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?

ANSWERS

Solutions were sent by Hakan Summakoğlu and Joe DeVincentis.

Here are the best known results.

Best Known Approximations Using 1's and ln's

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	1	π – 2.14		1	e – 1.72
2	1 + 1	π – 1.14		1 + 1	e – .718
3	1 + 1 + 1	π – .142		1 + 1 + 1	e + .282
4	1×1 + 1 + 1	π – .142		ln[11 + 1]	e – .233
5	ln[ (1 + 1) × 11]	π – .0506		1 + 1 + ln[1 + 1]	e – .0251
6	ln[ (1 + 1) × 11 + 1]	π – .00610		ln[111] – 1 – 1	e – .00875
7	1 + 1 + 1 / ln[ ln[11] ]	π + .00180		ln[111 + 1] – 1 – 1	e + .000217
8	1 + ln[11 – ln[11 + 1] ]	π + .000248		ln[111 + 1] – 1 – 1×1	e + .000217
9	ln[111 / (1+1) ln[11] ]	π + .000199		(1 + 1) / (1 – ln[1 + 1] ) ln[11]	e – .000155
10	ln[11 × (1 + ln[111 + 1] ) ] – 1	π + .00000896		1 + ln[ ln[ (111 – 1) ln[11] ] ]	e + .0000234
11	1 + 11 × 11 / (111 / (1+1) + 1)	π + .000000267		ln[ 11 / ln[1 + 1] – 1 / (ln[11] – 1) ]	e + .00000145
12	1 × 1 + 11 × 11 / (111 / (1+1) + 1)	π + .000000267	JD	ln[11×ln[ln[ln(1+1)×111]]–1]	e + .0000000275	JD

Best Known Approximations Using 2's and ln's

Complexity	Expression for π	Value	Expression for e	Value	Author
1	2	π – 1.14	2	e – .718
2	2 + 2	π + .858	2 + 2	e + 1.28
3	ln[22]	π – .0506	2 + ln[2]	e – .0251
4	ln[22 + 2]	π + .0365	(2 + 2) ln[2]	e + .0543
5	ln[22 + 2/2]	π – .00610	ln[22 ln[2] ]	e + .00625
6	ln[22 + 2 – 2/2]	π – .00610	2 + 2 / (2 × 2 ln[2] )	e + .00307
7	ln[22 + ln[ ln[22] ] ]	π – .000527	(2 + ln[2] ) / (ln[2 + ln[2] ] )	e + .000118
8	(2 – 2/22) × (2 + ln[2] ) – 2	π – .000130	2 + 2 / (ln[2 × 22] – 2/2)	e + .0000600
9	ln[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 + .000000364	JD

Best Known Approximations Using 3's and ln's

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	3	π – .142		3	e + .282
2	ln[3]	π – 2.04		ln[3]	e – 1.62
3	3 + 3 – 3	π – .142		3 / ln[3]	e + .0124
4	3 + 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
6	3 + 3 / (3 × (3+3) + 3)	π + .00126		3 × (3/3 – 3 ln[ ln[3] ] )	e – .000425
7	3 + 3/3 – 3 / ln[33]	π + .000408		3 / ln[3] ln[3 / ln[3] ]	e + .0000282
8	3 + (3 – 3 / ln[3] ) / (3 – ln[3] )	π + .0000314		(3 + 3) / ln[3×3 + 3/33]	e + .00000214
9	ln[3] × ln[ (3+3) × (3 – 3/33) ]	π – .000000844		(3 – 3 ln[ ln[3 / ln[3] ] ] ) / ln[3]	e – .0000000430
10	ln[3–(33–3)/3/(3–ln[33])]	π – .000000538	JD	3×(3/3–ln[ln[3/ln[3]]])/ln[3]	e – .0000000430	JD

Best Known Approximations Using 4's and ln's

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	4	π + .858		4	e + 1.28
2	ln[4]	π – 1.76		ln[4]	e – 1.33
3	4 – 4/4	π – .142		ln[4 × 4]	e – .0543
4	(4 + 4 + 4) / 4	π – .142		44 / (4 × 4)	e + .0317
5	ln[4×4 + 4 + 4]	π + .0365		ln[4 ln[44] ]	e – .00116
6	ln[4 × (4+4) / ln[4] ]	π – .00249		ln[4] + ln[ ln[44] ]	e – .00116
7	ln[4 + 4 + 4 ln[44] ]	π – .000170		(4 + 4) / (4 – 4/ln[44] )	e + .0000600
8	4 – ln[4 × (4+4) – 4/4] / 4	π – .0000894		4 / ln[ (4 × 4) / (4 – ln[ ln[4] ] ) ]	e + .0000691
9	ln[4 × 4 ln[4] + ln[4 – ln[4] ] ]	π + .0000340		ln[ (4 + (4 + 4/4) ln[4] ) ln[4] ]	e – .00000162
10	ln[(4+4–ln[4])/ln[ln[ln[44]]]]	π + .0000000426	JD	4/ln[4+(4+ln[4])/(4ln[44])]	e – .000000308	JD

Best Known Approximations Using 5's and ln's

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	5	π + 1.86		5	e + 2.28
2	ln[5]	π – 1.53		ln[5]	e – 1.11
3	5 / ln[5]	π – .0349		5 / ln[5]	e + .388
4	ln[5] + ln[5]	π + .0773		ln[5 + 5 + 5]	e – .0102
5	ln[25 – ln[5] ]	π + .0107		ln[5×5 – 5 – 5]	e – .0102
6	5 / ln[5 – 5/55 ]	π + .000910		ln[5 × 5] – 5 / (5+5)	e + .000594
7	5 / (ln[5] – 5 / (5 × 55) )	π + .000579		(5 + 5) / (5 – (5 + ln[5] ) / 5)	e + .000504
8	5 – ln[ ln[555 + 55] ]	π + .00000860		5 / ln[5] – ln[5 / (5 – ln[5] ) ]	e – .0000493
9	ln[5×5 – 5 / (5×5 – 5) – ln[5] ]	π – .00000564		5 + (5 – (5 + 5) / (ln[5] – 5/5) ) / 5	e + .00000570
10	ln[55]–5/(5–ln[ln[5+5]/5])	π + .000000450	JD	ln[5+5ln[5+5–5ln[ln[5]]]]	e – .000000204	JD

Best Known Approximations 6's and ln's

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	6	π + 2.86		6	e + 3.28
2	ln[6]	π – 1.35		ln[6]	e – .927
3	6 / ln[6]	π + .207		ln[6 + 6]	e – .233
4	ln[6] × ln[6]	π + .0688		6 – 6 / ln[6]	e – .0669
5	6 – ln[6 + 6 + 6]	π – .0320		ln[6 ln[6 + 6] ]	e – .0163
6	6 – (6 + 6) / ln[66]	π – .00579		(6 + 6 / ln[ ln[6] ] ) / 6	e – .00360
7	6 / (6/6 + ln[ ln[6 + 6] ] )	π – .000618		ln[ (6 + 6) / (ln[6] – 6/6) ]	e + .000122
8	6 / ln[6] – ln[6 + 6] / (6 + 6)	π – .00000445		ln[6 ln[6 + 6 + 6/(6+6) ] ]	e + .00000723
9	ln[(6 – ln[6 + 6] ) × (6 + ln[ ln[6] ] ) ]	π – .00000591		6 / ln[ (666 – 66) / 66]	e + .00000214
10	ln[6×6]–(ln[6+6]+6/(6×6))/6	π – .00000260	JD	(6+6–(6+6)×(ln[ln[66]/6]))/6	e + .000000450	JD

Best Known Approximations Using 7's and ln's

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	7	π + 3.86		7	e + 4.28
2	ln[7]	π – 1.20		ln[7]	e – .772
3	7 / ln[7]	π + .456		ln[7 + 7]	e – .0792
4	7 – ln[7 × 7]	π – .0334		7 – ln[77]	e – .0621
5	7 – 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] ) / 7	e + .00373
7	ln[7 ln[ (7+7) ln[7] ] ]	π – .000310		(7+7) ln[ ln[7 + 7] ] / 7	e – .000528
8	7 – ln[7×7 – 7 / ln[77] ]	π + .0000276		7 × (ln[ (7 + 7 + 7) × 77] – 7)	e + .0000132
9	ln[7 + 7 + (7+7) / ln[7] + ln[7] ]	π – .00000889		7 / (7 – ln[77 + 7 – 7/(7+7) ] )	e + .00000287
10	7–ln[7×7–7/(7–7/ln[7+7])]	π – .00000164	JD	ln[7]–(ln[7–ln[7+7/7]]–7)/7	e + .00000287	JD

Best Known Approximations Using 8's and ln's

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	8	π + 4.86		8	e + 5.28
2	ln[8]	π – 1.06		ln[8]	e – .639
3	ln[8 + 8]	π – .369		ln[8 + 8]	e + .0543
4	ln[8 + 8 + 8]	π + .0364		ln[8 ln[8] ]	e + .0933
5	8 – ln[8 × (8+8) ]	π + .00638		ln[8 + 8 – 8/8]	e – .0102
6	ln[8 × 8 / ln[8 + 8] ]	π – .00249		8 / ln[88/8 + 8]	e – .00130
7	ln[88] / ln[ ln[8 × 8] ]	π – .000145		ln[88/8 + ln[8 × 8] ]	e + .000305
8	ln[8 × (88 – ln[8] ) ] / ln[8]	π + .0000512		ln[8 + ln[ (8+8)×(88–8) ] ]	e + .0000233
9	8 – (8 + 8) / ln [(8×8 – 8) / ln[8] ]	π – .0000118		(8 – ln[8] ) / (ln[8 + 8 + 8] – 8/8)	e – .00000261
10	ln[8+8+8–8/(ln[8]×ln[88])]	π + .00000214	JD	ln[8]+8/(8+8+8/8–ln[88])	e + .00000146	JD

Best Known Approximations Using 9's and ln's

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	9	π + 5.86		9	e + 6.28
2	ln[9]	π – .944		ln[9]	e – .521
3	ln[9 + 9]	π – .251		ln[9 + 9]	e + .172
4	9 / ln[9 + 9]	π – .0278		ln[9 ln[9] ]	e + .266
5	9 / ln[9] – 9/9	π – .0455		9 / ln[9 + 9 + 9]	e + .0124
6	ln[999]/ ln[9]	π + .00181		ln[9 + 9 – ln[9 + 9] ]	e – .00295
7	9 – 9 ln[ ln[9 – ln[9] ] ]	π + .00000182		ln[9] + ln[99 + 9] / 9	e – .000820
8	9 ×(9/9 ln[ ln[9 – ln[9] ] ] )	π + .00000182		ln[9] / ln[ (9 + 9 + ln[9] ) / 9]	e – .0000231
9	9 – 9 ln[ ln[9 – ln[9] ] ] × 9/9	π + .00000182		ln[9 + 9 – ln[9 + 9 – ln[ ln[9] ] ] ]	e + .00000555
10	9×(9–9) + 9 – 9 ln[ ln[9 – ln[9] ] ]	π + .00000182	JD	(9+99)/ln[9+9/99]/(9+9)	e + .00000214	JD

Best Known Approximations Using Any Digits and ln's.

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	3	π – .142		3	e + .282
2	3 + 0	π – .142		8 / 3	e – .0516
3	22 / 7	π + .00126		19 / 7	e – .00400
4	3 + 9 / 64	π – .000968		87 / 32	e – .000468
5	3 × (5 ln[5] – 7)	π – .0000240		4 – (7 ln[3] / 6)	e – .00000383
6	3 + 7 / (45 ln[3] )	π + .000000115	JD	2 ln[894] / 5	e + .000000482	HS

Best Known Approximations Using 1's and Unlimited ln's

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	1	π – 2.14		1	e – 1.72
2	ln[11]	π – .744		ln[11]	e – .320
3	1 + 1 + 1	π – .142		ln[11 + 1]	e – .233
4	ln[11] + ln[1 + 1]	π – .0506		ln[1] – 1 / ln[ ln[2] ]	e + .0101
5	1 + 1 + 1 / ln[ ln[11] ]	π + .00180		ln[11] × (1 – ln[ ln[ ln[11] ] ] )	e + .000928
6	1+ln[11–ln[1+11]]	π + .000248	HS	1+1–ln[ln[1+ln[1+ln[ln[11]]]]]	e – .00000601	HS
7	1–ln[ln[ln[1+1]–ln[ln[ln[1+1+1]]]]–1]	π + .0000101	JD	(ln[ln[(ln[1]–1)/(ln[ln[1+1]])]]–1)/ln[ln[1+1]]	e – .0000000233	JD
8	ln[1]–ln[1–(ln[1+1])]/ln[1–ln[1+ln[ln[1+1]]]]	π – .00000102	JD	1×(ln[ln[(ln[1]–1)/(ln[ln[1+1]])]]–1)/ln[ln[1+1]]	e – .0000000233	JD

Best Known Approximations Using 2's and Unlimited ln's

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	2	π – 1.14		2	e – .718
2	ln[22]	π – .0506		2 + ln[2]	e – .0251
3	2 + ln[ ln[22] ]	π – .0131		ln[22] + ln [ ln[2] ]	e + .00625
4	ln[22+ln[ln[22]]]	π – .000527	HS	2/ln[ln[2]]/ln[ln[ln[2]]×ln[ln[2]]]	e + .0000188	JD
5	ln[22+ln[2+ln[ln[22]]]]	π – .00000589	HS	(ln[2]×ln[2])–(2×ln[ln[ln[2+2]]])	e – .000000404	HS
5	2+ln[2/ln[ln[ln[222]]]–ln[2]]	π + .000000524	JD	2+ln[2]–(ln[2]+2)×ln[ln[ln[2]+2]]	e + .000000364	JD

Best Known Approximations Using 3's and Unlimited ln's

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	3	π – .142		3	e + .281
2	3 + ln[ ln[3] ]	π – .0475		3 / ln[3]	e + .0124
3	ln[ ln[3] + ln[3] ] – ln[ ln[ ln[3] ] ]	π + .00955		3 – 3 ln[ ln[3] ]	e – .000425
4	ln[3/(ln[ln[3]–ln[ln[3]]/ln[ln[ln[3]]]])]	π + .000163	HS	(3/ln[3/ln[3]])/ln[3]	e + .0000282	HS
5	ln[3]–ln[ln[ln[3/ln[ln[3]]]]]/ln[ln[ln[ln[3]/ln[ln[3]]]]]	π + .00000203	HS	(3–3×ln[ln[3/ln[3]]])/ln[3]	e – .0000000430	HS
6	ln[3]+ln[ln[ln[ln[3–ln[ln[ln[3]]]]]]+ln[ln[3×3]]/ln[ln[3]]]	π + .00000000534	JD	(3–3ln[ln[3]])/ln[3–3ln[ln[3]]]	e + .0000000333	JD

Best Known Approximations Using 4's and Unlimited ln's

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	4	π + .858		4	e + 1.28
2	4 / ln[4]	π – .256		ln[4] + ln[4]	e + .0543
3	ln[4] / ln [ ln[4] ] + ln[ ln[ ln[4] ] ]	π – .0163		ln[4] + ln[ ln[44] ]	e – .00116
4	4+ln[4–ln[4]]/ln[ln[ln[4]]]	π – .000255	HS	ln[ln[ln[ln[4]]]/ln[ln[ln[(ln[ln[4]]+4/ln[ln[4]])]]]]	e – .00000745	HS
5	ln[(4+4–ln[4])/ln[ln[ln[44]]]]	π + .00000000426	HS	4–(4–ln[4])×ln[ln[4–ln[ln[ln[4]]]]]	e + .0000000300	HS
6	4ln[4/ln[4]–(ln[ln[4]]–ln[ln[ln[4]]]×ln[ln[4]])]	π + .000000105	JD	ln[4ln[44]–ln[ln[ln[4×4]]]/ln[ln[ln[4]]]]	e + .000000188	JD

Best Known Approximations Using 5's and Unlimited ln's

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	ln[5]	π – 1.53		ln[5]	e – 1.11
2	5 / ln[5]	π – .0349		ln[5] × ln[5]	e – .128
3	5 + ln[ ln[ ln[ ln[5] + ln[5] ] ] ]	π + .00164		ln[5 / ln[ ln[ ln[55] ] ] ]	e + .00604
4	ln[5×5+ln[ln[ln[ln[5×5]]]]]]	π + .000110	HS	ln[ln[ln[5]]]/ln[ln[ln[5×ln[5+ln[ln[5]]]]]]	e – .000289	HS
5	ln[ln[ln[5–ln[5]]]]×ln[ln[ln[ln[ln[ln[5]]/5]/ln[ln[ln[5]]]]]]	π – .00000104	HS	ln[ln[5×55]]–ln[ln[ln[5+ln[ln[ln[5]]]]]]	e – .00000475	HS
6	(ln[ln[5]–(ln[5]/ln[ln[ln[5]]])])×(ln[ln[ln[5]]]+5/ln[5])	π + .0000000218	JD	ln[ln[5]]–ln[5]/ln[ln[ln[5–(ln[ln[ln[5]+ln[5ln[ln[5]]]]])]]]	e – .000000122	JD

Best Known Approximations Using 6's and Unlimited ln's

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	ln[6]	π – 1.35		ln[6]	e – .927
2	ln[6] × ln[6]	π + .0688		ln[6 + 6]	e – .233
3	6 ln[ ln[6 – ln[ ln[6] ] ] ]	π + .00502		ln[6]–ln[ln[ln[6]]]/ln[ln[6]]	e – .00192	HS
4	ln[6]×ln[6–ln[ln[6×ln[ln[6]]]]]	π + .000275	HS	ln[ln[6]+(6+ln[6])/ln[ln[6]]]	e – .000139	HS
5	6+ln[ln[ln[6]]]–ln[6×ln[6]–ln[ln[6]]]	π – .00000351	HS	ln[ln[66]/ln[ln[ln[6+6×6]]]]	e + .00000339	HS
6	6/ln[6–(ln[ln[ln[ln[6]]–(ln[ln[ln[ln[ln[ln[6]]+66]]]])]])]	π – .0000000779	JD	ln[ln[ln[ln[ln[ln[6]]]×ln[ln[ln[6]]]]×ln[ln[ln[6]]]/6]/ln[ln[ln[ln[6]]+ln[6]]]]	e + .000000000530	JD

Best Known Approximations Using 7's and Unlimited ln's

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	ln[7]	π – 1.20		ln[7]	e – .772
2	ln[7] / ln[ ln[7] ]	π – .219		ln[7 + 7]	e – .0792
3	ln[ ln[7] / 7] / ln[ ln[ ln[7] ] ]	π + .00481		7 ln[ ln[7] ] – ln[7]	e – .00408
4	ln[7]×ln[ln[7]]+ln[7–ln[ln[7]]]	π – .000168	HS	ln[ln[ln[ln[7+ln[ln[ln[7]]]]]]/ln[ln[ln[7+7]]]]	e – .0000841	HS
5	ln[7]×ln[7–ln[7]/ln[ln[7]]]–ln[ln[ln[7]]]	π – .00000125	HS	ln[7+ln[7]]–ln[ln[ln[7]]]×ln[ln[7]]×ln[7]	e – .00000367	JD
6	ln[ln[7]]/ln[ln[ln[ln[ln[ln[7–ln[7ln[7]]]]]/ln[ln[ln[7+7]]]]]]	π – .0000000307	JD	ln[7]+ln[ln[7+ln[ln[7]]–ln[ln[ln[ln[7×(7+ln[7])]]]]]]	e + .00000000709	JD

Best Known Approximations Using 8's and Unlimited ln's

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	ln[8]	π – 1.06		ln[8]	e – .639
2	ln[8] / ln[ ln[8] ]	π – .301		ln[8 + 8]	e + .0543
3	ln[(ln[ln[8]]–8)/ln[ln[ln[8]]]]	π + .00714	HS	ln[ ln[8] ] – ln[8 – ln[ ln[8] ] ]	e – .00271
4	ln[ln[8–ln[ln[ln[8]]]]/(ln[ln[8]]/8)]	π + .00000948	HS	ln[8]/ln[ln[8+ln[ln[8–ln[8]]] ]]	e + .0000385	HS
5	ln[8ln[8ln[8]]–ln[ln[ln[8]]]×ln[8]]	π + .00000370	HS	ln[8+ln[ln[ln[ln[8×8]]+ln[ln[ln[8]]]]]/ln[ln[ln[8]]]]	e – .00000365	HS
6	ln[8–(8+8)/ln[ln[ln[ln[8]+ln[ln[ln[ln[8]]]+8]]]]]	π + .00000000155	JD	ln[8ln[ln[ln[ln[8]]]+8+ln[ln[ln[8–8/ln[8]]]]]]	e – .0000000234	JD

Best Known Approximations Using 9's and Unlimited ln's

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	ln[9]	π – .944		ln[9]	e – .521
2	ln[9] + ln[ ln[9] ]	π – .157		ln[9] / ln[ ln[9] ]	e + .0729
3	ln[9 / ln[ ln[ ln[9] + ln[9] ] ] ]	π – .00857		ln[9]–ln[ln[ln[9]]]×ln[9]	e + .00511	JD
4	9–9×ln[ln[9–ln[9]]]	π + .00000182	HS	ln[ln[ln[9]]] + ln[9+ln[ln[ln[9]]]] + ln[ln[9]]	e – .0000879	HS
5	ln[ln[9]/ln[ln[ln[9]+ln[ln[ln[9+ln[ln[9]]]]]/ln[ln[ln[9]]]]]]	π + .00000106	HS	ln[9]/ln[ln[9]+ln[ln[ln[9+9]]]×ln[ln[9]]]	e – .00000135	HS
6	ln[9ln[9–ln[9]–ln[ln[ln[ln[ln[ln[ln[ln[9]]–ln[ln[ln[9]]]]]/ln[ln[ln[9]]]]]]]]]	π + .0000000478	JD	ln[9–ln[ln[ln[9]/ln[ln[ln[ln[ln[9]]]+ln[9]×ln[9]]]]]]/ln[ln[9]]	e – .0000000466	JD

Best Known Approximations Using Any Digits and Unlimited ln's

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	3	π – .142	JD	ln[9]	e – .282	JD
2	ln[23]	π – .00610	HS	ln[5/ln[ln[4]]]	e + .0101	JD
3	ln[ln[2]–7/ln[ln[ln[8]]]]	π – .00000287	JD	ln[ln[ln[8]+ln[8]]]-ln[ln[ln[3]]]	e + .0000148	JD
4	ln[(8–ln[4])/ln[ln[ln[44]]]]	π + .00000000426	HS	1–ln[ln[ln[ln[7]+ln[4–ln[ln[3]]]]]]	e – .0000000122	JD

Best Known Approximations Using 1's and √'s

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	1	π – 2.142	HS	1	e – 1.718	HS
2	1+1	π – 1.142	HS	1+1	e – .718	HS
3	1+1+1	π – .142	HS	1+1+1	e + .282	HS
4	√[11–1]	π + .0207	HS	1+1+√1	e + .282	HS
5	√[11–√1]	π + .0207	HS	1+√[1+1+1]	e + .0138	HS
6	√[11–√√1]	π + .0207	HS	1+1+√[1/(1+1)]	e – .0112	HS
7	√[1+1+1]+√[1+1]	π + .00467	HS	(11–1–1)/√11	e – .00468	HS
8	√[1+1]×(11–1)–11	π + .000543	HS	√[11–√[1+1+11]]	e + .000992	HS
9	1+1+1–√[1+1]/(1–11)	π – .000171	HS	(1+11)/(1+1+1+√[1+1])	e + .000209	HS
10	1+1+√√[1+1–1/√11]	π + .0000118	JD	1+1+11/(√11+1+11)	e – .000108	JD
11	1+11×11/(1+111/(1+1))	π + .000000267	JD	√√[1+1]×(1+√111)-11	e – .000000309	JD
12	1×1+11×11/(1+111/(1+1))	π + .000000267	JD	1×√√[1+1]×(1+√111)-11	e – .000000309	JD

Best Known Approximations Using 2's and √'s

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	2	π – 1.142	HS	2	e – .0718	HS
2	2+2	π + .858	HS	2+2	e + 1.282	HS
3	1+2/2	π – .142	HS	2√2	e + .11	HS
4	2+√√2	π + .0476	HS	2+√2/2	e – .0112	HS
5	√[(22–2)/2]	π + .0207	HS	2+√[2/(2+2)]	e – .0112	HS
6	2+√[2–√2/2]	π – .00454	HS	2√√[2+√2]	e + .000364	HS
7	2/√[(2–√√2)/2]	π – .000433	HS	(2+2)/√√√22	e –. 000234	HS
8	2+√√[2–√[2/22]]	π + .0000118	HS	2+(√[2+√2/22])/2	e +. 0000987	HS
9	2–2/(2×22–2)+√√2	π – .00000459	HS	√[(22/(2+2))/(2–√2)–2]	e + .00000574	HS
10	2+√[2–√√[√2/(2+2×2)]]	π – .00000271	JD	√2+√√√[2+2/22]	e – .00000280	JD
11	(2+22×(22/(2+222/2)))/2	π + .000000267	JD	(2+√2)/2+√√[(2+2/22)/2]	e – .000000129	JD

Best Known Approximations Using 3's and √'s

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	3	π – .142	HS	3	e + .282	HS
2	√3	π – 1.41	HS	√3	e – .986	HS
3	√[3×3]	π – .142	HS	√[3+3]	e – .269	HS
4	3+3/33	π – .0507	HS	3/3+√3	e +. 0138	HS
5	√√[3×33]	π + .0127	HS	3–(√3/(3+3))	e – .00696	HS
6	3–((√3–3)/(3×3))	π – .000709	HS	√[((3+3)×√3)–3]	e + .000598	HS
7	3+3/(3+√333)	π – .000405	HS	(3+(√3/3))/√√3	e – .0000834	HS
8	√3+√√[3√√3]	π + .0000728	HS	√[33–√[3/33]]–3	e – .0000226	HS
9	√[3×3+3/(√[3+3]+3/3)]	π + .0000142	HS	3–(3×3)/33–3/333	e – .0000181	HS
10	3-3/(3–√[3×((3+3)×33–3)])	π + .00000512	JD	3–√[(3+3/3/3)/(3×3+33)]	e + .0000000866	JD

Best Known Approximations Using 4's and √'s

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	4	π + .858	HS	4	e + 1.282	HS
2	√4	π – 1.142	HS	√4	e – .718	HS
3	4 – 4/4	π – .142	HS	√[4+4]	e + .11	HS
4	4–√[4/4]	π – .142	HS	44/(4×4)	e +. 0317	HS
5	√[(44–4)/4]	π + .0207	HS	4–((4+4/4)/4)	e + .0317	HS
6	44/(4×4–√4)	π – .00126	HS	4–√[√44/4]	e – .00604	HS
7	4–√[4/(4+√√4)]	π – .00113	HS	(44–√4/4)/(4×4)	e + .000468	HS
8	4–(√√44/(4–4/4)	π – .0000958	HS	√[(44–4)/(4+√√4)]	e – .000201	HS
9	4/(4/(4×4)–4/(4/44–4))	π – .0000401	HS	4–√[4/(4–√√[4+√4])]	e + .0000132	HS
10	√[4√[4+44/√[444]]]	π + .0000102	JD	(√[4+4×44]–4)/√[4×4–4]	e + .000000979	JD
11	4/(4–44/(4×4+(4+√4)/44))	π + .000000267	JD	√[√√4+4+√(4+4/(4–44))]	e – .000000142	JD

Best Known Approximations Using 5's and √'s

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	5	π + 1.858	HS	5	e + 2.282	HS
2	√5	π – .906	HS	√5	e – .482	HS
3	√[5+5]	π + .0207	HS	5–√5	e + .0457	HS
4	5/5+√5	π + .0945	HS	√√55	e + .00499	HS
5	√[(55–5)/5]	π + .0207	HS	5/(5–√[5+5])	e + .00248	HS
6	5–5/(√[5+√5])	π – .000333	HS	5–√[5+5/(5×5)]	e + .00137	HS
7	5–√[5×5/(5+√5)]	π – .000333	HS	√√5+√√√5	e – .0000885	HS
8	5/√√[√55–5/5]	π + .00000495	HS	5/√√[(55+√5)/5]	e + .0000036	HS
9	5/√√[√55–√[5/5]]	π + .00000495	HS	5√[5/√[5×(55+√5)]]	e + .0000036	HS
10	5–√√[5√[5+5/(5+√5)]]	π + .00000000606	JD	√[55–5]–5+(√5+5/5)/5	e – .000000421	JD

Best Known Approximations Using 6's and √'s

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	6	π + 2.858	HS	6	e + 3.282	HS
2	√6	π – .692	HS	√6	e – .269	HS
3	√[6+6]	π + .323	HS	6/√6	e – .269	HS
4	(6+6+6)/6	π – .142	HS	√[6+6/6]	e –. 0725	HS
5	6–√√66	π + .00814	HS	6√6–6–6	e – .0213	HS
6	6–√6/6–√6	π – .000669	HS	√[6+(6+√6)/6]	e + .00353	HS
7	(6×6–6)/(6+6–√6)	π – .000399	HS	√[(66–6)/√66]	e – .000656	HS
8	6/6+√[6–√[(6+6)/6]]	π – .000148	HS	6–(√[666–6]–6)/6	e – .000026	HS
9	√[6√[(6+6)/(6–√√6)]]	π – .00000196	HS	√[6+√[66+√[6+6]]/6]	e + .0000054	HS
9	√[6√[6+6]/√[6–√√6]]	π – .00000196	JD	6–√√[66×(6–√[6+6+6])]	e + .00000118	JD

Best Known Approximations Using 7's and √'s

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	7	π + 3.858	HS	7	e + 4.282	HS
2	√7	π – .496	HS	√7	e – .0725	HS
3	7/√7	π – .496	HS	7/√7	e – .0725	HS
4	√[7+√7]	π – .0358	HS	√√[7×7]	e – .0725	HS
5	7/(7+7)+√7	π + .00416	HS	√[7+√7/7]	e – .00204	HS
6	(77+77)/7×7	π + .00126	HS	7/((7+7)×7)+√7	e – .0011	HS
7	√√[7+7+7]+7/7	π – .000898	HS	√[7+7/(77/7+7)]	e – .0000308	HS
8	√[7+7/7+7/√[7+7]]	π + .000195	HS	√[7+7/((77+7×7)/7)]	e – .0000308	HS
9	√7+√[√√√77/7]	π + .0000159	HS	√7+((7×7)/77)/√77	e – .0000102	HS
10	√[7–7/(7+7)+√[7+7–√7]]	π + .000000351	JD	7/(7/7/7+√√[7×7–7–7])	e – .000000352	JD

Best Known Approximations Using 8's and √'s

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	8	π + 4.858	HS	8	e + 5.282	HS
2	√8	π – .313	HS	√8	e + .11	HS
3	8/√8	π – .313	HS	8/√8	e + .11	HS
4	√√88	π – .0788	HS	√[8–8/8]	e – .0725	HS
5	√√[88+8]	π – .0114	HS	√8–8/(8×8)	e – .0149	HS
6	√[√8–8/8+8]	π – .00656	HS	√8–8/((8×8)+8)	e – .000966	HS
7	√√[88+√88]	π – .000228	HS	√[(88–8)/(8+√8)]	e – .000201	HS
8	√[(8–√√8)×√8–8]	π + .000157	HS	88/(8+√√8/8)–8	e +. 0000614	HS
9	8/(√8–√8/8)–8/88	π – .0000136	HS	8/√[8+√[(8–8/8)/(8+8)]]	e +. 00000319	HS
10	√[88/√[88–8–8/(8+8)]]	π – .00000231	JD	√[8×8/(8+√[(8–8/8)/(8+8)])]	e +. 000000400	JD

Best Known Approximations Using 9's and √'s

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	9	π + 5.858	HS	9	e + 6.282	HS
2	√9	π – .142	HS	√9	e + .282	HS
3	9/√9	π – .142	HS	9/√9	e + .282	HS
4	√√99	π + .0127	HS	√[9–9/9]	e +. 11	HS
5	√[(99–9)/9]	π + .0207	HS	9√[9/99]	e – .00468	HS
6	√[9–(9/9–9)/9]	π + .00307	HS	9√9/√99	e – .00468	HS
7	√[9+(9/√[9+99])]	π – .00057	HS	√[√[9+99]–√9]	e + .00598	HS
8	√[9+9/(9+(9+√9)/9)]	π + .000217	HS	√[9–(9+99/(9+9))/9]	e – .0000308	HS
9	9√[9+9]/(9+√√99)	π – .0000186	HS	√[9–9/9–99/(9×(9+9))]	e – .0000308	HS
10	√[9+9/(9+√√√[99/9])]	π + .000000400	JD	√[9+9–√9]–√[9+99]/9	e + .000000979	JD

Best Known Approximations Using Any Digits and √'s.

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	3	π – .142	HS	3	e + .282	HS
2	√9	π – .142	HS	8 / 3	e – .0516	HS
3	22 / 7	π + .00126	HS	19 / 7	e – .00400	HS
4	√51 – 4	π – .000164	HS	87 / 32	e – .000468	HS
5	(9 + √45) / 5	π + .0000481	HS	3 – √[5 / 63]	e + .0000000866	HS
6	9 / (2 + 9 / 52) – 1	π + .000000267	HS	9 – 6 – √[5 / 63]	e + .0000000866	HS
7	√√[97 + 9 / 22]	π – .00000000101	HS	2 + √[7 / (8 + √31)]	e – .00000000945	HS

Best Known Approximations Using 1's and Decimal Points

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	1	π – 2.142	HS	1	e – 1.718	HS
2	1+1	π – 1.142	HS	1+1	e – .718	HS
3	1+1+1	π – .142	HS	1+1+1	e + .282	HS
4	1×(1+1+1)	π – .142	HS	1×(1+1+1)	e + .282	HS
5	1+1+1.1	π – .0416	HS	1–.1+1+1	e + .182	HS
6	1+1–1/(.1–1)	π – .0305	HS	(1+1+1)/1.1	e + .00899	HS
7	1+1+1/(1–.11)	π – .0180	HS	(1+1+1)×(1/1.1)	e + .00899	HS
8	11/(11/(1+1)–1–1)	π + .00126	HS	(1+1)/1.1+1–.1	e – .000100	HS
9	11×(1/(11/(1+1)–1–1))	π + .00126	HS	((1+1)/.1)/11+1–.1	e – .000100	HS
10	(1–.1+1+1)×(1/(1+11)+1)	π + .0000740	HS	(1+1+1)/1.1–1/111	e – .0000181	HS
11	1+11×11/(1+111/(1+1))	π + .000000267	JD	(1+1)/(1.1)+1/(1.111)	e – .0000100	JD
12	1×1+11×11/(1+111/(1+1))	π + .000000267	JD	(1+1+1/(1–.1))/1.1–.11	e + .00000100	JD
13	1×1×1+11×11/(1+111/(1+1))	π + .000000267	JD	1+(1+1)/1.1–(1–.1)×.111	e – .0000000103	JD

Best Known Approximations Using 2's and Decimal Points

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	2	π – 1.142	HS	2	e – .718	HS
2	2+2	π + .858	HS	2+2	e + 1.282	HS
3	2+2/2	π – .142	HS	2+2/2	e + .282	HS
4	(2+2+2)/2	π – .142	HS	2+2/(2+2)	e – .218	HS
5	2–2/(.2–2)	π – .0305	HS	22/(2×(2+2))	e + .0317	HS
6	2+2/(2–.22)	π – .0180	HS	(2+2+2)/2.2	e + .00899	HS
7	2–2/(2/(2×(2+2))–2)	π + .00126	HS	2/(2+2)+2.22	e + .00172	HS
8	22×(2/(22–2×(2+2)))	π + .00126	HS	2+2+2/(2×.22–2)	e – .000333	HS
9	2+(2/(2+22)+2.2)/2	π + .0000740	HS	2–(.2–2)/2–(2+2)/22	e – .000100	HS
10	(2.2+2+2+2/(2+22))/2	π + .0000740	HS	(2+2+2)/2.2–2/222	e – .0000181	HS
11	(2+22×22/(2+222/2))/2	π + .000000267	JD	2+(22–2)/(22–.22)-.2	e – .00000818	JD
12	2+22×22/(2×2+222)–2/2	π + .000000267	JD	2–2/(2×2×2–2/2.22)+2/2	e – .00000772	JD

Best Known Approximations Using 3's and Decimal Points

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	3	π – .142	HS	3	e + .282	HS
2	3/3	π – 2.142	HS	3/3	e – 1.718	HS
3	3+3–3	π – .142	HS	3–.3	e – .0183	HS
4	3+.3/3	π – .0416	HS	.3×3×3	e – .0183	HS
5	3+3/(3×(3+3))	π + .0251	HS	3×3/3.3	e + .00899	HS
6	3+3/(3+3×(3+3))	π + .00126	HS	3×(3/3–3/33)	e + .00899	HS
7	(33/(3+3/(3+3)))/3	π + .00126	HS	3–(.3+3×3)/33	e – .000100	HS
8	3+3/(3+(3+3)/.33)	π + .0000382	HS	3–(3+3+3.3)/33	e – .000100	HS
9	3+3/(3+(3×3–3×3)/.33)	π + .0000382	HS	3×3/3.3–3/333	e – .0000181	HS
10	3–33/(333×(.3–3/3))	π – .0000225	HS	3×(3/(3+3/333)–3/33)	e + .00000886	HS
11	(3+33–3/(3+3))/(.3+33/3)	π + .000000267	JD	3–.3×.333–(3+3)/33	e – .0000000103	JD

Best Known Approximations Using 4's and Decimal Points

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	4	π + .858	HS	4	e + 1.282	HS
2	4/4	π – 2.142	HS	4/4	e – 1.718	HS
3	4 – 4/4	π – .142	HS	4 – 4/4	e + .282	HS
4	.4×(4+4)	π + .0584	HS	44/(4×4)	e + .0317	HS
5	4+(.4–4)/4	π – .0416	HS	4–(4+4/4)/4	e + .0317	HS
6	44/(4+4/.4)	π + .00126	HS	(44–.4)/(4×4)	e + .00672	HS
7	4/(4/4–.44)–4	π + .00126	HS	(44–4/(4+4))/(4×4)	e + .000468	HS
8	.4/(4+.4×4)×44	π + .00126	HS	4+(4+4/4)/(.4/4–4)	e – .000333	HS
9	4/(4/(4×4)–4/(4/44–4))	π – .0000401	HS	((4+44)/.4–.4)/44	e – .000100	HS
10	4–4/(4.4/4+4–.44)	π + .0000382	HS	4/(4+444/(.4–4×44))	e – .0000156	HS
11	(4+44/(4+(4+4/4)/4.4))/4	π + .000000267	JD	4+(4+4/4)/(4/(44.4–4)–4)	e – .00000772	JD

Best Known Approximations Using 5's and Decimal Points

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	5	π + 1.858	HS	5	e + 2.282	HS
2	5/5	π – 2.142	HS	5/5	e – 1.718	HS
3	.5×5	π – .642	HS	.5×5	e – .218	HS
4	(5+5+5)/5	π – .142	HS	.55×5	e + .0317	HS
5	(55/5+5)/5	π + .0584	HS	.5×5.5	e + .0317	HS
6	5–55/(5×5+5)	π + .0251	HS	(5+5+5)/5.5	e + .00899	HS
7	55/(5+.5×5×5)	π + .00126	HS	.5–(5+5)/(.5–5)	e + .00394	HS
8	5/(.5+(5+55)/55)	π + .00126	HS	(5/.55–.5+5)/5	e – .000100	HS
9	5–(5/(5×5)+5/.55)/5	π – .000226	HS	.5×5+((5+55)/5)/55)	e – .000100	HS
10	(5+5.5)/5–5/(5/(5×5)–5)	π + .0000740	HS	5/((55+(5+5)/55)/(5+5×5))	e + .00000483	HS
11	(5+55/(5+(5+5×.5)/55))/5	π + .000000267	JD	5×(5/(5+5/555)–5×5/55)	e – .00000191	JD
12	5–(5+5.5)/(5.55+.5/5)	π + .000000267	JD	5×(5/(5+5/555)–5/(5+5+5/5))	e – .00000191	JD

Best Known Approximations Using 6's and Decimal Points

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	6	π + 2.858	HS	6	e + 3.282	HS
2	6/6	π – 2.142	HS	6/6	e – 1.718	HS
3	.6×6	π + .458	HS	(6+6)/6	e – .718	HS
4	(6+6+6)/6	π – .142	HS	(6+6+6)/6	e + .282	HS
5	6/.66–6	π – .0507	HS	(6+6/.6)/6	e – .0516	HS
6	6×(6/(6–.6+6))	π + .0163	HS	6–6×6/66×6	e + .00899	HS
7	6/(6/6+6/6.6)	π + .00126	HS	(6+6)/(6+6/6)+6/6	e – .00400	HS
8	6×(6/(6–.66)–.6)	π – .0000196	HS	(6×(6×6–6)–.6)/66	e – .000100	HS
9	6/((6–.66)/6)–.6×6	π – .0000196	HS	6–6×((.6/6+6×6)/66)	e – .000100	HS
10	6×(6+6/(6–.66)–6.6)	π – .0000196	HS	(6+6/(.6–(6+6)/666))/6	e – .0000156	HS
11	6/6/(.6–(6+6)/(6.6+6×6))	π + .000000267	JD	6×(6/(6+6/666)–6×6/66)	e – .00000460	JD

Best Known Approximations Using 7's and Decimal Points

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	7	π + 3.858	HS	7	e + 4.282	HS
2	7/7	π – 2.142	HS	7/7	e – 1.718	HS
3	(7+7)/7	π – 1.142	HS	(7+7)/7	e – .718	HS
4	7/.7–7	π – .142	HS	7/.7–7	e + .282	HS
5	7+7–77/7	π – .142	HS	.7+(7+7)/7	e – .0183	HS
6	(77+77)/(7×7)	π + .00126	HS	7×(7/(7+77/7))	e + .00394	HS
7	(7+.7/7)/.7–7	π + .00126	HS	7–77/(7+77/7)	e + .00394	HS
8	7+7–77/(7+7/77)	π – .000567	HS	.777/(((7+7)/7)/7)	e + .00122	HS
9	(7+77/777)/.7–7	π – .0000225	HS	(7+7)/7.7–(.7–7)/7	e – .000100	HS
10	((77+77)/7–7/777)/7	π – .0000225	HS	7/((7+(77+(7+7)/77)/7)/7)	e + .0000179	HS
11	7–(7×7–.77)/(7+77/(7+7))	π + .00000735	JD	(7+7)/(7/7+7×.77×.77)	e + .00000642	JD

Best Known Approximations Using 8's and Decimal Points

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	8	π + 4.858	HS	8	e + 5.282	HS
2	8/8	π – 2.142	HS	8/8	e – 1.718	HS
3	(8+8)/8	π – 1.142	HS	(8+8)/8	e – .718	HS
4	88/8–8	π – .142	HS	88/8–8	e + .282	HS
5	(88–8×8)/8	π – .142	HS	8/(88/8–8)	e – .0516	HS
6	(88+8/8)/8–8	π – .0166	HS	(8+8+8)/8.8	e + .00899	HS
7	(8+8)/8–8/(8/8–8)	π + .00126	HS	(8+88/(.8×8))/8	e + .000468	HS
8	(88+(8+8/8)/8)/8–8	π – .000968	HS	(.8+88/(8×8))/.8	e + .000468	HS
9	(88+8/(8×.88))/8–8	π + .000453	HS	(8+8)/8–(.8–8×8)/88	e – .000100	HS
10	(.8+88+8/(8+8+8))/8–8	π + .0000740	HS	(8+8+8)/8.8–8/888	e – .0000181	HS
11	88/8–8/(8–8×8–8/(8+8))–8	π + .000000267	JD	(8+8)/8.8+8/8.888	e – .0000100	JD

Best Known Approximations Using 9's and Decimal Points

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	9	π + 5.858	HS	9	e + 6.282	HS
2	9/9	π – 2.142	HS	9/9	e – 1.718	HS
3	(9+9)/9	π – 1.142	HS	(9+9)/9	e – .718	HS
4	(9+9+9)/9	π – .142	HS	(9+9+9)/9	e + .282	HS
5	(9+99)/9–9	π – .142	HS	.9+(9+9)/9	e + .182	JD
6	((9+9)+9/.9)/9	π – .0305	HS	(9+9+9)/9.9	e + .00899	HS
7	.9×(9–99/(9+9))	π + .00841	HS	.9+(9+9)/9.9	e – .000100	HS
8	(99/(9–99/(9+9)))/9	π + .00126	HS	.9+(99+9×9)/99	e – .000100	HS
9	(9+9/(9×9))/(.9+(9+9)/9)	π + .000170	HS	9/.9–9×(.9–9/99)	e – .000100	HS
10	99/(9–99/(9+9+99))–9	π – .0000832	HS	(9+9+9)/9.9–9/999	e – .0000181	HS
11	(9+99)/(9–(9–9/(9+9))/9/9)–9	π – .0000243	JD	.9+.99+(9×9+9/9)/99	e + .00000100	JD

Best Known Approximations Using Any Digits and Decimal Points

Complexity	Expression for π	Value	Author	Expression for e	Value	Author
1	3	π – .142	JD	3	e + .282	JD
2	3+0	π – .142	JD	8/3	e – .0516	JD
3	22/7	π + .00126	JD	19/7	e – .00400	JD
4	3+9/64	π – .000968	JD	3–9/32	e + .000468	JD
5	3+.85/6	π + .0000740	HS	3–2/7.1	e + .0000280	JD
6	5/(1+6/(9+8/7))	π + .000000267	JD	2+5/(7–3/77)	e + .00000175	HS
7	6/(5/6+8/(9+4))–1	π + .000000267	JD	2/77+35/13	e – .000000110	HS

Best Known Approximations Using 1's and Unlimited Decimal Points

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	1	π – 2.142	JD	1	e – 1.718	JD
2	1+1	π – 1.142	JD	1+1	e – .718	JD
3	1+1+1	π – .142	JD	1+1+1	e + .282	JD
4	1+1+1.1	π – .0416	JD	1+1+1–.1	e + .182	JD
5	1+1+1/(1–.1)	π – .0305	JD	(1+1+1)/1.1	e + .00899	JD
6	1/(.1+.1+.1)–.1–.1	π – .00826	JD	1+(.1+.1)/.11–.1	e – .000100	JD
7	11/((1–.1)/(.1+.1)–1)	π – .00126	JD	1×1+(.1+.1)/.11–.1	e – .000100	JD
8	(.1+(.1+.1+.1)/.11)/(1–.1)	π – .000179	JD	(.1+.1+.1/(1+.1×.1))/.11	e – .0000100	JD
9	1/(.1+.1+.1)–1/(11–.1)–.1	π – .00000244	JD	(.1+.1+.1/(1–.1))/.11–.11	e – .00000100	JD
10	1×1/(.1+.1+.1)–1/(11–.1)–.1	π – .00000244	JD	1–.1×(1–.1×.1×.1–(1+1)/.11)	e – .0000000103	JD

Best Known Approximations Using 2's and Unlimited Decimal Points

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	2	π – 1.142	JD	2	e – .718	JD
2	2+2	π + .858	JD	2.2	e – .518	JD
3	2+2/2	π – .142	JD	2+2/2	e + .282	JD
4	2+2/(2–.2)	π – .0305	JD	2+.2×2×2	e + .0817	JD
5	2/(.2+.2+.2)–.2	π – .00826	JD	2+(2–.2)×(.2+.2)	e + .00172	JD
6	22/((2–.2)/.2–2)	π + .00126	JD	2×(2+.2×(2–.2))–2	e + .000172	JD
7	.2+2/.2/(2×(2–.2)–.2)	π – .000416	JD	2+(2–.2)/(2–.2×.2)–.2	e + .0000855	JD
8	2+2/.2/.2/(2×22–.2)	π – .0000401	JD	(.2+.2/.22)/(2+.2×.2)/.2	e + .0000782	JD
9	.22×(.2+2/.2)×(2–.2–.2–.2)	π + .00000735	JD	2×(.2+2.2/(2–.2/(2–.2×.2)))	e – .00000226	JD

Best Known Approximations Using 3's and Unlimited Decimal Points

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	3	π – .142	JD	3	e + .282	JD
2	3.3	π + .158	JD	3–.3	e – .0183	JD
3	3+.3/3	π – .158	JD	3×3×.3	e – .0183	JD
4	3+.3/(3–.3)	π – .0305	JD	3×.3/.33	e + .00899	JD
5	3+.3/(3–3×.3)	π + .00126	JD	3+.3×.3×.3–.3	e + .00872	JD
6	3+.3/(3.–3–.3–.3)	π + .00126	JD	.3×3+(.3+.3)/.33	e – .000100	JD
7	3+3/(3+(3+3)/.33)	π + .0000382	JD	.3×(3/.33–.3×.3/3)	e – .00000910	JD
8	3/(.3–33)–(.3–3/.3)/3	π – .00000244	JD	.3/3×(3×3/.33–.3×.3)	e – .00000910	JD
9	(.3+3/(.3+(.3+3)/.3))/(.3+.3)/.3	π + .000000267	JD	.3×((3+.3/3)/.33–.333)	e – .0000000103	JD

Best Known Approximations Using 4's and Unlimited Decimal Points

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	4	π + .858	JD	4	e + 1.282	JD
2	4–.4	π + .458	JD	4–.4	e + .882	JD
3	4–.4–.4	π – .058	JD	4/4/.4	e – .218	JD
4	4–.4–.44	π + .0184	JD	.44/.4/.4	e – .0317	JD
5	4/(.4+.4×.4)–4	π + .00126	JD	4–.4×.4×(4+4)	e + .00172	JD
6	4/.4/(.4+4/4)–4	π + .00126	JD	4/.4/(4–.4×(.4+.4))	e – .000891	JD
7	4–.4–4.4/(4/.4–.4)	π + .0000740	JD	((4+44)/.4–.4)/44	e – .000100	JD
8	4–.44+4/(.44–4/.4)	π – .00000269	JD	4–4/(4–.4×.4)–.4+.4×.4	e + .0000515	JD
9	4–.44+.4×4/(.4×.44–4)	π – .00000269	JD	4/(.44–4/(.44/(4–.4)–4))	e – .00000159	JD

Best Known Approximations Using 5's and Unlimited Decimal Points

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	5	π + 1.85	JD	5	e + 2.28	JD
2	.5×5	π – .642	JD	.5×5	e – .218	JD
3	.5+5×.5	π – .142	JD	.5×5.5	e + .0317	JD
4	5.5+.55	π – .0916	JD	.5×(5–.5)+.5	e + .0317	JD
5	.5+.5×(5+.5×.5)	π – .0166	JD	.5+(5+5)/(5–.5)	e + .00394	JD
6	.5/(.5×.5–5/55)	π + .00126	JD	(5/.55+5–.5)/5	e – .000100	JD
7	.5×(5–5/(5.5/5–5))	π – .000567	JD	5–(.5+5×5×5)/55	e – .000100	JD
8	(.5+5/(5–.5))×(5×.5–.55)	π + .0000740	JD	5×(.5+5/(55/.5+5–.5))	e + .000000587	JD
9	5–(5+5+.5)/(.5/5+5+.55)	π + .000000267	JD	.5+5/(.5×(5–.5)+.5/5/5/5)	e – .00000321	JD
10	(5+5)/(.5×.5+5/(5+.5/(5–.5)))–5	π + .000000267	JD	((5–5/(5–.5))/.5/.55–.55)/5	e + .000000100	JD

Best Known Approximations Using 6's and Unlimited Decimal Points

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	.6	π – 2.54	JD	.6	e – 2.12	JD
2	.6×6	π + .458	JD	.6×6	e + .882	JD
3	.6×(6–.6)	π – .0984	JD	.6×6–.6	e + .282	JD
4	6/.66–6	π – .0507	JD	6–.6×(6–.6)	e + .0417	JD
5	.6×(6–.6)–.6/6	π – .00159	JD	6/(.6+.6+6/6)	e + .00899	JD
6	6×(6/(6–.66)–.6)	π – .0000196	JD	(6/.6/.6–.6×.6)/6	e – .000504	JD
7	6×6/(6–.66)–6×.6	π – .0000196	JD	6–(.6+6×6×6)/66	e – .000100	JD
8	6–.6×(.6+.6+.66×(6–.6))	π – .00000735	JD	.6×(6–.6–6/((6–.6)/6+6))	e – .0000210	JD
9	.6/(.6×.6+6/(.6–6×6–.6/6))	π + .000000267	JD	((.6+.6+6/.6)/.66–.66)/6	e + .00000100	JD

Best Known Approximations Using 7's and Unlimited Decimal Points

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	.7	π – 2.44	JD	.7	e – 2.02	JD
2	.7+.7	π – 1.74	JD	.7+.7	e – 1.32	JD
3	7/.7–7	π – .142	JD	7/.7–7	e + .282	JD
4	(7–.7)×.7×.7	π – .0546	JD	7×.7×.7–.7	e + .0117	JD
5	(7+.7/7)/.7–7	π + .00126	JD	7–(7/.7–7))/.7	e – .00400	JD
6	7/(.7+(.7+7/.7)/7)	π – .000567	JD	7/(.7+7/7)–.7–.7	e – .000635	JD
7	.7×.7×(7–.7+.7/(7–.7))	π – .000148	JD	((7+7)/.7–.7)/(7+.7/7)	e + .0000280	JD
8	(.7+7×7)/(7+(.7+.7)×(7–.7))	π + .000000267	JD	7/.7–7–(7+7)/(.7+7×7)	e + .0000280	JD
9	7/(7×(.7–(7+7)/(.7+7×7))–.7)	π + .000000267	JD	(7–.7)×(.7×.77–.7/(7–.7×.7))	e – .00000118	JD

Best Known Approximations Using 8's and Unlimited Decimal Points

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	.8	π – 2.34	JD	.8	e – 1.92	JD
2	.8+.8	π – 1.54	JD	.8+.8	e – 1.12	JD
3	.8+.8+.8	π – .742	JD	.8+.8+.8	e – .318	JD
4	.8+.8+.8+.8	π + .0584	JD	.8+(8+8)/8	e + .0817	JD
5	(8+8)/.8/.8/8	π – .0166	JD	.8+.8×(.8+.8+.8)	e + .00172	JD
6	8.8/(.8+8/.8–8)	π + .00126	JD	(.8+.88/.8/.8)/.8	e + .000468	JD
7	8–.88×(.8×8–.88)	π + .000807	JD	.8+(.8+.888)/.88	e – .000100	JD
8	(.88+.8+8/(.8+8.8))/.8	π + .0000740	JD	8–(8–.8)/(.8+.8×.8×.88)	e + .0000280	JD
9	.88×(.8+.88×8×8)/(8+8)	π + .00000735	JD	(8×8–.8)/((8+8)/.8/.8/.8–8)	e – .00000226	JD

Best Known Approximations Using 9's and Unlimited Decimal Points

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	.9	π – 2.24	JD	.9	e – 1.82	JD
2	.9+.9	π – 1.34	JD	.9+.9	e – .918	JD
3	.9+.9+.9	π – .442	JD	.9+.9+.9	e – .0183	JD
4	(.9+.9)×(.9+.9)	π + .0984	JD	.9+.9+.99	e + .0717	JD
5	.9+(9+9)/(9–.9)	π – .0194	JD	.9+(.9+.9)/.99	e – .000100	JD
6	(.9+.9)×(.9+.9)–.9/9	π – .00159	JD	.9+.9+9/(9+.9×.9)	e – .000851	JD
7	(9–(9/.9/.9/(.9+.9)))/.9	π – .000303	JD	.9+.9+9/.99/9.9	e – .00000818	JD
8	.9–((.9+9)/(.9×(.9+.9)–9)–.9)	π – .000129	JD	.9+.9+(9+.9/(.9+9))/(.9+9)	e – .00000818	JD
9	.9+(.99+9/(.9+9/.9))/.9/.9	π – .00000244	JD	.9+.9+9/(9+.9×.9–.9/99)	e + .000000334	JD

Best Known Approximations Using Any Digits and Unlimited Decimal Points

Digits	Expression for π	Value	Author	Expression for e	Value	Author
1	3	π – .142	JD	3	e + .282	JD
2	3.1	π – .0416	JD	2.7	e – .0183	JD
3	22/7	π + .00126	JD	2.72	e + .0172	JD
4	(.9+7/.6)/4	π + .0000740	JD	3–2/(8–.9)	e + .0000280	JD
5	1/(.6–2/(8–.9))	π + .000000267	JD	2.9/.93–.4	e – .00000226	JD

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.