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Alex. Lupas
science forum Guru Wannabe

Joined: 06 May 2005
Posts: 245

Posted: Tue May 24, 2005 11:54 pm    Post subject: Napier's constant,continued fractions,electrical networks...

Let M be a positive integer , M=< 100 . Using only M perfect one ohm

resistors, construct a resistance of E ohms , were E approximate ,, e"

(Napier's constant) with at least five decimals (only series-parallel

circuits are allowed).

See:
[1] Problem E2459 proposed by A.A.Mullin in Amer.Math.Monthly (1974)
[2] Problem 393 from Matematicki Vesnik 13(2(1976),solution in
15(1978).
John Bailey

Joined: 05 May 2005
Posts: 72

Posted: Sun May 29, 2005 10:06 am    Post subject: Re: Napier's constant,continued fractions,electrical networks...

On 24 May 2005 18:54:37 -0700, "Alex. Lupas"
<alexandru.lupas@ulbsibiu.ro> wrote:

 Quote: Let M be a positive integer , M=< 100 . Using only M perfect one ohm resistors, construct a resistance of E ohms , were E approximate ,, e" (Napier's constant) with at least five decimals (only series-parallel circuits are allowed).

Nice! Finally, using Excel, able to compute a resistor ladder which
mimics the continued fraction for e. The breakthrough was realizing
that many of the web given series for the continued fraction for e are
misleading if not wrong. After computing my own series, all came out
good.
John Bailey
Alex. Lupas
science forum Guru Wannabe

Joined: 06 May 2005
Posts: 245

Posted: Sun May 29, 2005 10:50 pm    Post subject: Re: Napier's constant,continued fractions,electrical networks...

John Bailey wrote:
 Quote: On 24 May 2005 18:54:37 -0700, "Alex. Lupas" alexandru.lupas@ulbsibiu.ro> wrote: Let M be a positive integer , M=< 100 . Using only M perfect one ohm resistors, construct a resistance of E ohms , were E approximate ,, e" (Napier's constant) with at least five decimals (only series-parallel circuits are allowed). Nice! Finally, using Excel, able to compute a resistor ladder which mimics the continued fraction for e. The breakthrough was realizing that many of the web given series for the continued fraction for e are misleading if not wrong. After computing my own series, all came out good. John Bailey

Thank you for interest!
How many resistors do you have used ?/Alex
John Bailey

Joined: 05 May 2005
Posts: 72

Posted: Mon May 30, 2005 9:30 am    Post subject: Re: Napier's constant,continued fractions,electrical networks...

On 29 May 2005 17:50:31 -0700, "Alex. Lupas"
<alexandru.lupas@ulbsibiu.ro> wrote:

 Quote: John Bailey wrote: On 24 May 2005 18:54:37 -0700, "Alex. Lupas" alexandru.lupas@ulbsibiu.ro> wrote: Let M be a positive integer , M=< 100 . Using only M perfect one ohm resistors, construct a resistance of E ohms , were E approximate ,, e" (Napier's constant) with at least five decimals (only series-parallel circuits are allowed). Nice! Finally, using Excel, able to compute a resistor ladder... How many resistors do you have used ?/Alex The ladder used 75 resistors. The next rung would have required 74

more so I had to quit. The result was: 2.718281755 The rungs were:
1,1,4(in parallel),1,1,8(in parallel),1,1,12(in parallel),1,3(in
parallel) and 1. The series resistors were 2,2,1,1,6,1,1,10,1,1,
1,1,2,3 and 1

John Bailey
phil kenny
science forum beginner

Joined: 02 Jun 2005
Posts: 1

Posted: Thu Jun 02, 2005 4:58 am    Post subject: Re: Napier's constant,continued fractions,electrical networks...

"Alex. Lupas" wrote:
 Quote: Let M be a positive integer , M=< 100 . Using only M perfect one ohm resistors, construct a resistance of E ohms , were E approximate ,, e" (Napier's constant) with at least five decimals (only series-parallel circuits are allowed). See: [1] Problem E2459 proposed by A.A.Mullin in Amer.Math.Monthly (1974) [2] Problem 393 from Matematicki Vesnik 13(2(1976),solution in 15(1978).

Here is a circuit which uses 22 resistors:

o--- 2 ---o---- 2 ---o---- 1 ---o---- 1 ----o---- 1 ----o---- 6 ----o
| | | | | |
| | | | | |
1 1 1/4 1 1 1
| | | | | |
| | | | | |
o---------o----------o----------o-----------o-----------o-----------o

The 1/4 ohm is realized by 4 parallel 1 ohm resistors.

The resistance as seen looking into the left-hand terminals is
1457/536 ohms or 2.71828358 ohms

It was generated using the continued fraction representation for e.

phil
Alex. Lupas
science forum Guru Wannabe

Joined: 06 May 2005
Posts: 245

 Posted: Thu Jun 02, 2005 8:00 am    Post subject: Re: Napier's constant,continued fractions,electrical networks... Nice rematrk /solution/ . It's 22 the minimal number of resistors ? /Thank you, Alex
phil
science forum beginner

Joined: 02 Jun 2005
Posts: 2

Posted: Thu Jun 02, 2005 12:44 pm    Post subject: Re: Napier's constant,continued fractions,electrical networks...

"Alex. Lupas" wrote:
 Quote: Nice rematrk /solution/ . It's 22 the minimal number of resistors ? /Thank you, Alex

Hello Alex,

Although I can't give a formal proof, I suspect that my solution
represents the minimum possible network.

o--- Z_1 ---o--- Z_3 ---o--- Z_5 ---o---- ....
| | |
| | |
Y_2 Y_4 Y_6
| | |
| | |
o-----------o-----------o---------o---- ....

It can be shown that the impedance, Z_eq, looking into the input
terminals of this network may be written as:

Z_eq =

1
Z_1 + ----------------------------------------
1
Y_2 + ----------------------------------
1
Z_3 + ---------------------------
1
y_4 + -------------------
1
Z_5 + -----------
Y_6 + ...

where Z_i is the impedance of the i_th series element and Y_i is
the admittance of the i_th shunt element. In this case we are dealing
with purely resistive components, so impedance and admittance become
resistance and conductance.

Notice that this expression for Z_eq has the identical form as a
continued fraction expansion.

Thanks for posing this problem.

phil
phil
science forum beginner

Joined: 02 Jun 2005
Posts: 2

Posted: Thu Jun 02, 2005 5:13 pm    Post subject: Re: Napier's constant,continued fractions,electrical networks...

phil kenny wrote:
 Quote: "Alex. Lupas" wrote: Let M be a positive integer , M=< 100 . Using only M perfect one ohm resistors, construct a resistance of E ohms , were E approximate ,, e" (Napier's constant) with at least five decimals (only series-parallel circuits are allowed). See: [1] Problem E2459 proposed by A.A.Mullin in Amer.Math.Monthly (1974) [2] Problem 393 from Matematicki Vesnik 13(2(1976),solution in 15(1978). Here is a circuit which uses 22 resistors: o--- 2 ---o---- 2 ---o---- 1 ---o---- 1 ----o---- 1 ----o---- 6 ----o | | | | | | | | | | | | 1 1 1/4 1 1 1 | | | | | | | | | | | | o---------o----------o----------o-----------o-----------o-----------o

The circuit I drew should have been:

A circuit which uses 20 resistors:

o--- 2 ---o---- 2 ---o---- 1 ---o---- 1 ----o---- 6 ----o
| | | | |
| | | | |
1 1 1/4 1 1
| | | | |
| | | | |
o---------o----------o----------o-----------o-----------o

An extra section accidently crept into my initial answer. Other
than that, the rest is correct (I hope).

 Quote: The 1/4 ohm is realized by 4 parallel 1 ohm resistors. The resistance as seen looking into the left-hand terminals is 1457/536 ohms or 2.71828358 ohms It was generated using the continued fraction representation for e. phil
Timothy Little
science forum Guru Wannabe

Joined: 30 May 2005
Posts: 295

Posted: Fri Jun 03, 2005 2:31 am    Post subject: Re: Napier's constant,continued fractions,electrical networks...

Alex. Lupas wrote:
 Quote: It's 22 the minimal number of resistors ?

No, there's a network of 16 resistors that yields e to 5 decimal
places (found by exhaustive search). Unfortunately I failed to
instruct the program to allocate space for the diagram records, and so
it crashed upon attempting to print the diagram.

I'll repost tomorrow with the actual network diagram. Unless somebody
else beats me to it.

- Tim

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