000100HS
10 (2.2+2+2+2/(2+22))/2 π + .0000740HS (2+2+2)/2.2–2/222 e – .0000181HS
11 (2+22×22/(2+222/2))/2 π + .000000267JD 2+(22–2)/(22–.22)-.2 e – .00000818JD
12 2+22×22/(2×2+222)–2/2 π + .000000267JD 2–2/(2×2×2–2/2.22)+2/2 e – .00000772JD

Best Known Approximations Using 3's and Decimal Points
ComplexityExpression for πValueAuthorExpression for eValueAuthor
1 3 π – .142HS 3 e + .282HS
2 3/3 π – 2.142HS 3/3 e – 1.718HS
3 3+3–3 π – .142HS 3–.3 e – .0183HS
4 3+.3/3 π – .0416HS .3×3×3 e – .0183HS
5 3+3/(3×(3+3)) π + .0251HS 3×3/3.3 e + .00899HS
6 3+3/(3+3×(3+3)) π + .00126HS 3×(3/3–3/33) e + .00899HS
7 (33/(3+3/(3+3)))/3 π + .00126HS 3–(.3+3×3)/33 e – .000100HS
8 3+3/(3+(3+3)/.33) π + .0000382HS 3–(3+3+3.3)/33 e – .000100HS
9 3+3/(3+(3×3–3×3)/.33) π + .0000382HS 3×3/3.3–3/333 e – .0000181HS
10 3–33/(333×(.3–3/3)) π – .0000225HS 3×(3/(3+3/333)–3/33) e + .00000886HS
11 (3+33–3/(3+3))/(.3+33/3) π + .000000267JD 3–.3×.333–(3+3)/33 e – .0000000103JD

Best Known Approximations Using 4's and Decimal Points
ComplexityExpression for πValueAuthorExpression for eValueAuthor
1 4 π + .858HS 4 e + 1.282HS
2 4/4 π – 2.142HS 4/4 e – 1.718HS
3 4 – 4/4 π – .142HS 4 – 4/4 e + .282HS
4 .4×(4+4) π + .0584HS 44/(4×4) e + .0317HS
5 4+(.4–4)/4 π – .0416HS 4–(4+4/4)/4 e + .0317HS
6 44/(4+4/.4) π + .00126HS (44–.4)/(4×4) e + .00672HS
7 4/(4/4–.44)–4 π + .00126HS (44–4/(4+4))/(4×4) e + .000468HS
8 .4/(4+.4×4)×44 π + .00126HS 4+(4+4/4)/(.4/4–4) e – .000333HS
9 4/(4/(4×4)–4/(4/44–4)) π – .0000401HS ((4+44)/.4–.4)/44 e – .000100HS
10 4–4/(4.4/4+4–.44) π + .0000382HS 4/(4+444/(.4–4×44)) e – .0000156HS
11 (4+44/(4+(4+4/4)/4.4))/4 π + .000000267JD 4+(4+4/4)/(4/(44.4–4)–4) e – .00000772JD

Best Known Approximations Using 5's and Decimal Points
ComplexityExpression for πValueAuthorExpression for eValueAuthor
1 5 π + 1.858HS 5 e + 2.282HS
2 5/5 π – 2.142HS 5/5 e – 1.718HS
3 .5×5 π – .642HS .5×5 e – .218HS
4 (5+5+5)/5 π – .142HS .55×5 e + .0317HS
5 (55/5+5)/5 π + .0584HS .5×5.5 e + .0317HS
6 5–55/(5×5+5) π + .0251HS (5+5+5)/5.5 e + .00899HS
7 55/(5+.5×5×5) π + .00126HS .5–(5+5)/(.5–5) e + .00394HS
8 5/(.5+(5+55)/55) π + .00126HS (5/.55–.5+5)/5 e – .000100HS
9 5–(5/(5×5)+5/.55)/5 π – .000226HS .5×5+((5+55)/5)/55) e – .000100HS
10 (5+5.5)/5–5/(5/(5×5)–5) π + .0000740HS 5/((55+(5+5)/55)/(5+5×5)) e + .00000483HS
11 (5+55/(5+(5+5×.5)/55))/5 π + .000000267JD 5×(5/(5+5/555)–5×5/55) e – .00000191JD
12 5–(5+5.5)/(5.55+.5/5) π + .000000267JD 5×(5/(5+5/555)–5/(5+5+5/5)) e – .00000191JD

Best Known Approximations Using 6's and Decimal Points
ComplexityExpression for πValueAuthorExpression for eValueAuthor
1 6 π + 2.858HS 6 e + 3.282HS
2 6/6 π – 2.142HS 6/6 e – 1.718HS
3 .6×6 π + .458HS (6+6)/6 e – .718HS
4 (6+6+6)/6 π – .142HS (6+6+6)/6 e + .282HS
5 6/.66–6 π – .0507HS (6+6/.6)/6 e – .0516HS
6 6×(6/(6–.6+6)) π + .0163HS 6–6×6/66×6 e + .00899HS
7 6/(6/6+6/6.6) π + .00126HS (6+6)/(6+6/6)+6/6 e – .00400HS
8 6×(6/(6–.66)–.6) π – .0000196HS (6×(6×6–6)–.6)/66 e – .000100HS
9 6/((6–.66)/6)–.6×6 π – .0000196HS 6–6×((.6/6+6×6)/66) e – .000100HS
10 6×(6+6/(6–.66)–6.6) π – .0000196HS (6+6/(.6–(6+6)/666))/6 e – .0000156HS
11 6/6/(.6–(6+6)/(6.6+6×6)) π + .000000267JD 6×(6/(6+6/666)–6×6/66) e – .00000460JD

Best Known Approximations Using 7's and Decimal Points
ComplexityExpression for πValueAuthorExpression for eValueAuthor
1 7 π + 3.858HS 7 e + 4.282HS
2 7/7 π – 2.142HS 7/7 e – 1.718HS
3 (7+7)/7 π – 1.142HS (7+7)/7 e – .718HS
4 7/.7–7 π – .142HS 7/.7–7 e + .282HS
5 7+7–77/7 π – .142HS .7+(7+7)/7 e – .0183HS
6 (77+77)/(7×7) π + .00126HS 7×(7/(7+77/7)) e + .00394HS
7 (7+.7/7)/.7–7 π + .00126HS 7–77/(7+77/7) e + .00394HS
8 7+7–77/(7+7/77) π – .000567HS .777/(((7+7)/7)/7) e + .00122HS
9 (7+77/777)/.7–7 π – .0000225HS (7+7)/7.7–(.7–7)/7 e – .000100HS
10 ((77+77)/7–7/777)/7 π – .0000225HS 7/((7+(77+(7+7)/77)/7)/7) e + .0000179HS
11 7–(7×7–.77)/(7+77/(7+7)) π + .00000735JD (7+7)/(7/7+7×.77×.77) e + .00000642JD

Best Known Approximations Using 8's and Decimal Points
ComplexityExpression for πValueAuthorExpression for eValueAuthor
1 8 π + 4.858HS 8 e + 5.282HS
2 8/8 π – 2.142HS 8/8 e – 1.718HS
3 (8+8)/8 π – 1.142HS (8+8)/8 e – .718HS
4 88/8–8 π – .142HS 88/8–8 e + .282HS
5 (88–8×8)/8 π – .142HS 8/(88/8–8) e – .0516HS
6 (88+8/8)/8–8 π – .0166HS (8+8+8)/8.8 e + .00899HS
7 (8+8)/8–8/(8/8–8) π + .00126HS (8+88/(.8×8))/8 e + .000468HS
8 (88+(8+8/8)/8)/8–8 π – .000968HS (.8+88/(8×8))/.8 e + .000468HS
9 (88+8/(8×.88))/8–8 π + .000453HS (8+8)/8–(.8–8×8)/88 e – .000100HS
10 (.8+88+8/(8+8+8))/8–8 π + .0000740HS (8+8+8)/8.8–8/888 e – .0000181HS
11 88/8–8/(8–8×8–8/(8+8))–8 π + .000000267JD (8+8)/8.8+8/8.888 e – .0000100JD

Best Known Approximations Using 9's and Decimal Points
ComplexityExpression for πValueAuthorExpression for eValueAuthor
1 9 π + 5.858HS 9 e + 6.282HS
2 9/9 π – 2.142HS 9/9 e – 1.718HS
3 (9+9)/9 π – 1.142HS (9+9)/9 e – .718HS
4 (9+9+9)/9 π – .142HS (9+9+9)/9 e + .282HS
5 (9+99)/9–9 π – .142HS .9+(9+9)/9 e + .182JD
6 ((9+9)+9/.9)/9 π – .0305HS (9+9+9)/9.9 e + .00899HS
7 .9×(9–99/(9+9)) π + .00841HS .9+(9+9)/9.9 e – .000100HS
8 (99/(9–99/(9+9)))/9 π + .00126HS .9+(99+9×9)/99 e – .000100HS
9 (9+9/(9×9))/(.9+(9+9)/9) π + .000170HS 9/.9–9×(.9–9/99) e – .000100HS
10 99/(9–99/(9+9+99))–9 π – .0000832HS (9+9+9)/9.9–9/999 e – .0000181HS
11 (9+99)/(9–(9–9/(9+9))/9/9)–9 π – .0000243JD .9+.99+(9×9+9/9)/99 e + .00000100JD

Best Known Approximations Using Any Digits and Decimal Points
ComplexityExpression for πValueAuthorExpression for eValueAuthor
1 3 π – .142JD 3 e + .282JD
2 3+0 π – .142JD 8/3 e – .0516JD
3 22/7 π + .00126JD 19/7 e – .00400JD
4 3+9/64 π – .000968JD 3–9/32 e + .000468JD
5 3+.85/6 π + .0000740HS 3–2/7.1 e + .0000280JD
6 5/(1+6/(9+8/7)) π + .000000267JD 2+5/(7–3/77) e + .00000175HS
7 6/(5/6+8/(9+4))–1 π + .000000267JD 2/77+35/13 e – .000000110HS

Best Known Approximations Using 1's and Unlimited Decimal Points
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 1 π – 2.142JD 1 e – 1.718JD
2 1+1 π – 1.142JD 1+1 e – .718JD
3 1+1+1 π – .142JD 1+1+1 e + .282JD
4 1+1+1.1 π – .0416JD 1+1+1–.1 e + .182JD
5 1+1+1/(1–.1) π – .0305JD (1+1+1)/1.1 e + .00899JD
6 1/(.1+.1+.1)–.1–.1 π – .00826JD 1+(.1+.1)/.11–.1 e – .000100JD
7 11/((1–.1)/(.1+.1)–1) π – .00126JD 1×1+(.1+.1)/.11–.1 e – .000100JD
8 (.1+(.1+.1+.1)/.11)/(1–.1) π – .000179JD (.1+.1+.1/(1+.1×.1))/.11 e – .0000100JD
9 1/(.1+.1+.1)–1/(11–.1)–.1 π – .00000244JD (.1+.1+.1/(1–.1))/.11–.11 e – .00000100JD
10 1×1/(.1+.1+.1)–1/(11–.1)–.1 π – .00000244JD 1–.1×(1–.1×.1×.1–(1+1)/.11) e – .0000000103JD

Best Known Approximations Using 2's and Unlimited Decimal Points
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 2 π – 1.142JD 2 e – .718JD
2 2+2 π + .858JD 2.2 e – .518JD
3 2+2/2 π – .142JD 2+2/2 e + .282JD
4 2+2/(2–.2) π – .0305JD 2+.2×2×2 e + .0817JD
5 2/(.2+.2+.2)–.2 π – .00826JD 2+(2–.2)×(.2+.2) e + .00172JD
6 22/((2–.2)/.2–2) π + .00126JD 2×(2+.2×(2–.2))–2 e + .000172JD
7 .2+2/.2/(2×(2–.2)–.2) π – .000416JD 2+(2–.2)/(2–.2×.2)–.2 e + .0000855JD
8 2+2/.2/.2/(2×22–.2) π – .0000401JD (.2+.2/.22)/(2+.2×.2)/.2 e + .0000782JD
9 .22×(.2+2/.2)×(2–.2–.2–.2) π + .00000735JD 2×(.2+2.2/(2–.2/(2–.2×.2))) e – .00000226JD

Best Known Approximations Using 3's and Unlimited Decimal Points
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 3 π – .142JD 3 e + .282JD
2 3.3 π + .158JD 3–.3 e – .0183JD
3 3+.3/3 π – .158JD 3×3×.3 e – .0183JD
4 3+.3/(3–.3) π – .0305JD 3×.3/.33 e + .00899JD
5 3+.3/(3–3×.3) π + .00126JD 3+.3×.3×.3–.3 e + .00872JD
6 3+.3/(3.–3–.3–.3) π + .00126JD .3×3+(.3+.3)/.33 e – .000100JD
7 3+3/(3+(3+3)/.33) π + .0000382JD .3×(3/.33–.3×.3/3) e – .00000910JD
8 3/(.3–33)–(.3–3/.3)/3 π – .00000244JD .3/3×(3×3/.33–.3×.3) e – .00000910JD
9 (.3+3/(.3+(.3+3)/.3))/(.3+.3)/.3 π + .000000267JD .3×((3+.3/3)/.33–.333) e – .0000000103JD

Best Known Approximations Using 4's and Unlimited Decimal Points
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 4 π + .858JD 4 e + 1.282JD
2 4–.4 π + .458JD 4–.4 e + .882JD
3 4–.4–.4 π – .058JD 4/4/.4 e – .218JD
4 4–.4–.44 π + .0184JD .44/.4/.4 e – .0317JD
5 4/(.4+.4×.4)–4 π + .00126JD 4–.4×.4×(4+4) e + .00172JD
6 4/.4/(.4+4/4)–4 π + .00126JD 4/.4/(4–.4×(.4+.4)) e – .000891JD
7 4–.4–4.4/(4/.4–.4) π + .0000740JD ((4+44)/.4–.4)/44 e – .000100JD
8 4–.44+4/(.44–4/.4) π – .00000269JD 4–4/(4–.4×.4)–.4+.4×.4 e + .0000515JD
9 4–.44+.4×4/(.4×.44–4) π – .00000269JD 4/(.44–4/(.44/(4–.4)–4)) e – .00000159JD

Best Known Approximations Using 5's and Unlimited Decimal Points
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 5 π + 1.85JD 5 e + 2.28JD
2 .5×5 π – .642JD .5×5 e – .218JD
3 .5+5×.5 π – .142JD .5×5.5 e + .0317JD
4 5.5+.55 π – .0916JD .5×(5–.5)+.5 e + .0317JD
5 .5+.5×(5+.5×.5) π – .0166JD .5+(5+5)/(5–.5) e + .00394JD
6 .5/(.5×.5–5/55) π + .00126JD (5/.55+5–.5)/5 e – .000100JD
7 .5×(5–5/(5.5/5–5)) π – .000567JD 5–(.5+5×5×5)/55 e – .000100JD
8 (.5+5/(5–.5))×(5×.5–.55) π + .0000740JD 5×(.5+5/(55/.5+5–.5)) e + .000000587JD
9 5–(5+5+.5)/(.5/5+5+.55) π + .000000267JD .5+5/(.5×(5–.5)+.5/5/5/5) e – .00000321JD
10 (5+5)/(.5×.5+5/(5+.5/(5–.5)))–5 π + .000000267JD ((5–5/(5–.5))/.5/.55–.55)/5 e + .000000100JD

Best Known Approximations Using 6's and Unlimited Decimal Points
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 .6 π – 2.54JD .6 e – 2.12JD
2 .6×6 π + .458JD .6×6 e + .882JD
3 .6×(6–.6) π – .0984JD .6×6–.6 e + .282JD
4 6/.66–6 π – .0507JD 6–.6×(6–.6) e + .0417JD
5 .6×(6–.6)–.6/6 π – .00159JD 6/(.6+.6+6/6) e + .00899JD
6 6×(6/(6–.66)–.6) π – .0000196JD (6/.6/.6–.6×.6)/6 e – .000504JD
7 6×6/(6–.66)–6×.6 π – .0000196JD 6–(.6+6×6×6)/66 e – .000100JD
8 6–.6×(.6+.6+.66×(6–.6)) π – .00000735JD .6×(6–.6–6/((6–.6)/6+6)) e – .0000210JD
9 .6/(.6×.6+6/(.6–6×6–.6/6)) π + .000000267JD ((.6+.6+6/.6)/.66–.66)/6 e + .00000100JD

Best Known Approximations Using 7's and Unlimited Decimal Points
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 .7 π – 2.44JD .7 e – 2.02JD
2 .7+.7 π – 1.74JD .7+.7 e – 1.32JD
3 7/.7–7 π – .142JD 7/.7–7 e + .282JD
4 (7–.7)×.7×.7 π – .0546JD 7×.7×.7–.7 e + .0117JD
5 (7+.7/7)/.7–7 π + .00126JD 7–(7/.7–7))/.7 e – .00400JD
6 7/(.7+(.7+7/.7)/7) π – .000567JD 7/(.7+7/7)–.7–.7 e – .000635JD
7 .7×.7×(7–.7+.7/(7–.7)) π – .000148JD ((7+7)/.7–.7)/(7+.7/7) e + .0000280JD
8 (.7+7×7)/(7+(.7+.7)×(7–.7)) π + .000000267JD 7/.7–7–(7+7)/(.7+7×7) e + .0000280JD
9 7/(7×(.7–(7+7)/(.7+7×7))–.7) π + .000000267JD (7–.7)×(.7×.77–.7/(7–.7×.7)) e – .00000118JD

Best Known Approximations Using 8's and Unlimited Decimal Points
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 .8 π – 2.34JD .8 e – 1.92JD
2 .8+.8 π – 1.54JD .8+.8 e – 1.12JD
3 .8+.8+.8 π – .742JD .8+.8+.8 e – .318JD
4 .8+.8+.8+.8 π + .0584JD .8+(8+8)/8 e + .0817JD
5 (8+8)/.8/.8/8 π – .0166JD .8+.8×(.8+.8+.8) e + .00172JD
6 8.8/(.8+8/.8–8) π + .00126JD (.8+.88/.8/.8)/.8 e + .000468JD
7 8–.88×(.8×8–.88) π + .000807JD .8+(.8+.888)/.88 e – .000100JD
8 (.88+.8+8/(.8+8.8))/.8 π + .0000740JD 8–(8–.8)/(.8+.8×.8×.88) e + .0000280JD
9 .88×(.8+.88×8×8)/(8+8) π + .00000735JD (8×8–.8)/((8+8)/.8/.8/.8–8) e – .00000226JD

Best Known Approximations Using 9's and Unlimited Decimal Points
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 .9 π – 2.24JD .9 e – 1.82JD
2 .9+.9 π – 1.34JD .9+.9 e – .918JD
3 .9+.9+.9 π – .442JD .9+.9+.9 e – .0183JD
4 (.9+.9)×(.9+.9) π + .0984JD .9+.9+.99 e + .0717JD
5 .9+(9+9)/(9–.9) π – .0194JD .9+(.9+.9)/.99 e – .000100JD
6 (.9+.9)×(.9+.9)–.9/9 π – .00159JD .9+.9+9/(9+.9×.9) e – .000851JD
7 (9–(9/.9/.9/(.9+.9)))/.9 π – .000303JD .9+.9+9/.99/9.9 e – .00000818JD
8 .9–((.9+9)/(.9×(.9+.9)–9)–.9) π – .000129JD .9+.9+(9+.9/(.9+9))/(.9+9) e – .00000818JD
9 .9+(.99+9/(.9+9/.9))/.9/.9 π – .00000244JD .9+.9+9/(9+.9×.9–.9/99) e + .000000334JD

Best Known Approximations Using Any Digits and Unlimited Decimal Points
DigitsExpression for πValueAuthorExpression for eValueAuthor
1 3 π – .142JD 3 e + .282JD
2 3.1 π – .0416JD 2.7 e – .0183JD
3 22/7 π + .00126JD 2.72 e + .0172JD
4 (.9+7/.6)/4 π + .0000740JD 3–2/(8–.9) e + .0000280JD
5 1/(.6–2/(8–.9)) π + .000000267JD 2.9/.93–.4 e – .00000226JD

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 4/10/14.