Problem of the Month (August 2004)

This month's problem is to approximate famous mathematical constants using only the first n digits (each used exactly once) and the mathematical symbols + 每 * / ( ) . and ^. For each n, what is the best approximation you can get to the following constants?

What other constants can you approximate well in this manner?

ANSWERS

Richard Sabey, Joseph DeVincentis, Jeremy Galvagni, Bill Clagett, Bertrand Winter, Berend Jan van der Zwaag, Ambrus Zsb芍n, and Gerrit de Blaauw sent answers this month, with Richard Sabey and Gerrit de Blaauw furnishing the vast majority of results.

Approximations of 污 using 1每n
n expression value error author
1 1 1. .42 EF
2 .1.2 .6 .053 BC
3 3每1/2 .5773 .00013 EF
4 (4每.3+.1)2 .57722 1.04E每5 RS
5 (4每3+.1.5)每2 .5772153 每2.72E每7 RS
6 (.6((.1+.4)每.3)每.2).5 .577215656 每8.38E每9 GB
7 3^(每.5每2^(每4每.7^(.1每6))) .577215666 1.21E每9 RS
8 (.4^((6+3*.8)^每7)+.1^.5)^每2 .5772156648 每8.10E每12 RS
9 .8^(.2674^9每1/3)每.5 .5772156648 每4.25E每12 RS

Approximations of e using 1每n
n expression value error author
1 1 1. 每1.7 EF
2 1+2 3. .28 EF
3 3每.1每.2 2.70 每.018 EF
4 2(.1+.3)每.4 2.71829 9.16E每6 RS
5 3.5+4每(.12) 2.718284 1.68E每6 GB
6 ((.4+6.2)*5每1)/3 2.71828180 每2.66E每8 BC
7 2^((.3+.1)^每.4每5^每7.6) 2.718281812 每1.57E每8 RS
8 (1+2^每76)^(4^38+.5) 2.718281828 3.96E每47 RS
9 (1+9每47*6)3285 2.718281828 每2.01E每18457734525360901453873570 RS

Approximations of 羽 using 1每n
n expression value error author
1 1 1. 每2.1 EF
2 1+2 3.0 每.14 EF
3 (1/.3)每.2 3.13 每.0082 BC
4 .4每(2.1).3 3.14156 每3.26E每5 GB
5 3+.2/(.4.1+.5) 3.141598 5.92E每6 BC
6 (4+6/(3(.2)(.1))).5 3.14159263 每1.38E每8 BC
7 3+1/(7+4(6每5每2)) 3.14159269 4.33E每8 RS
8 2^(5^.4)每.6每.1^(38/7) 3.141592654 5.15E每10 RS
9 2^5^.4每.6每(.3^9/7)^(.8^.1) 3.14159265359 6.60E每13 RS

Approximations of Feigenbaum's constant 4.66920160910299 using 1每n
n expression value error author
1 1 1. 每3.6 RS
2 1/.2 5. .33 BC
3 (.1)每2/3 4.64 每.027 RS
4 4*1+2/3 4.666 每.0025 RS
5 (.3每.1+.4)2/.5 4.66923 2.94E每5 RS
6 (.3每(.61.5)每.4)2 4.6692017 1.27E每7 GB
7 .2 + (4 + .5(每13/6)).7 4.66920161 8.59E每9 GB
8 4+.8^(3*.6)每.2^(7+.1^.5) 4.66920161 1.01E每8 RS
9 4+.8^(2*.9)每(7*6)^每3.15 4.669201608 每3.04E每10 RS

Corey Plover suggested the following variant: use the first n decimal digits of a constant instead of the digits 1 through n. He asks, for a given constant, what is the smallest n that gives a closer approximation than the trivial decimal expansion. Here are some answers.

constant digits expression value error author
污 .57721 (1+(7/.7)^每.5)^每2 .5772153 每2.72E每7 RS
e 2.71 (2每.1)/.7 2.714 每3.99E每3 RS
羽 3.141 (.4^.1)/.3+.1 3.1414 每1.14E每4 RS

This got me thinking. What is the best approximation to these constants using ANY n digits? Here's the best known:

Approximations of 污 using any digits
expression value error author
1 .6 .6 .023 AZ
2 3每.5 .5773 .00013 EF
3 3每1/2 .5773 .00013 EF
4 (1+.1^.5)^每2 .5772154 每2.72E每7 GB
5 5*.3^(.6^每(.8^每.6)) .5772154 每2.57E每9 GB
6 ((6每(.1^(每.7)))/((.1^(每.4))每.8)) .5772156642 每6.27E每10 AZ

Approximations of e using any digits
expression value error author
1 3 3. .28 EF
2 2.7 2.70 每.018 EF
3 2.4每.4 2.71829 9.16E每6 RS
4 (.2+6^.2)/.6 2.71828180 每2.66E每8 GB
5 (1+9每9)99 2.718281825 每3.50E每9 RS
6 (1+9每9)99+.5 2.718281828 1.50E每18 RS

Approximations of 羽 using any digits
expression value error author
1 3 3. 每.14 EF
2 .1每.5 3.16 .026 RS
3 .8每4+.7 3.1414 每1.86E每4 RS
4 25.4每.6 3.141596 3.72E每6 RS
5 .8^每(.1^每.1+5)每.9 3.14159268 3.36E每8 GB
6 ((7每(.3^(每.7)))^(.7^(.8^.8))) 3.141592656 每3.14E每9 AZ

Approximations of Feigenbaum's constant 4.66920160910299 using any digits
expression value error author
1 5 5. .33 GB
2 9^.7 4.656 每.014 GB
3 6^.86 4.668 3.50E每4 GB
4 9每6^(.8^.9) 4.669198 每3.33E每6 GB
5 (8^.7)^(.6^每.6每.3) 4.669201600 9.50E每10 GB
6 8^((.6^(.1每.7)每.3)*.7) 4.669201600 9.50E每10 GB

And this got Richard Sabey thinking. What is the best approximation to these constants using the digits 1 through n in that order? Here's the best known:

Approximations of 污 using 1每n in order
n expression value error author
1 1 1. .42 RS
2 .1^.2 .6 .053 RS
3 1.2^每3 .578 .0014 RS
4 (1每2^.3)/每.4 .5778 6.45E每4 RS
5 (1+.2^.3/4)*.5 .5771 每8.64E每5 RS
6 1/(2每.3+.4^.5每.6) .5772153 每2.72E每7 RS
7 (.1每(23^每.45每.6))^.7 .5772155 每7.13E每8 GB
8 ((.1+(2^(3^.4))*5*.6)^.7)/8 .57721567 1.34E每8 GB
9 .1^每(.2/(.3每4^每(.5^每((6*7)^每.8))每.9)) .5772156644 每5.01E每10 GB

Approximations of e using 1每n in order
n expression value error author
1 1 1. 每1.7 RS
2 1+2 3. .28 RS
3 1+2每.3 2.70 每.018 RS
4 .1+2+.3^.4 2.717 每4.80E每4 RS
5 每.1+(每.2+.3)^每.45 2.7183 1.01E每4 RS
6 (1+2)^(.3+.4^5+.6) 2.718284 2.30E每6 RS
7 每((.1每2)/.3^.4每.5+6/7) 2.71828185 2.17E每8 GB
8 ((.1^每.2)每((3每.4)^(.5每.6))/.7)^每.8 2.718281825 每2.77E每9 GB
9 (1+2^(每3*(4+5)))^(.6*.7+8^9) 2.718281826 每1.62E每9 RS

Approximations of 羽 using 1每n in order
n expression value error author
1 1 1. 每2.1 RS
2 1+2 3.0 每.14 RS
3 .1^(每.2每.3) 3.16 .020 RS
4 (.1+2^.3)^4 3.13 每.0018 RS
5 (.1/.2^.3^4)^每.5 3.1417 1.39E每4 RS
6 .1^(每2/3)+4.5每6 3.14158 每3.81E每6 RS
7 (1+2^每.3)^.4^(每.5/.6^.7) 3.1415924 每1.88E每7 RS
8 .1^(每2/3)+(4/.5)^每6每.7每.8) 3.14159264 每5.28E每9 GB
9 (.1+2^每(3每(4^每((.5+6^每7)^8))))*9 3.14159262 2.43E每8 GB

Approximations of Feigenbaum's constant 4.66920160910299 using 1每n in order
n expression value error author
1 1 1. 每3.6 RS
2 1/.2 5. .33 RS
3 (.1)每2/3 4.64 每.027 RS
4 1*2/3+4 4.666 每.0025 RS
5 .1^.2*(.3每4)/每.5 4.6690 每1.17E每4 RS
6 1/(.2每3^每4)每.5^.6 4.6691 每8.19E每6 RS
7 .12*(.3+(4+.5^每.6)*7) 4.6692019 3.07E每7 GB
8 .1^每(2/((3/4.5)^(6每.7每8))) 4.669201630 2.04E每8 GB
9 1+2^((3每.4*(.5每.6^7)每.8)^.9) 4.669201613 4.02E每9 GB

Richard Sabey also sent results for expressions without using the decimal point, both in any order and in ascending order. Joseph DeVincentis sent results that use the digits 0每n. And thanks to Berend Jan van der Zwaag for finding a bunch of typos.

In 2008, I extended these results by considering the best approximations to various constants using n copies of a given digit. Here are the best results.

Best Approximations to 羽 with n Copies of the Digit k
n \ k 2 3 4 5 6 7
1 1 + 1
= 2 		1 + 1 + 1
= 3 		(. 1)每1/(1+1)
= 3.162 		1 + (1 + 1)1.1
= 3.143 		.1^-(.1+(.1^.1)/(1+1))
= 3.1416 (GB) 		1.1^(11+(1-.1)^-.1)
= 3.141598 (GB)
2 2 + 2
= 4 		2 + 2(. 2)
= 3.149 		22(. 2)(. 2)
= 3.144 		2 + 2 每 2每(. 22)
= 3.1414 		2*(.2+.2+2.2^.2)
= 3.1416 (GB) 		2^(2-.2^-(.2^.2-.2^-.2))
= 3.14158 (GB)
3 3.3
= 3.3 		3 + (. 3)/3
= 3.1 		333(. 3)
= 3.140 		3 ℅ [(3 每 . 3)(. 3) 每 . 3]
= 3.1413 		.3^-((3-.3)^-((3-.3)^-3))
= 3.14157 (GB) 		3/(.33/(.3-3^-3)-.3)
= 3.1415929 (GB)
4 4 每 (. 4)
= 3.6 		4 每 (. 4) 每 (. 4)
= 3.2 		(. 4)每4(. 4)℅(. 4)
= 3.139 		4 每 4每(. 44)/4
= 3.1414 		((.4+.4)^-(.4^-(.4^4)))/.4
= 3.141594 (GB) 		(4-4/(.4-.4*4^.4))^.4
= 3.141593 (GB)
5 5 ℅ (. 5)
= 2.5 		(5 + 5)(. 5)
= 3.162 		5 / [5 ℅ (. 5)](. 5)
= 3.162 		5 每 (. 5)每[5每(. 5)/(. 5)]
= 3.1411 		.5+(5^.5)^(.5+.5^.5)
= 3.141591 (GB) 		.5+.5+.5^-((.5+.5^.5)^.5)
= 3.14159268 (GB)
6 6(. 6)
= 2.930 		6.6(. 6)
= 3.103 		[6 + (. 6)(. 6)](. 6)
= 3.140 		(. 6)[6 ℅(. 6) 每(. 6)每(. 6)]
= 3.142 		6*6*.666^6
= 3.1415 (GB) 		(6^.6)/(.6^-(6^-(.6/6))-.6)
= 3.141593 (GB)
7 7(. 7)
= 3.905 		7 每 7(. 7)
= 3.095 		[(. 7)每7 每 7](. 7)
= 3.147 		77/[7+7℅(. 7)]
= 3.1413 		(7*7)^((7*.7)^-.77)
= 3.14158 (GB) 		7^-(.7-7^(.7^(7-.7^-.7)))
= 3.1415927 (GB)
8 (. 8) + (. 8)
= 1.6 		(. 8)每8(. 8)
= 3.247 		(. 8)每8℅(. 8)℅(. 8)
= 3.135 		8(. 8) / [(. 8) + (. 88)]
= 3.1416 		(8/((.8+.8)/.8^.8))^.8
= 3.1416 (GB) 		(8^-((.8+.8)^-(8^(.8^.8))))^-8
= 3.1415929 (GB)
9 (. 9)每9
= 2.581 		(. 9)每9/(. 9)
= 2.868 		9 ℅ (. 9) ℅ (. 9)9
= 3.138 		(. 9) ℅ [(. 9)(. 9) + (. 9)每9]
= 3.1416 		(9-.99*.9)*.9^9
= 3.1415927 (GB) 		(.9-.9*(.99-9))*.9^9
= 3.1415927 (GB)

Best Approximations to e with n Copies of the Digit k
n \ k 2 3 4 5 6 7
1 1 + 1
= 2 		1 + 1 + 1
= 3 		1 + 1 + (. 1)(. 1)
= 2.794 		(1 + 1) ℅ [(. 1) + (. 1)每(. 1)]
= 2.717 		1-.1+(.1+.1)/.11
= 2.7181 (GB) 		(1+(.1+.1)^-.1)/(1-.1-.1)
= 2.71827 (GB)
2 2 + (. 2)
= 2.2 		2 + (. 2)(. 2)
= 2.725 		(. 2)[(.2)每(. 2)每2]
= 2.713 		2[2℅(. 2)]每2℅(. 2)
= 2.71829 		2^(((2^.2)*.2^.2)^-2)
= 2.71829 (GB) 		2+(2*(2+2^-2))^-.22
= 2.71827 (GB)
3 3 每 (. 3)
= 2.7 		3 ℅ 3 ℅ (. 3)
= 2.7 		3 ℅ 3每(. 3)℅(. 3)
= 2.717 		[(. 3)每3 每 3 ℅ 3](. 3)
= 2.7184 		(3-.3+.3^-3)^.3-.3
= 2.71827 (GB) 		3-.3^(.333+3^-.3)
= 2.7182817 (GB)
4 4 每 (. 4)
= 3.6 		4 ℅ (. 4)(. 4)
= 2.772 		(. 4)每[(. 4)+(. 4)(. 4)]
= 2.722 		[(4 + 4) / 4](. 4)每(. 4)
= 2.71829 		.4+4^(4^-((.4^-.4)/4))
= 2.718287 (GB) 		4^((4-.4/4)^-(.4-.4*.4))
= 2.7182815 (GB)
5 5 ℅ (. 5)
= 2.5 		(. 5) + 5(. 5)
= 2.736 		55(. 5)℅(. 5)
= 2.723 		5 ℅ (. 55) 每 (. 5)5
= 2.7187 		.5^-((5*.5)^(.5-.5/5))
= 2.71829 (GB) 		(.5+.5+5^-5)^(.5+5^5)
= 2.71828185 (GB)
6 6(. 6)
= 2.930 		[6 每 (. 6)](. 6)
= 2.751 		(. 6)每(. 6) + (. 6)每(. 6)
= 2.717 		(. 6)每[(. 6)℅6(. 66)]
= 2.7183 		(6/6+6^-6)^(6^6)
= 2.71825 (GB) 		(.6+.6+6^(.6+.6))/(6*.6)
= 2.71828180 (GB)
7 7(. 7)
= 3.905 		(. 7) / 7每(. 7)
= 2.733 		7.7(. 7)℅(. 7)
= 2.7188 		7[(. 77)每7每(. 7)]
= 2.71822 		.7^-((7+7*7)^(7^-.7))
= 2.7183 (GB) 		(.7^-((7+7)^(7^-.77)))/.7
= 2.7182817 (GB)
8 (. 8) + (. 8)
= 1.6 		8 每 8(. 8)
= 2.722 		(. 8)[(. 8)每8(. 8)]
= 2.716 		(. 8) + 8 ℅ (. 8)8℅(. 8)
= 2.7180 		.8*(.8+.8^-8)^(.8*.8)
= 2.7181 (GB) 		(8/(8-8^-8))^(8/8^-8)
= 2.71828183 (GB)
9 (. 9)每9
= 2.581 		(. 9) + (. 9) + (. 9)
= 2.7 		9每(. 9) + (. 9)每9
= 2.719 		(. 9) + (. 9) + (. 9)(. 9)℅(. 9)
= 2.7181 		9*(.9+.9)-(9+9)^.9
= 2.7182815 (GB) 		(.9/(.9+9^-9))^-(.9/9^-9)
= 2.718281824 (GB)

Best Approximations to 污 with n Copies of the Digit k
n \ k 2 3 4 5 6 7
1 (. 1)(. 1)
= .794 		[(. 1) ℅ (. 1)](. 1)
= .630 		(1 + 1)每(. 1)(. 1)
= .576 		(. 1) / (. 11)(. 1)(. 1)
= .5773 		.1*((.1+.1)^-1.1-.1)
= .5773 (GB) 		(1+.1^(1/(1+1)))^-(1+1)
= .57723 (GB)
2 (. 2)(. 2)
= .724 		2(. 2) / 2
= .574 		2 每 (. 2)每(. 22)
= .575 		[(. 2) / (2 每 (. 2))]2每2
= .5773 		.2+.2^(.2^(.2^(.2^.2)))
= .577218 (GB) 		2^-((.2^-.2-.2^-(.2^2))^.2)
= .5772151 (GB)
3 (. 3) + (. 3)
= .6 		(3 + 3)每(. 3)
= .584 		3每3/[3+3]
= .5773 		(. 3)每(. 3) 每 [(. 3) + (. 3)](. 3)
= .5771 		3*(.3^(.3^.3))-3^-.3
= .57722 (GB) 		.3/.3^((3+(.3+.3)^-3)^-.3)
= .5772157 (GB)
4 4每(. 4)
= .574 		(. 4) / (. 4)(. 4)
= .5770 		(. 4)4℅(. 4) / (. 4)
= .5770 		[(. 4)(. 4) 每 (. 44)](. 4)
= .57722 		.4*(.4+(4/(4-.4))^.4)
= .577217 (GB) 		(44+4)^-((4-.4)^-4)-.4
= .5772154 (GB)
5 (. 55)
= .55 		(. 555)
= .555 		[5 ℅ (. 5) + (. 5)]&ndash(. 5)
= .5773 		(. 5) / (. 5)[(. 5)(. 5)每(. 5)]
= .5771 		.5^(.5+(.5^-.55)/5)
= .5772158 (GB) 		.5/(.5+5^-((5^(.5^.5))/5))
= .5772158 (GB)
6 (. 66)
= .66 		6每(. 6) / (. 6)
= .569 		[6 / 6 每 (. 6)](. 6)
= .5770 		[66每6 每 (. 6)](. 6)
= .5771 		.6-(6^.6)*((.6^6)/6)
= .5772151 (GB) 		(.6+(6-(.6^-.6)/6)^.6)/6
= .5772158 (GB)
7 (. 7) ℅ (. 7)
= .49 		7 ℅ 7(. 7)
= .576 		[(. 7) ℅ (. 7)](. 77)
= .5773 		(. 7) 每 [(. 7) / (7 + 7)](. 7)
= .5771 		(.7^(.7*(.7+.7)))^-.7-.7
= .577211 (GB) 		.7-((7-.7)^-(7^-(.7^-.7)))/7
= .5772158 (GB)
8 (. 8) ℅ (. 8)
= .64 		(. 8) 每 8每(. 8)
= .610 		8每8每(. 8)℅(. 8)
= .57723 		[(. 8) ℅ (. 8)]8(. 8)/8
= .57727 		(.8*8^(.8^8.8))^-8
= .577216 (GB) 		((8^-((.8^8)*(.8^.8)))/.8)^8
= .577216 (GB)
9 (. 9)9
= .387 		[(. 9) + (. 9)]每(. 9)
= .589 		[(. 9) ℅ (. 9)](. 9)每9
= .580 		[(. 9) + 9每(. 9)] / [(. 9) + (. 9)]
= .576 		(9^(.9-.9^(9+9)))/9
= .57723 (GB) 		(9-.9/9)/(9+9-.9^-9)
= .577216 (GB)

Best Approximations to Feigenbaum's Constant with n Copies of the Digit k
n \ k 2 3 4 5 6 7
1 1 + 1
= 2 		1 / [(. 1) + (. 1)]
= 5 		1 / [(. 11) + (. 1)]
= 4.762 		(1 + 1)每1.1 / (. 1)
= 4.665 		.1^-((.1^.1-.1)^1.1)
= 4.67 (GB) 		.1+.1+1/(1-.1^.11)
= 4.66921 (GB)
2 2 + 2
= 4 		22.2
= 4.594 		2每(. 2)/2 / (. 2)
= 4.665 		2 + 2 + (. 2)2每2
= 4.668 		.2+(2*2^.2)^(2-.2)
= 4.6691 (GB) 		.2^-(.2^(2^-(2+.2^-(.2^.2))))
= 4.669203 (GB)
3 3 + 3
= 6 		3(. 3) / (. 3)
= 4.634 		(. 33)每3(. 3)
= 4.671 		3[3 + (. 3) ℅ (. 3)](. 3)
= 4.6697 		3*(.3+.3+.3^(3^-3))
= 4.6691 (GB) 		.3^-((.3+.3)^(.3^3))+3^.3
= 4.6692014 (GB)
4 4 + (. 4)
= 4.4 		4 + (. 4)(. 4)
= 4.693 		4 + (. 4)(. 44)
= 4.668 		4 / [(. 4) + (. 4)](. 4)(. 4)
= 4.6690 		4+.4^(.4/(.4^(.4/4)))
= 4.6691 (GB) 		.44-(4/.4^.4-4/.4)
= 4.669200 (GB)
5 5 每 (. 5)
= 4.5 		(. 5)每5(. 5)
= 4.711 		5 ℅ (. 5)(. 5)/5
= 4.665 		[5 每 (. 5)每(. 5)每(. 5)] / (. 5)
= 4.6697 		5-(5^-(.5*5^.5))/.5
= 4.669205 (GB) 		.5^5+5.5^((5-.5)/5)
= 4.669203 (GB)
6 6 每 (. 6)
= 5.4 		6 每 (. 6)每(. 6)
= 4.641 		6 每 (. 6) 每 (. 6)(. 6)
= 4.663 		6 每 6(. 6)6℅(.6)
= 4.670 		6^(6/(6+.6^(.6^6)))
= 4.6691 (GB) 		6*(.6^((6-.6)/66))^6
= 4.66921 (GB)
7 7 ℅ (. 7)
= 4.9 		(. 7) + 7(. 7)
= 4.604 		(. 77) + 7(. 7)
= 4.674 		(. 7) ℅ [7 每 (7℅(. 7))每(. 7)]
= 4.6698 		7^(.7^(.7^(.7+.7*.7)))
= 4.6691 (GB) 		((.77+.7^.7)/(7+7))^-.7
= 4.6692019 (GB)
8 8(. 8)
= 5.278 		(. 8) / (. 8)8
= 4.768 		(. 88) / 8每(. 8)
= 4.644 		[8 / (. 8)](. 8)℅(. 8)(. 8)
= 4.668 		.8*(8/(.8+.8)+.8^.8)
= 4.669209 (GB) 		8/(8^.8-.8^-(8.8^.8))
= 4.6691 (GB)
9 (. 9)每9
= 2.581 		9 / [(. 9) + (. 9)]
= 5 		[(. 9) + (. 9)] / 9(. 9)
= 4.646 		(. 9) + (. 9) + (. 9)每9 / (. 9)
= 4.667 		9^.9-.99*.9^-9
= 4.6693 (GB) 		.9/(.99+.9-(.9+.9)^.9)
= 4.669207 (GB)

If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 5/15/08.