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Is There Something to CS Theory?. Alan Kaylor Cline July 10, 2009. Can one do computer science without doing programming?. Can one do computer science without doing programming?. No and Yes. Problem: How many games are required to determine the winner of - PowerPoint PPT Presentation
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Is There Something to CS Theory?
Alan Kaylor ClineJuly 10, 2009
Can one do computer science without doing programming?
Can one do computer science without doing programming?
No and Yes
Problem:
How many games are required to determine the winner of
a single elimination tournament with 18 teams?
Single Elimination Tournament: a tournament in which each game involves exactly two teams and after which exactly one is the winner of the game
and one is the loser. The tournament proceeds until all but one team has lost.
The Pigeonhole Principle
The Pigeonhole PrincipleStatement
Children’s Version: “If k > n, you can’t stuff k pigeons in n holes
without having at least two pigeons in the same hole.”
The Pigeonhole PrincipleStatement
Children’s Version: “If k > n, you can’t stuff k pigeons in n holes
without having at least two pigeons in the same hole.”
Smartypants Version: “No injective function exists mapping a set of higher cardinality into a set of
lower cardinality.”
The Pigeonhole PrincipleExample
Twelve people are on an elevator and they exit on ten different
floors. At least two got of on the same floor.
The ceiling function:
For a real number x, the ceiling(x) equals the smallest integer greater than or equal to x
Examples:
ceiling(3.7) = 4
ceiling(3.0) = 3
ceiling(0.0) = 0
If you are familiar with the truncation function, notice that
the ceiling function goes in the opposite direction –
up not down.If you owe a store 12.7 cents and they make you pay 13 cents,they have used the ceiling function.
The Extended (i.e. coolguy)Pigeonhole Principle
Statement
Children’s Version: “If you try to stuff k pigeons in n holes there must be at least ceiling (n/k)
pigeons in some hole.”
The Extended (i.e. coolguy)Pigeonhole Principle
Statement
Children’s Version: “If you try to stuff k pigeons in n holes there must be at least ceiling (n/k)
pigeons in some hole.”Smartypants Version: “If sets A and B are
finite and f:A B, then there is some element b of B so that cardinality(f -1(b))
is at least ceiling (cardinality(B)/ cardinality(A).”
The Extended (i.e. coolguy)Pigeonhole Principle
Example
Twelve people are on an elevator and they exit on five different
floors. At least three got off on the same floor.
(since the ceiling(12/5) = 3)
The Extended (i.e. coolguy)Pigeonhole Principle
Example
Twelve people are on an elevator and they exit on five different
floors. At least three got off on the same floor.
(since the ceiling(12/5) = 3)Example of even cooler
“continuous version”
If you travel 12 miles in 5 hours, you must have traveled at least 2.4
miles/hour at some moment.
Application 1: Among any group of six acquaintances
there is either a subgroup of three mutual friends or three mutual enemies.
Application 1: Among any group of six acquaintances
there is either a subgroup of three mutual friends or three mutual enemies.
Application 1: Among any group of six acquaintances
there is either a subgroup of three mutual friends or three mutual enemies.
FF
F
Application 1: Among any group of six acquaintances
there is either a subgroup of three mutual friends or three mutual enemies.
E
E
E
Application 1: Among any group of six acquaintances
there is either a subgroup of three mutual friends or three mutual enemies.
Application 1: Among any group of six acquaintances
there is either a subgroup of three mutual friends or three mutual enemies.
How would you solve this?
You could write down every possibleacquaintanceship relation.
Application 1: Among any group of six acquaintances
there is either a subgroup of three mutual friends or three mutual enemies.
How would you solve this?
You could write down every possibleacquaintanceship relation.
There are 15 pairs of individuals.
Each pair has two possibilities: friends or enemies.
That’s 215 different relations.
Application 1: Among any group of six acquaintances
there is either a subgroup of three mutual friends or three mutual enemies.
How would you solve this?
You could write down every possibleacquaintanceship relation.
There are 15 pairs of individuals.
Each pair has two possibilities: friends or enemies.
That’s 215 different relations.
By analyzing one per minute, you could prove this in 546 hours.
Application 1: Among any group of six acquaintances
there is either a subgroup of three mutual friends or three mutual enemies.
Could the pigeonhole principle be applied to this?
Application 1: Among any group of six acquaintances
there is either a subgroup of three mutual friends or three mutual enemies.
Could the pigeonhole principle be applied to this?
I am glad you asked.Yes.
Begin by choosing one person:
*
*
Begin by choosing one person:
*
Five acquaintances remain
These five must fall into two classes:friends and enemies
*
Begin by choosing one person:
*
Five acquaintances remain
These five must fall into two classes:friends and enemies
The extended pigeonhole principle
says that at least three must be in the same class -
that is: three friends or three enemies
*
*
?
??
Suppose the three are friends of :*
??
Either at least two of the three are friends of each other…
*
Suppose the three are friends of :*
??
Either at least two of the three are friends of each other…
In which case we have three mutual friends.
*
Suppose the three are friends of :*
Suppose the three are friends of :
*
Either at least two of the three are friends of each other…or none of the three are friends
*
Suppose the three are friends of :
*
Either at least two of the three are friends of each other…or none of the three are friends
In which case we have three mutual enemies.
*
Similar argument if we suppose the three are enemies of .
?
??
*
*
Application 2: Given twelve coins – exactly eleven of which have equal weight
determine which coin is different and whether it is heavy or light in a minimal number of weighings using a
three position balance.
H
OUR SOLUTION
1 1
1
1 1
1
5
5
5 5
5
5 22
2
2 2
6
6
6
6
6
63
3 3
3 3
7
7 7
7
7
7
4
4 4
4 4
8
8
8
8
8
8
99
9
9
9
1212
12
11
1111
10
10
1010
10
H HH HHH HH HH H HL LL L L LL L LLL L
1
1
2
2
3
3
4
Application 3: In any sequence of n2+1 distinct integers, there is
a subsequence of length n+1 that is either strictly increasing or strictly decreasing
n=2: 3,5,1,2,4 2,3,5,4,1
n=3: 2,5,4,6,10,7,9,1,8,3 10,1,6,3,8,9,2,4,5,7
n=4: 7,9,13,3,22,6,4,8,25,1,2,16,19,26,10,12,15,20,23,5,24,11,14,21,18,17
Application 3: In any sequence of n2+1 distinct integers, there is
a subsequence of length n+1 that is either strictly increasing or strictly decreasing
n=2: 3,5,1,2,4 2,4,5,3,1
n=3: 2,5,4,6,10,7,9,1,8,3 10,1,6,3,8,9,2,4,5,7
n=4: 7,9,13,3,22,6,4,8,25,1,2,16,19,26,10,12,15,20,23,5,24,11,14,21,18,17
Application 3: In any sequence of n2+1 distinct integers, there is
a subsequence of length n+1 that is either strictly increasing or strictly decreasing
Idea: Could we solve this by considering cases?
For sequences of length 2: 2 cases
For sequences of length 5: 120 cases For sequences of length 10: 3,628,800 cases
For sequences of length 17: 3.6 1014 cases
For sequences of length 26: 4.0 10 26 cases
For sequences of length 37: 1.4 10 43 cases
10 m-15
The Limits of Computation
Speed: speed of light = 3 10 8 m/s
Distance: proton width = 10 –15 m
With one operation being performed inthe time light crosses a proton
there would be 3 1023 operations per second.
Compare this with current serial processor speeds of 6 1011 operations per second
The Limits of Computation
With one operation being performed inthe time light crosses a proton
there would be 3 1023 operations per second.
Big Bang: 14 Billion years ago… that’s 4.4 1017 seconds ago
So we could have done 1.3 1041
operations since the Big Bang.
So we could not have proved this (using enumeration)even for the case of subsequences of length 7 from
sequences of length 37.
But with the pigeon hole principle we can prove it in two minutes.
We can apply the pigeon hole principle to a sorting
problem
We can apply the pigeon hole principle to a sorting
problem
How many comparisons are required to sort n distinct
elements?
Think of a sorting algorithm as determining in which permutation that an array is presented
Think of a sorting algorithm as determining in which permutation that an array is presented
The array: 2, -5, 6, 0, 12
is presented as the permutation:3, 1, 4, 2, 5
Think of a sorting algorithm as determining in which permutation that an array is presented
The array: 2, -5, 6, 0, 12
is presented as the permutation:3, 1, 4, 2, 5
Thus the problem of sorting is equivalent to
determining in which of the n! permutations an array is presented
In terms of functions, the sorting algorithm is a function with n! possible different inputs that must act differently on each
In terms of functions, the sorting algorithm is a function with n! possible different inputs that must act differently on each
Think of the various permutations as leaves at the bottomof a binary tree with comparisons at the internal nodes of the tree.
A tree of height k (i.e. with at most k comparisons to reach any leaf)
has at most 2k leaves.
In terms of functions, the sorting algorithm is a function with n! possible different inputs that must act differently on each
Think of the various permutations as leaves at the bottomof a binary tree with comparisons at the internal nodes of the tree.
A tree of height k (i.e. with at most k comparisons to reach any leaf)
has at most 2k leaves.
Thus, we must have2 !k n
In terms of functions, the sorting algorithm is a function with n! possible different inputs that must act differently on each
Think of the various permutations as leaves at the bottomof a binary tree with comparisons at the internal nodes of the tree.
A tree of height k (i.e. with at most k comparisons to reach any leaf)
has at most 2k leaves.
Thus, we must have2 !k n or
equivalently
2log ( !)k n
We have shown that the minimum number of comparisons for a comparison based sorting algorithm is
2log ( !)n
We have shown that the minimum number of comparisons for a comparison based sorting algorithm is
2log ( !)n
Have you seen this presented as
2log ( )?n n
We have shown that the minimum number of comparisons for a comparison based sorting algorithm is
2log ( !)n
Have you seen this presented as
2log ( )?n n
For large values of n these are close, but they are certainly not identical
2log ( !)n vs. 2log ( )?n n vs. mergesort
Problem: What is the volume of the largest
sphere that can be fit between a unit radius cylinder
adjoining a floor and a wall?
Radius is
2 1
2 1so the volume is
34 2 13 2 1
Problem: What is the volume of the largest
sphere that can be fit between a unit radius cylinder
adjoining a floor and a wall?
How should we calculate
32 1
2 1
if we use 2 1.41?
How should we calculate
32 1
2 1Some alternatives:
expression computed as relative error
double precision error multiple
0.00491 2.51% 103.
0.00475 5.95% 137.
0.00583 15.47% 339.
0.30000 5839.85% 71,791.
0.00510 1.05% 35.
0.00493 0.44% 18.
0.00505 0.15% 1.
32 1
2 1
6
2 1
3
3 2 2
99 70 2
6
1
2 1
3
1
3 2 2
1
99 70 2
if we use 2 1.41?
How should we calculate
32 1
2 1?
Some alternatives:expression computed as relative
errordouble precision error multiple
0.00491 2.51% 103.
0.00475 5.95% 137.
0.00583 15.47% 339.
0.30000 5839.85% 71,791.
0.00510 1.05% 35.
0.00493 0.44% 18.
0.00505 0.15% 1.
32 1
2 1
6
2 1
3
3 2 2
99 70 2
6
1
2 1
3
1
3 2 2
1
99 70 2
How should we calculate
32 1
2 1?
Some alternatives:expression computed as relative
errordouble precision error multiple
0.00491 2.51% 103.
0.00475 5.95% 137.
0.00583 15.47% 339.
0.30000 5839.85% 71,791.
0.00510 1.05% 35.
0.00493 0.44% 18.
0.00505 0.15% 1.
32 1
2 1
6
2 1
3
3 2 2
99 70 2
6
1
2 1
3
1
3 2 2
1
99 70 2
How should we calculate
32 1
2 1?
Some alternatives:expression computed as relative
errordouble precision error
multiple
0.00491 2.51% 103.
0.00475 5.95% 137.
0.00583 15.47% 339.
0.30000 5839.85% 71,791.
0.00510 1.05% 35.
0.00493 0.44% 18.
0.00505 0.15% 1.
32 1
2 1
6
2 1
3
3 2 2
99 70 2
6
1
2 1
3
1
3 2 2
1
99 70 2
An easy way to approximate pi is to construct a sequence of inscribed
regular polygons of 2n sides
We have:
2 2 2p
and for n = 2, 3, … :
21 2 2(1 1 ( / 2 ) )n n
n np p
2 2 2.2.828427124746190828427124746190
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
29 29 4.0000000000000004.000000000000000
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
29 29 4.0000000000000004.000000000000000
30 30 0.0000000000000000.000000000000000
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
29 29 4.0000000000000004.000000000000000
30 30 0.0000000000000000.000000000000000
31 31 0.0000000000000000.000000000000000
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
29 29 4.0000000000000004.000000000000000
30 30 0.0000000000000000.000000000000000
31 31 0.0000000000000000.000000000000000
32 32 0.0000000000000000.000000000000000
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459416548490545941
6 6 3.143.1403311569547390331156954739
7 7 3.1413.141277250932757277250932757
8 8 3.14153.14151380114414513801144145
9 9 3.14153.14157294036788372940367883
10 10 3.141583.1415877252799617725279961
11 11 3.141593.1415914215046351421504635
12 12 3.1415923.141592345611077345611077
13 13 3.14159253.14159257654500476545004
14 14 3.14159263.14159263346324833463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
29 29 4.0000000000000004.000000000000000
30 30 0.0000000000000000.000000000000000
31 31 0.0000000000000000.000000000000000
32 32 0.0000000000000000.000000000000000
0?
Let’s use this
algebraically equivalentversion of the formula for :
2 2 2p
and for n = 2, 3, … :
1np
21 2/ (1 1 ( / 2 ) )n
n n np p p
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459465484905459411
6 6 3.143.1403311569547033115695473939
7 7 3.1413.141277250932727725093275757
8 8 3.14153.14151380114413801144145145
9 9 3.14153.14157294036772940367883883
10 10 3.141583.1415877252777252799619961
11 11 3.141593.1415914215142150463504635
12 12 3.1415923.141592345634561107711077
13 13 3.14159253.14159257657654500445004
14 14 3.14159263.14159263334632483463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
29 29 4.0000000000000004.000000000000000
30 30 0.0000000000000000.000000000000000
31 31 0.0000000000000000.000000000000000
32 32 0.0000000000000000.000000000000000
2.2.828427124746190828427124746190
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459465484905459411
6 6 3.143.1403311569547033115695473939
7 7 3.1413.141277250932727725093275757
8 8 3.14153.14151380114413801144145145
9 9 3.14153.14157294036772940367883883
10 10 3.141583.1415877252777252799619961
11 11 3.141593.1415914215142150463504635
12 12 3.1415923.141592345634561107711077
13 13 3.14159253.14159257657654500445004
14 14 3.14159263.14159263334632483463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
29 29 4.0000000000000004.000000000000000
30 30 0.0000000000000000.000000000000000
31 31 0.0000000000000000.000000000000000
32 32 0.0000000000000000.000000000000000
2.2.828427124746190828427124746190
3.03.06146745892071961467458920719
..
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459465484905459411
6 6 3.143.1403311569547033115695473939
7 7 3.1413.141277250932727725093275757
8 8 3.14153.14151380114413801144145145
9 9 3.14153.14157294036772940367883883
10 10 3.141583.1415877252777252799619961
11 11 3.141593.1415914215142150463504635
12 12 3.1415923.141592345634561107711077
13 13 3.14159253.14159257657654500445004
14 14 3.14159263.14159263334632483463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
29 29 4.0000000000000004.000000000000000
30 30 0.0000000000000000.000000000000000
31 31 0.0000000000000000.000000000000000
32 32 0.0000000000000000.000000000000000
2.2.828427124746190828427124746190
3.03.06146745892071961467458920719
3.13.12144515225805321445152258053
..
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459465484905459411
6 6 3.143.1403311569547033115695473939
7 7 3.1413.141277250932727725093275757
8 8 3.14153.14151380114413801144145145
9 9 3.14153.14157294036772940367883883
10 10 3.141583.1415877252777252799619961
11 11 3.141593.1415914215142150463504635
12 12 3.1415923.141592345634561107711077
13 13 3.14159253.14159257657654500445004
14 14 3.14159263.14159263334632483463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
29 29 4.0000000000000004.000000000000000
30 30 0.0000000000000000.000000000000000
31 31 0.0000000000000000.000000000000000
32 32 0.0000000000000000.000000000000000
2.2.828427124746190828427124746190
3.03.06146745892071961467458920719
3.13.12144515225805321445152258053
3.133.1365484905459406548490545940
3.143.1403311569547530331156954753
3.1413.141277250932773277250932773
3.14153.14151380114430113801144301
3.14153.14157294036709172940367091
3.141583.1415877252771607725277160
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459465484905459411
6 6 3.143.1403311569547033115695473939
7 7 3.1413.141277250932727725093275757
8 8 3.14153.14151380114413801144145145
9 9 3.14153.14157294036772940367883883
10 10 3.141583.1415877252777252799619961
11 11 3.141593.1415914215142150463504635
12 12 3.1415923.141592345634561107711077
13 13 3.14159253.14159257657654500445004
14 14 3.14159263.14159263334632483463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
29 29 4.0000000000000004.000000000000000
30 30 0.0000000000000000.000000000000000
31 31 0.0000000000000000.000000000000000
32 32 0.0000000000000000.000000000000000
2.2.828427124746190828427124746190
3.03.06146745892071961467458920719
3.13.12144515225805321445152258053
3.133.1365484905459406548490545940
3.143.1403311569547530331156954753
3.1413.141277250932773277250932773
3.14153.14151380114430113801144301
3.14153.14157294036709172940367091
3.141583.1415877252771607725277160
3.141593.1415914215112001421511200
3.1415923.141592345570118345570118
3.14159253.14159257658487276584872
3.14159263.14159263433856334338563
3.141592643.1415926487769858776985
3.141592653.1415926523865912386591
2 2 2.2.828427124746190828427124746190
3 3 3.03.06146745892071961467458920719
4 4 3.13.12144515225805321445152258053
5 5 3.133.1365484905459465484905459411
6 6 3.143.1403311569547033115695473939
7 7 3.1413.141277250932727725093275757
8 8 3.14153.14151380114413801144145145
9 9 3.14153.14157294036772940367883883
10 10 3.141583.1415877252777252799619961
11 11 3.141593.1415914215142150463504635
12 12 3.1415923.141592345634561107711077
13 13 3.14159253.14159257657654500445004
14 14 3.14159263.14159263334632483463248
15 15 3.141592653.1415926548075894807589
16 16 3.141592643.1415926453212155321215
17 17 3.14159263.14159260737572007375720
18 18 3.1415923.141592910939673910939673
19 19 3.141593.1415941251951914125195191
20 20 3.141593.1415965537048206553704820
21 21 3.141593.1415965537048206553704820
22 22 3.1413.141674265021758674265021758
23 23 3.1413.141829681889202829681889202
24 24 3.143.1424512724941342451272494134
25 25 3.143.1424512724941342451272494134
26 26 3.13.16227766016838062277660168380
27 27 3.13.16227766016838062277660168380
28 28 3.3.464101615137754464101615137754
29 29 4.0000000000000004.000000000000000
30 30 0.0000000000000000.000000000000000
31 31 0.0000000000000000.000000000000000
32 32 0.0000000000000000.000000000000000
2.2.828427124746190828427124746190
3.03.06146745892071961467458920719
3.13.12144515225805321445152258053
3.133.1365484905459406548490545940
3.143.1403311569547530331156954753
3.1413.141277250932773277250932773
3.14153.14151380114430113801144301
3.14153.14157294036709172940367091
3.141583.1415877252771607725277160
3.141593.1415914215112001421511200
3.1415923.141592345570118345570118
3.14159253.14159257658487276584872
3.14159263.14159263433856334338563
3.141592643.1415926487769858776985
3.141592653.1415926523865912386591
3.1415926533.141592653288992288992
3.14159265353.14159265351459314593
3.14159265353.14159265357099370993
3.141592653583.1415926535850935093
3.1415926535883.141592653588618618
3.1415926535893.141592653589499499
3.14159265358973.14159265358971919
3.14159265358973.14159265358977474
3.141592653589783.1415926535897888
3.141592653589793.1415926535897922
3.1415926535897933.141592653589793
3.1415926535897933.141592653589793
3.1415926535897933.141592653589793
3.1415926535897933.141592653589793
3.1415926535897933.141592653589793
3.1415926535897933.141592653589793
The steps of scientific computation
1.Observation of nature
2.Construction of a mathematical model
3.Selection of a computational method to solve the mathematical
formulation
4.Program the method
5.Execute the program and display the results
6.Interpret the results and compare to observations
7.(Possibly) Refine
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