Adi Cox Mathematics 25-6-15

Mathematics By Adi Cox

CONTENT(01) *Symmetries Of Checkered Cross Sections (02) n-simplex numbers(03) Wallpaper Classification Algorithm(04) Groups Within Mansissas(05) Where The Sum And The Product Are Equal(06) Where Multiplication And Addition Meet(07) The Quartic Polynomial, The Quintic Polynomial And Its Unsolvability(08) Number Dictionary(09) What Is An Adi Polytope?(10) Adi 5 Space Polytope(11) Stifled Squares And (nth roots)(12) Large Stifled Numbers (Where the decimal parent number has 10 digits)(13) 1,000,000,000 As Stifled Numbers(14) The w Value Of Stifled Numbers And A List Of Null Stifled Numbers.(15) The Stifled Numbers From 1 To 2023(16) Antisymmetry(17) Investigating The Iterative Formula: x = kx(1 - x)(18) Matrices(19) The 4 x 4 Magic Square(20) Finding Fractions In Other Bases(21) Encryption(22) Noughts And Crosses, Orbits And Stabilizers(23) The Hour And The Minute Time Problem(24) The Family Of Adi Polytopes(25) The Investigation Of A Tetrahedron(26) (S-1) Stifled Minus One Numbers(27) Chromogeometry(28) Applications And Non Euclidean Geometry(29) Factorizing Cubic Equations Of The Form Ax^3 + Bx^2 + Cx + D = 0(30) Problem 1 - Find two 3-digit numbers(32) problem 2 - 500 Factors(33) The 4 x 4 Magic Square(34) An Introduction To Stifled Numbers(35) Sir Isaac Newton 1642 1727

*work in progress

Symmetries Of Checkered Cross SectionsWritten By Adi Cox 25th June 2015

Indroduction

This paper is analogous to looking at the cross sections of a cone to find second degree polynomial curves in the form of parabolas, hyperbolas and ellipses.

The cone is replaced by the checkered cube and the various second degree polynomial curves are replaced by cross sections that have some of the seventeen types of wallpaper symmetries. Just how many is what this paper intends to find out.

Below is a checkered cube 10 x 10

If I was to cut through the cube from the base corner at the front at pi/4 radians (45 degrees) to the horizontal then we would get this cross section :

If we look at the wallpaper classification algorithm below then we see that thecross section has the same symmetry as p3m1

Wallpaper Classification Algorithm

There are seventeen types of wallpaper patterns. To use the algorithm first determine the highest order of rotational symmetry and then follow the appropriate diagram below.

Standard forms and notation for wallpaper groups.

n Notation Point group

1 p1 C1

2 pm D1

3 pg D1

4 cm D1

5 p2 C1

6 p2mm or pmm D1

7 p2mg D2

8 p2gg or pgg D2

9 c2mm or cmm D2

10 p3 C3

11 p3m1 D3

12 p31m D3

13 p4 C4

14 p4mm or p4m D4

15 p4gm or p4g D4

16 p6 C6

17 p6mm or p6m D6

All the cross sections of the checkered cube will be flat planes which will be determined by three factors only:

(1) The angle x that is horizontal to the x axis and arcs up to the vertical z axis.

(2) The angle y that is horizontal to the y axis and arcs up to the vertical z axis.

(3) Translation t that is a fraction of the check cube side length, and t is vertical along the z axis.

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n-simplex numbers

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written by Adi Cox 9th June 2015

What is an n-simplex number?

An n-simplex has a visual representation with n vertices and all these vertices are connected to every other vertex. So if we number the vertices from zero to n we get a number if we follow the path of all the connections. Sometimes the connections are used twice. This is when n is even as in the case for 4-simplices.

1 o----o 2 |\ /| | \/ | | /\ | |/ \| 0 o----o 3

where 2 and 1 connections are used twice:01230213 base 4 = 6951 base 10

where 2 and 3 connections are used twice:01230231 base 4 = 6957 base 10

All even n-simplex numbers have odd order vertices and as such all even n-simplex numbers use one connection path twice, unless n = 2.

All odd n-simplex numbers have even order vertices and as such all odd n-simplex numbers use all connection paths once only and as such this makes odd n-simplices Eulerian, unless n = 0 where there are no connection paths.

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0-simplex number

therefore set 0-SN{NULL} is the set of 0-simplex numbers.

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1-simplex number

0 o

therefore set 1-SN{0} is the set of 1-simplex numbers.

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2-simplex numbers

0 o-----o 1

01 base 2 = 1 base ten10 base 2 = 2 base ten

therefore set 2-SN{1,2} is the set of 2-simplex numbers.

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3-simplex numbers

1 o / \ / \0 o-----o 2

0120 base 3 = 15 base 100210 base 3 = 21 base 10



therefore set 3-SN{15,21,34,46,59,65} is the set of 3-simplex numbers.

the inverse | the converse_______________________________0120 and 2102 | 0120 and 02101021 and 1201 | 1021 and 12012012 and 0210 | 2012 and 2102

The group of '3-simplex numbers and *equal digit numbers' is an order 9 cyclic group. The 'equal digit numbers' make the order 3 cyclic subgroup:

0 = 0000* a = 0120 b = 0210 c = 1111* d = 1021 e = 1201 f = 2222* g = 2012 h = 2102

|+3 | 0 a b c d e f g h_________________________________________________ | 0 | 0 a b c d e f g h | a | a b 0 e c d g h f | b | b 0 a d e c h f g | c | c e d f h g 0 a b | d | d c e h g f b 0 a | e | e d c g f h a b 0 | f | f g h 0 b a c e d | g | g h f a 0 b e d c | h | h f g b a 0 d c e

rearranging we get:

|+3 | 0* c* f* a b d e g h_________________________________________________ | 0*| 0* c* f* a b d e g h | c*| c* f* 0* e d h g a b | f*| f* 0* c* g h b a e d | a | a e g b 0 g d h f | b | b d h 0 a f c f g | d | d h b g f g f 0 a | e | e g a d c f h b 0 | g | g a e h f 0 b d c | h | h b d f g a 0 c e

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4-simplex numbers

1 o----o 2 |\ /| | \/ |

| /\ | |/ \| 0 o----o 3

Using the programs that I have written below we find that 4-simplex numbers have eight digits and that the 4-simplex number set 4-SN{} has order 192.

Below is the full list of all 192, 4-simplex numbers

001 01203123 base 4 = 6363 base 10002 01203132 base 4 = 6366 base 10003 01203213 base 4 = 6375 base 10004 01203231 base 4 = base 10005 01230123 base 4 = base 10006 01230132 base 4 = base 10007 01230213 base 4 = base 10008 01230231 base 4 = base 10009 01302123 base 4 = base 10010 01302132 base 4 = base 10011 01302312 base 4 = base 10012 01302321 base 4 = base 10013 01320123 base 4 = base 10014 01320132 base 4 = base 10015 01320312 base 4 = base 10016 01320321 base 4 = base 10017 02103123 base 4 = base 10018 02103132 base 4 = base 10019 02103213 base 4 = base 10020 02103231 base 4 = base 10021 02130123 base 4 = base 10022 02130132 base 4 = base 10023 02130213 base 4 = base 10024 02130231 base 4 = base 10025 02301213 base 4 = base 10026 02301231 base 4 = base 10027 02301312 base 4 = base 10028 02301321 base 4 = base 10029 02310213 base 4 = base 10030 02310231 base 4 = base 10031 02310312 base 4 = base 10032 02310321 base 4 = base 10033 03102123 base 4 = base 10034 03102132 base 4 = base 10035 03102312 base 4 = base 10036 03102321 base 4 = base 10037 03120123 base 4 = base 10038 03120132 base 4 = base 10039 03120312 base 4 = base 10040 03120321 base 4 = base 10041 03201213 base 4 = base 10042 03201231 base 4 = base 10043 03201312 base 4 = base 10044 03201321 base 4 = base 10045 03210213 base 4 = base 10046 03210231 base 4 = base 10047 03210312 base 4 = base 10

048 03210321 base 4 = base 10049 10213023 base 4 = base 10050 10213032 base 4 = base 10051 10213203 base 4 = base 10052 10213230 base 4 = base 10053 10231023 base 4 = base 10054 10231032 base 4 = base 10055 10231203 base 4 = base 10056 10231230 base 4 = base 10057 10312023 base 4 = base 10058 10312032 base 4 = base 10059 10312302 base 4 = base 10060 10312320 base 4 = base 10061 10321023 base 4 = base 10062 10321032 base 4 = base 10063 10321302 base 4 = base 10064 10321320 base 4 = base 10065 12013023 base 4 = base 10066 12013032 base 4 = base 10067 12013203 base 4 = base 10068 12013230 base 4 = base 10069 12031023 base 4 = base 10070 12031032 base 4 = base 10071 12031203 base 4 = base 10072 12031230 base 4 = base 10073 12301203 base 4 = base 10074 12301230 base 4 = base 10075 12301302 base 4 = base 10076 12301320 base 4 = base 10077 12310203 base 4 = base 10078 12310230 base 4 = base 10079 12310302 base 4 = base 10080 12310320 base 4 = base 10081 13012023 base 4 = base 10082 13012032 base 4 = base 10083 13012302 base 4 = base 10084 13012320 base 4 = base 10085 13021023 base 4 = base 10086 13021032 base 4 = base 10087 13021302 base 4 = base 10088 13021320 base 4 = base 10089 13201203 base 4 = base 10090 13201230 base 4 = base 10091 13201302 base 4 = base 10092 13201320 base 4 = base 10093 13210203 base 4 = base 10094 13210230 base 4 = base 10095 13210302 base 4 = 31026 base 10096 13210320 base 4 = 31032 base 10097 20123013 base 4 = 34503 base 10098 20123031 base 4 = 34509 base 10099 20123103 base 4 = 34515 base 10100 20123130 base 4 = 34524 base 10101 20132013 base 4 = 34695 base 10102 20132031 base 4 = base 10103 20132103 base 4 = base 10104 20132130 base 4 = base 10105 20312013 base 4 = base 10106 20312031 base 4 = base 10107 20312301 base 4 = base 10

108 20312310 base 4 = base 10109 20321013 base 4 = base 10110 20321031 base 4 = base 10111 20321301 base 4 = base 10112 20321310 base 4 = base 10113 21023013 base 4 = base 10114 21023031 base 4 = base 10115 21023103 base 4 = base 10116 21023130 base 4 = base 10117 21032013 base 4 = base 10118 21032031 base 4 = base 10119 21032103 base 4 = base 10120 21032130 base 4 = base 10121 21302103 base 4 = base 10122 21302130 base 4 = base 10123 21302301 base 4 = base 10124 21302310 base 4 = base 10125 21320103 base 4 = base 10126 21320130 base 4 = base 10127 21320301 base 4 = base 10128 21320310 base 4 = base 10129 23012013 base 4 = base 10130 23012031 base 4 = base 10131 23012301 base 4 = base 10132 23012310 base 4 = base 10133 23021013 base 4 = base 10134 23021031 base 4 = base 10135 23021301 base 4 = base 10136 23021310 base 4 = base 10137 23102103 base 4 = base 10138 23102130 base 4 = base 10139 23102301 base 4 = base 10140 23102310 base 4 = base 10141 23120103 base 4 = base 10142 23120130 base 4 = base 10143 23120301 base 4 = base 10144 23120310 base 4 = base 10145 30123012 base 4 = base 10146 30123021 base 4 = base 10147 30123102 base 4 = base 10148 30123120 base 4 = base 10149 30132012 base 4 = base 10150 30132021 base 4 = base 10151 30132102 base 4 = base 10152 30132120 base 4 = base 10153 30213012 base 4 = base 10154 30213021 base 4 = base 10155 30213201 base 4 = base 10156 30213210 base 4 = base 10157 30231012 base 4 = base 10158 30231021 base 4 = base 10159 30231201 base 4 = base 10160 30231210 base 4 = base 10161 31023012 base 4 = base 10162 31023021 base 4 = base 10163 31023102 base 4 = base 10164 31023120 base 4 = base 10165 31032012 base 4 = base 10166 31032021 base 4 = base 10167 31032102 base 4 = base 10

168 31032120 base 4 = base 10169 31203102 base 4 = base 10170 31203120 base 4 = base 10171 31203201 base 4 = base 10172 31203210 base 4 = base 10173 31230102 base 4 = base 10174 31230120 base 4 = base 10175 31230201 base 4 = base 10176 31230210 base 4 = base 10177 32013012 base 4 = base 10178 32013021 base 4 = base 10179 32013201 base 4 = base 10180 32013210 base 4 = base 10181 32031012 base 4 = base 10182 32031021 base 4 = base 10183 32031201 base 4 = base 10184 32031210 base 4 = base 10185 32103102 base 4 = base 10186 32103120 base 4 = base 10187 32103201 base 4 = base 10188 32103210 base 4 = base 10189 32130102 base 4 = base 10190 32130120 base 4 = base 10191 32130201 base 4 = 59169 base 10192 32130210 base 4 = 59172 base 10

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The program below converts the 8 digit base 4, 4-simplex number to a base 10, 4-simplex number._____________________________________________________________________

A for loop

document.write("n base b = x base 10")

n=1203123;b=4;

m1=n-10*Math.floor(n/10);

m2=(((n-100*Math.floor(n/100))-(n-10*Math.floor(n/10)))/10);







x1=m1;x2=m2*b;x3=m3*b*b;x4=m4*b*b*b;x5=m5*b*b*b*b;x6=m6*b*b*b*b*b;x7=m7*b*b*b*b*b*b;x8=m8*b*b*b*b*b*b*b;

x=x1+x2+x3+x4+x5+x6+x7+x8;

document.write(n," base ",b," = ",x," base 10","");

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We get the output below_____________________________________________________________________

n base b = x base 10

1203123 base 4 = 6363 base 10

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5-simplex numbers

3 o

4 o o 2

o o 0 1

This is where it gets a little more difficult. I have struggled to write a program to work out all the 5-simplex numbers, so I have done what I can by hand.

It does appear that in 5-simplex numbers they alway start and end with the same number, as with the other odd number that we have looked at the 3-simplex numbers.

Without any proof I conjecture that as 3-simplex numbers in base 3

are one number twice, which is always the first and the last number and one of each of the other two numbers. This will give a number like 1021.

So a 5-simplex number in base 5 are one number three times, which is always the first, the last and one in the middle somewhere and then two of the other four numbers. This will give an 11 digit number like 01203423140.

I believe that there will be 2340, 5-simplex numbers and this is how I have worked it out:

012031423400120314324001203241340012032341300120341324001203423140012041324300120413423001204231430012042341300120431423001204324130

0123024134001230243140012304134200123043142001231402430012314034200123142034001231420430012314302400123143042001234024130012340314200123413024001234130420

01240231430012402341300124034132001241320340012413204300124134023001241340320012430231400124304132001243140230012431403200124320314001243204130

Above are the 39 numbers that are 012xxxxxxx0. We can multiply 39 by 12 for all the 5-simplex numbers that begin and end in zero:

012xxxxxxx0013xxxxxxx0014xxxxxxx0

021xxxxxxx0023xxxxxxx0024xxxxxxx0031xxxxxxx0032xxxxxxx0034xxxxxxx0041xxxxxxx0042xxxxxxx0043xxxxxxx0

And then finally we multiply 12 times 39 by 5 for each of the different begining and end numbers 0,1,2,3,4. So we get:

12 x 39 x 5 = 2340

therefore the order of the set for 5-simplex numbers, set 5-SN{} has order 2340.

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n-simplex numbers

The number of digits of an n-simplex number in its n base is:

for even n numbers n(n/2)for odd n numbers (n(n-1)/2)+1

even example where n = 4:(n/2)n ---> ([4]/2)[4] = 8

e.g. 01230213 base 4

odd example where n = 5:(((n-1)/2)n)+1 ---> ((([5]-1)/2)[5])+1 = 11

e.g. 01234031420 base 5

| number of digits | number of n-simplex n | in base n | numbers in set n-SN{}_____________________________________________________________________ 00 | 000 | 000 01 | 001 | 001 02 | 002 | 002 03 | 004 | 006 04 | 008 | 192 05 | 011 | 2340 ? 06 | 018 | ???? 07 | 022 | 08 | 032 | 09 | 037 | 10 | 050 | 11 | 056 | 12 | 072 | 13 | 079 | 14 | 098 |

15 | 106 | 16 | 128 | 17 | 137 | 18 | 162 | 19 | 172 | 20 | 200 |

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Below is the program for finding 3-simplex numbers

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A for loop

document.write("x1, x2, x3, x4")//B=28;//D=-160;u=0;n=3;for (x1 = u; x1 < n; ++x1){for (x2 = u; x2 < n; ++x2){for (x3 = u; x3 < n; ++x3){for (x4 = u; x4 < n; ++x4){if((x1+x2+x3+x4)

2, 1, 0, 2

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Below is the program for finding 4-simplex numbers

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A for loop

document.write("x1, x2, x3, x4, x5, x6, x7, x8 ")

u=0;n=4;for (x1 = u; x1 < n; ++x1){for (x2 = u; x2 < n; ++x2){for (x3 = u; x3 < n; ++x3){for (x4 = u; x4 < n; ++x4){for (x5 = u; x5 < n; ++x5){for (x6 = u; x6 < n; ++x6){for (x7 = u; x7 < n; ++x7){for (x8 = u; x8 < n; ++x8){

if((x1+x2+x3+x4+x5+x6+x7+x8)==(12)&&(x2!=x1)&&(x3!=x1)&&(x3!=x2)&&(x4!=x2)&&(x4!=x3)&&(x5!=x3)&&(x5!=x4)&&(x5!=x2)&&(x6!=x4)&&(x6!=x5)&&(x7!=x1)&&(x7!=x6)&&(x8!=x7))

{document.write(x1, x2, x3, x4, x5, x6, x7, x8, "");}

}}}}}}}}

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Below is the output of the program above

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x1, x2, x3, x4, x5, x6, x7, x80120312301203132012032130120323101230123012301320123021301230231013021230130213201302312013023210132012301320132013203120132032102103123021031320210321302103231021301230213013202130213021302310230121302301231023013120230132102310213023102310231031202310321031021230310213203102312031023210312012303120132031203120312032103201213032012310320131203201321032102130321023103210312032103211021302310213032102132031021323010231023

102310321023120310231230103120231031203210312302103123201032102310321032103213021032132012013023120130321201320312013230120310231203103212031203120312301230120312301230123013021230132012310203123102301231030212310320130120231301203213012302130123201302102313021032130213021302132013201203132012301320130213201320132102031321023013210302132103202012301320123031201231032012313020132013201320312013210320132130203120132031203120312301203123102032101320321031203213012032131021023013

210230312102310321023130210320132103203121032103210321302130210321302130213023012130231021320103213201302132030121320310230120132301203123012301230123102302101323021031230213012302131023102103231021302310230123102310231201032312013023120301231203103012301230123021301231023012312030132012301320213013210230132120302130123021302130213201302132103023101230231021302312013023121031023012310230213102310231023120310320123103202131032102310321203120310231203120312032013120321031230102

31230120312302013123021032013012320130213201320132013210320310123203102132031201320312103210310232103120321032013210321032130102321301203213020132130210

Wallpaper Classification Algorithm

There are seventeen types of wallpaper patterns. To use the algorithm first determine the highest order of rotational symmetry and then follow the appropriate diagram below.

Standard forms and notation for wallpaper groups.

n Notation Point group

1 p1 C1

2 pm D1

3 pg D1

4 cm D1

5 p2 C1

6 p2mm or pmm D1

7 p2mg D2

8 p2gg or pgg D2

9 c2mm or cmm D2

10 p3 C3

11 p3m1 D3

12 p31m D3

13 p4 C4

14 p4mm or p4m D4

15 p4gm or p4g D4

16 p6 C6

17 p6mm or p6m D6

(1) Wallpaper p1

(2) Wallpaper pm

(3) Wallpaper pg

(4) Wallpaper cm

(5) Wallpaper p2

(6) Wallpaper p2mm or pmm

(7) Wallpaper p2mg or pmg

(8) Wallpaper p2gg or pgg

(9) Wallpaper c2mm or cmm

(10) Wallpaper p3

(11) Wallpaper p3m1

(12) Wallpaper p31m

(13) Wallpaper p4

(14) Wallpaper p4mm or p4m

(15) Wallpaper p4gm or p4g

(16) Wallpaper p6

(17) Wallpaper p6mm or p6m

Groups Within Mansissas

Written by Adi Cox

14th May 2015

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Definition Of A Group

A group is a set, G, together with an operation * (called the group law of G) that combines any two elements a and b to form another element, denoted a * b or ab. To qualify as a group, the set and operation, (G, *), must satisfy four requirements known as the group axioms:

(1) Closure

For all a, b in G, the result of the operation, a * b, is also in G.

(2) Associativity

For all a, b and c in G, (a * b) * c = a * (b * c).

(3) Identity element

There exists an element e in G, such that for every element a in G, the equation e * a = a * e = a holds. Such an element is unique (see below), and thus one speaks of the identity element.

(4) Inverse element

For each a in G, there exists an element b in G such that a * b = b * a = e, where e is the identity element.

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What Is A Mantissa?

The mantissa is the digits on the right of the point.

e.g. the mantissa of 123.456 is .456

In the cases below the mantissa is in base 5 and the units are unary, so 1.0 == 0.0

therefore 444444/444444 base5 == 1.0 == 0.0 == 000000/444444 base5

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Group Example (1)

0.4rec in base5 = 1

000000/444444 base5 = 0.000000rec base5 = 0/7 base10032412/444444 base5 = 0.032412rec base5 = 1/7 base10120324/444444 base5 = 0.120324rec base5 = 2/7 base10

203241/444444 base5 = 0.203241rec base5 = 3/7 base10241203/444444 base5 = 0.241203rec base5 = 4/7 base10324120/444444 base5 = 0.324120rec base5 = 5/7 base10412032/444444 base5 = 0.412032rec base5 = 6/7 base10444444/444444 base5 = 0.444444rec base5 = 7/7 base10

All the values in the order 7 Cayley table below are recuring quintals and so the addition is base 5 addition +5.

|+5 |.032412 .120324 .203241 .241203 .324120 .412032 .444444_____________________________________________________________________ |.032412 |.120324 .203241 .241203 .324120 .412032 .444444 .032412 |.120324 |.203241 .241203 .324120 .412032 .444444 .032412 .120324 |.203241 |.241203 .324120 .412032 .444444 .032412 .120324 .203241 |.241203 |.324120 .412032 .444444 .032412 .120324 .203241 .241203 |.324120 |.412032 .444444 .032412 .120324 .203241 .241203 .324120 |.412032 |.444444 .032412 .120324 .203241 .241203 .324120 .412032 |.444444 |.032412 .120324 .203241 .241203 .324120 .412032 .444444 |

The above group is isomorphic to the group (Z7,+7). This group has one self inverse and is abelian.

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Group Example (2)

0.Drec in base 14 = 1

0.Drec base14 = 0.9rec base10 = 1/10.7rec base14 = 0.5 base10 = 1/20.37rec base14 = 0.25 base10 = 1/40.1A7rec base14 = 0.125 base10 = 1/80.0C37 base14 = 0.0625 base10 = 1/160.061A7 base14 = 0.03125 base10 = 1/320.030B37 base14 = 0.015625 base10 = 1/64

All the values in the infinite order Cayley table below are recuring hexaquarternals and so the multiplication is base 14 ( X14 ).

|X14 |.'D' .'7' .'37' .'1A7' .0C37 .061A7 .030B37 ..._____________________________________________________________________ |.'D' |.'D' .'7' .'37' .'1A7' .0C37 .061A7 .030B37 ... |.'7' |.'7' .'37' .'1A7' .0C37 .061A7 .030B37 .'D' ... |.'37' |.'37' .'1A7' .0C37 .061A7 .030B37 .'D' .'7' ...

|.'1A7' |.'1A7' .0C37 .061A7 .030B37 .'D' .'7' .'37' ... |.0C37 |.0C37 .061A7 .030B37 .'D' .'7' .'37' .'1A7' ... |.061A7 |.061A7 .030B37 .'D' .'7' .'37' .'1A7' .0C37 ... |.030B37 |.030B37 .'D' .'7' .'37' .'1A7' .0C37 .061A7 ... . | . . . . . . . . | . . . . . . . . | . . . . . . .

The above group is isomorphic to the Q* infinite group under multiplication.

Where The Sum And The Product Are Equal

Written By Adi Cox.

13th May 2015

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Example:

0.5 + 0.5 + 3 + 4 + 4 = 120.5 * 0.5 * 3 * 4 * 4 = 12

Because there are five values this example is of order 5

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Examples of order 5 as fractions:

1/2, 1/2, 6/2, 8/2, 8/2 = 12

1/2, 2/2, 3/2, 7/2, 8/2 = 10.5

1/1, 1/1, 1/1, 2/1, 5/1 = 10

1/2, 2/2, 4/2, 5/2, 8/2 = 10

1/3, 5/3, 6/3, 9/3, 9/3 = 10

1/2, 3/2, 4/2, 4/2, 6/2 = 9

2/2, 2/2, 3/2, 3/2, 8/2 = 9

1/1, 1/1, 1/1, 3/1, 3/1 = 92/2, 2/2, 2/2, 6/2, 6/2 = 9 3/3, 3/3, 3/3, 9/3, 9/3 = 9

3/3, 3/3, 5/3, 5/3, 9/3 = 8.333333333333334

1/1, 1/1, 2/1, 2/1, 2/1 = 82/2, 2/2, 4/2, 4/2, 4/2 = 83/3, 3/3, 6/3, 6/3, 6/3 = 84/4, 4/4, 8/4, 8/4, 8/4 = 8

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The smallest answer that we get when we multiply or add the five values which turn out to be surds and not a fractions:

x^5 = 5x=> x^4 = 5=> x = 5^(1/4) = 1.495348781 to 9 d.p.

5^.25 + 5^.25 + 5^.25 + 5^.25 + 5^.25 = 5^.25 x 5^.25 x 5^.25 x 5^.25 x 5^.25

= 7.476743906

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So what does the answer measure?

The smallest answer is when all five values are equal and so the more varied the five values are then the greater the answer. It is important to say that the answer measures how much the values vary both as values of multiplication as well as values of addition.

Multiplication has the identity of 1 because anything multiplied by 1 is unchanged and the identity of addition is zero because the addition of zero leaves the answer unchanged. So what we are doing here is finding a balance to satisfy both identities to the same degree at the same time.

Using the above information we can use the multiplication identity of 1 to change the addition without changing the multiplication:

e.g. 2 x 3 = 6 2 + 3 = 5

To make the above addition equal the multiplication we can make it into order 3.

e.g. 1 x 2 x 3 = 6 1 + 2 + 3 = 6

So what the order measures is how much the multiplicative identity and the additive identity varies.

e.g. 2 x 2 x 5 = 20 2 + 2 + 5 = 9

1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 x 2 x 2 x 5 = 20 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 2 + 2 + 5 = 20

To make the above addition equal the multiplication we can make it into order 14.

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Where Multiplication And Addition Meet

Written By Adi Cox

12th May 2015.

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It is a trivial example to say that 2 x 2 = 2 + 2

We can express the the example above as:

x^2 = 2x

Rearranging the equation by dividing both sides by x:

we get x = 2

With two operands we have a binary operation as with the 2 times 2 and the 2 plus 2, but what about a trinary operation where x plus x plus x equals 3 times x. This can be expressed as follows:

x^3 = 3x

Rearranging the equation to find the value of x we divide both sides by x and then square root both sides of the equation to get:

x = 3^(1/2) x equals the square root of 3.

If we check the answer we will find that:

3 * 3^(1/2) = 3^(1/2) + 3^(1/2) + 3^(1/2) = 5.196152423 to 9 d.p.

Next we look at expressing a quaternary operation:

x^4 = 4x

And get the solution for x:

x = 4^(1/3) x equals the cube root of 4.

If we check the answer we will find that:

4 * 4^(1/3) = 4^(1/3) + 4^(1/3) + 4^(1/3) = 6.349604208 to 9 d.p.

So x plus x plus x plus x = 4 times x when x = 4^(1/3)

So if we look we will see a pattern:

x^y = yx => x^y - yx = 0

Solving the above equation we find that x = y^(1/y-1),

And so y.y^(1/y-1) = y^(y/y-1), where y is the order of operation.

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If we only consider positive integers then we get these results for both multiplication and addition:

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order 2{2,2} = 4

order 3{1,2,3} = 6

order 4{1,1,2,4} = 8

order 5{1,1,1,2,5} = 10{1,1,1,3,3} = 9{1,1,2,2,2} = 8

order 6{1,1,1,1,2,6} = 12

order 7{1,1,1,1,1,2,7} = 14{1,1,1,1,1,3,4} = 12

order 8{1,1,1,1,1,1,2,8} = 16{1,1,1,1,1,2,2,3} = 12

order 9{1,1,1,1,1,1,1,2,9} = 18{1,1,1,1,1,1,1,3,5} = 15

order 10{1,1,1,1,1,1,1,1,2,10} = 20{1,1,1,1,1,1,1,1,4,4} = 16

order 11{1,1,1,1,1,1,1,1,1,2,11} = 22{1,1,1,1,1,1,1,1,1,3,6} = 18{1,1,1,1,1,1,1,1,2,2,4} = 16

order 12{1,1,1,1,1,1,1,1,1,1,2,12} = 24{1,1,1,1,1,1,1,1,2,2,2,2} = 16

order 13{1,1,1,1,1,1,1,1,1,1,1,2,13} = 26{1,1,1,1,1,1,1,1,1,1,1,3,7} = 21{1,1,1,1,1,1,1,1,1,1,1,4,5} = 20{1,1,1,1,1,1,1,1,1,1,2,3,3} = 18

order 14{1,1,1,1,1,1,1,1,1,1,1,1,2,14} = 28{1,1,1,1,1,1,1,1,1,1,1,2,2,5} = 20

order 15{1,1,1,1,1,1,1,1,1,1,1,1,1,2,15} = 30{1,1,1,1,1,1,1,1,1,1,1,1,1,3,8} = 24

order 16{1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,16} = 32{1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,6} = 24

order 17{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,17} = 34{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,9} = 27{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,5} = 25{1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,6} = 24

order 18{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,18} = 36{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,4} = 24

order 19{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,19} = 38{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,10} = 30{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,7} = 28{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,3} = 24

order 20{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,20} = 40{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,7} = 28

order 21{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,21} = 42{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,11} = 33{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,6} = 30{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3} = 27

order 22{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,22} = 44{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,8} = 32

order 23{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,23} = 46{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,12} = 36{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,8} = 32{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,5} = 30

order 24{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,24} = 48

order 25{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,25} = 50{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,13} = 39{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,9} = 36{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,7} = 35{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,4} = 32

order 26{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,26} = 52

{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,6} = 36{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,9} = 36{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,4} = 32

order 27{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,27} = 54{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,14} = 42{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2} = 32

order 28{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,28} = 56{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,10} = 40{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,6} = 36

order 29{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,29} = 58{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,15} = 45{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,8} = 40{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,10} = 40{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,4} = 36

order 30{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,30} = 60{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3} = 36

order 31{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,31} = 62{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,16} = 48{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,4,11} = 44{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,7} = 42

order 32{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,32} = 64{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,11} = 44{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,5} = 40

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Pattern 1

The most obvious pattern for any order where r is any order:

{(r-2)*1,2,r} where r is an integer > 0.

e.g. where r=5 we get {1,1,1,2,5}

It is interesting to note that order 24 only has this one pattern.

Pattern 2

{(r-2)*1,3,2+n} where r = 2n+3 and n is an integer > 0.

e.g. where r=5 we get {1,1,1,3,3}

Pattern 3

{(r-3)*1,2,2,1+n} where r = 3n+2 and n is an integer > 0.

e.g. where r=5 we get {1,1,2,2,2}

Pattern 4

{(r-2)*1,4,3+n} where r = 3n+7 and n is an integer > 0.

e.g. where r=10 we get {1,1,1,1,1,1,1,1,4,4}

Pattern 5

{(r-4)*1,2,2,2,1+n} where r = 7n+5 and n is an integer > 0.

e.g. where r=12 we get {1,1,1,1,1,1,1,1,2,2,2,2}

Pattern 6

{(r-2)*1,5,4+n}

where r = 4n+13 and n is an integer > 0.

e.g. where r=17 we get {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,5,5}

Pattern 7

{(r-3)*1,2,3,3+n}


e.g. where r=18 we get {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,3,4}

Pattern 8

{(r-3)*1,3,3,2+n}


e.g. where r=21 we get {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,3}

Pattern 9

{(r-3)*1,2,4,2+n}


e.g. where r=25 we get:

{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,4,4}

Pattern 10

{(r-2)*1,6,5+n}



{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,6,6}

Pattern 11

{(r-5)*1,2,2,2,2,1+n}

where r = ? and n is an integer > 0.


{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}

Pattern 12

{(r-4)*1,2,2,3,2+n}

where r = ? and n is an integer > 0.


{1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3}

MORE MATHEMATICAL MUSINGS LATER. TIME TO END THIS SESSION : (

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An example of the program used to get the order 5 results:

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A for loop

document.write("a * b * c * d * e = a + b + c + d + e = z")

u=1;v=9;for (a = u; a

1 * 3 * 3 * 1 * 1 = 1 + 3 + 3 + 1 + 1 = 91 * 5 * 1 * 1 * 2 = 1 + 5 + 1 + 1 + 2 = 101 * 5 * 1 * 2 * 1 = 1 + 5 + 1 + 2 + 1 = 101 * 5 * 2 * 1 * 1 = 1 + 5 + 2 + 1 + 1 = 102 * 1 * 1 * 1 * 5 = 2 + 1 + 1 + 1 + 5 = 102 * 1 * 1 * 2 * 2 = 2 + 1 + 1 + 2 + 2 = 82 * 1 * 1 * 5 * 1 = 2 + 1 + 1 + 5 + 1 = 102 * 1 * 2 * 1 * 2 = 2 + 1 + 2 + 1 + 2 = 82 * 1 * 2 * 2 * 1 = 2 + 1 + 2 + 2 + 1 = 82 * 1 * 5 * 1 * 1 = 2 + 1 + 5 + 1 + 1 = 102 * 2 * 1 * 1 * 2 = 2 + 2 + 1 + 1 + 2 = 82 * 2 * 1 * 2 * 1 = 2 + 2 + 1 + 2 + 1 = 82 * 2 * 2 * 1 * 1 = 2 + 2 + 2 + 1 + 1 = 82 * 5 * 1 * 1 * 1 = 2 + 5 + 1 + 1 + 1 = 103 * 1 * 1 * 1 * 3 = 3 + 1 + 1 + 1 + 3 = 93 * 1 * 1 * 3 * 1 = 3 + 1 + 1 + 3 + 1 = 93 * 1 * 3 * 1 * 1 = 3 + 1 + 3 + 1 + 1 = 93 * 3 * 1 * 1 * 1 = 3 + 3 + 1 + 1 + 1 = 95 * 1 * 1 * 1 * 2 = 5 + 1 + 1 + 1 + 2 = 105 * 1 * 1 * 2 * 1 = 5 + 1 + 1 + 2 + 1 = 105 * 1 * 2 * 1 * 1 = 5 + 1 + 2 + 1 + 1 = 105 * 2 * 1 * 1 * 1 = 5 + 2 + 1 + 1 + 1 = 10

The Quartic Polynomial, The Quintic Polynomial And Its Unsolvability

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Written By Adi Cox

11th May 2015

It is intriguing to learn that the quintic polynomial is not always solvable by radicals. So I have looked into the structure of polynomials to see what I can find:_____________________________________________________________________

Quartic Polynomial

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Ax^4 + Bx^3 + Cx^2 + Dx^1 + Ex^0 = 0

A = 1B = a+b+c+dC = ab+ac+ad+bc+bd+cdD = abc+abd+acd+bcdE = abcd

where {-a,-b,-c,-d} are the four solutions to the quartic polynomial.

i | | | -1 ------------0------------ 1 | | | -i

Example: where the solutions are: {-1, i, i, 1}

we change the signs:{a=1, b=-i, c=-i, d=-1}

A = 1

B = (1)+(-i)+(-i)+(-1) = -2i

C = (1)(-i)+(1)(-i)+(1)(-1)+(-i)(-i)+(-i)(-1)+(-i)(-1) = -i - i - 1 - 1 + i + i = -2

D = (1)(-i)(-i)+(1)(-i)(-1)+(1)(-i)(-1)+(-i)(-i)(-1) = -1 + i + i + 1 = 2i

E = (1)(-i)(-i)(-1) = 1

And so

x^4 - 2ix^3 - 2x^2 + 2ix + 1 = 0

=> (x+1)(x-i)(x-i)(x-1)

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Quintic Polynomial

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Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex^1 + Fx^0 = 0

A = 1B = a+b+c+d+eC = ab+ac+ad+ae+bc+bd+be+cd+ce+deD = abc+abd+abe+acd+ace+ade+bcd+bce+bde+cdeE = abcd+abce+abde+acde+bcdeF = abcde

where {-a,-b,-c,-d,-e} are the five solutions to the quintic polynomial.

y= 1 ^ | | -1 --0----2--3----5----7--------11--> x | | y= -1

Example: where the solutions are: {2, 3, 5, 7, 11}

we change the signs:{a=-2, b=-3, c=-5, d=-7, e=-11}

A = 1

B = (-2)+(-3)+(-5)+(-7)+(-11) = -28

C = (-2)(-3)+(-2)(-5)+(-2)(-7)+(-2)(-11)+(-3)(-5)+(-3)(-7)+(-3)(-11) +(-5)(-7)+(-5)(-11)+(-7)(-11) = 6 + 10 + 14 + 22 + 15 + 21 + 33 + 35 + 55 + 77 = 288

D = (-2)(-3)(-5)+(-2)(-3)(-7)+(-2)(-3)(-11)+(-2)(-5)(-7) +(-2)(-5)(-11)+(-2)(-7)(-11)+(-3)(-5)(-7)+(-3)(-5)(-11) +(-3)(-7)(-11)+(-5)(-7)(-11) = - 30 - 42 - 66 - 70 - 110 - 154 - 105 - 165 - 231 - 385 = -1358

E = (-2)(-3)(-5)(-7)+(-2)(-3)(-5)(-11)+(-2)(-3)(-7)(-11) +(-2)(-5)(-7)(-11)+(-3)(-5)(-7)(-11) = 210 + 330 + 462 + 770 + 1155 = 2927

F = (-2)(-3)(-5)(-7)(-11) = -2310

And so

x^5 - 28x^4 + 288x^3 - 1358x^2 + 2927x^1 - 2310x^0 = 0

=> (x-2)(x-3)(x-5)(x-7)(x-11)

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So What Happens With x^5 - x^1 + 1 = 0

The above quintic polynomial is unsolvable by radicals, so let us see what we get when we apply the previous methods for solving polynomials:

Ax^5 + Bx^4 + Cx^3 + Dx^2 + Ex^1 + Fx^0 = 0

A = 1B = a+b+c+d+e = 0C = ab+ac+ad+ae+bc+bd+be+cd+ce+de = 0D = abc+abd+abe+acd+ace+ade+bcd+bce+bde+cde = 0E = abcd+abce+abde+acde+bcde = -1F = abcde = 1

where {-a,-b,-c,-d,-e} are the five solutions to the quintic polynomial.

x^5 - x^1 + 1 = 0

=> 1x^5 - 1x^1 + 1x^0 = 0

=> (1)x^5 + (0)x^4 + (0)x^3 + (0)x^2 + (-1)x^1 + (1)x^0 = 0

So the problem above is going to take some satisfying. Nothing short of inventing a new type of number with specific properties to satisfy this puzzle.

I'M WORKING ON IT!

Number Dictionary18-4-15

by Adi Cox

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Forward

Numbers are my friends. if I was a number I would want to be happy and lucky. The prime number 367 is both lucky and happy but being a prime number means that it is deficient and it is stifled in base 7.

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A

Algebraic number Any number that is the root of a non-zero polynomial with rational coefficients is an algebraic number.

AmicableThe smallest pair of amicable numbers is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220.{(220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368), ...}

C

CompositeA natural number greater than 1 that is not a prime number is called a composite number.

D

DecagonalThe n-th decagonal number is given by the formula d = 4n^2 - 3n. So the 5th decagonal number is: 4[5]^2-3[5] = 85{0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, ...}

DodecagonalThe dodecagonal number for n is given by the formula 5n^2 - 4n where n > 0. So the 3rd dodecagonal is 5[3]^2-4[3] = 45-12 = 33{1, 12, 33, 64, 105, 156, 217, 288, 369, ...}

E

EvenAn even number is an integer of the form n = 2k, where k is an integer.{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, ...}

F

FactorialThe nth factorial is the product of all n digits: 0!=1, 1!=1, 2!=2x1=2, 3!=3x2x1=6, 4!=4x3x2x1=24, 5!=5x4x3x2x1=120, ...

{1, 1, 2, 6, 24, 120, 720, ...}

FigurateA member of the subset of the sets above containing only triangular numbers, square numbers, pentagonal numbers, hexagonal numbers, heptagonal numbers, octagonal numbers, nonagonal numbers, decagonal numbers, dodecagonal numbers, pyramidal numbers, and their analogs in other dimensions.

FractionSee rational number.

FriendlyThe smallest friendly number is 6, forming for example the friendly pair 6 and 28 with abundancy s(6) / 6 = (1+2+3+6) / 6 = 2, the same as s(28) / 28 = (1+2+4+7+14+28) / 28 = 2.

H

HappyA happy number goes to one when the digits of the numbers are squared and added together:2^2+3^2=13 --> 1^2+3^2=10 --> 1^2+0^2=1{1, 7, 10, 13, 19, 23, 28, ...}

HarshadThe number 18 is a Harshad number in base 10, because the sum of the digits 1 and 8 is 9 (1+8=9), and 18 is divisible by 9.7864 is a Harshad number in base 2: 7864=1111010111000. The sum of the digits equal 8 and 7864/8=983.

HeptagonalThe n-th heptagonal number is given by the formula (5n^2-3n)/2. So the 7th heptagonal number is: (5[7]^2-3[7])/2 = (5x49-21)/2 = 112{1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, ...}

HexagonalThe formula for the nth hexagonal number is: 2n^2-n. So the 5th hexagonal number is: 2[5]^2-[5] = 2x25-5 = 45{1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, ...}

I

IntegerAn integer is a whole number which also includes negative numbers.{... -10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,10, ...}

N

NaturalA natural number is a non negative integer:{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20, ...}

NonagonalThe nonagonal number for n is given by the formula: (n(7n-5))/2 So the 6th nonogonal number is: ([6](7[6]-5))/2 = (6x37)/2 = 111{1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, 474, ...}

O

OctagonalThe octagonal number for n is given by the formula 3n2 - 2n, where n > 0. The first few octagonal numbers are:{1, 8, 21, 40, 65, 96, 133, 176, 225, 280, ...}

OddAn odd number is an integer of the form n = 2k + 1 where k is an integer.{1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, ...}

P

PentagonalIs given by the formula: pentagonal number p = (3n^2-n)/2 for n greater than 1. So the fifth pentagonal number is:(3[5]^2-[5])/2 = (75-5)/2 = 35{1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, ...}

PerfectThe first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6.{6, 28, 496, 8128, ...}

PolygonalThese are numbers that can be represented as dots that are arranged in the shape of a regular polygon see: Triangular, Square, Pentagonal, Hexagonal, Heptagonal, Octagonal, Nonagonal, Decagonal, Dodecagonal.

PrimeA prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Q

QuotientSee rational number.

R

RationalA rational number is a number that is represented as a quotient or fraction. Where the denominator is not zero. p/q is a fraction where p is the numerator and q is the denominator.

S

SadA sad number or an unhappy number is a number that is not happy. see happy numbers.{2,3,4,5,6,7,89,11,12,14,15,16,17,18,20, ...}

Sociable1264460 = 2^2.5.17.3719, the sum of the proper divisors = 15478601547860 = 2^2.5.193.401, the sum of the proper divisors = 17276361727636 = 2^2.521.829, the sum of the proper divisors = 13051841305184 = 2^5.40787, the sum of the proper divisors = 1264460

SquareIs an integer that is a product of itself. Examples are: 1x1=1,

2x2=4, 3x3=9, 4x4=16, 5x5=25, 6x6=36, 7x7=49, 8x8=64, ...{1,2,4,9,16,25,36,49,64,81,100,121,144, ...}

StifledA number is stifled when the digits of that number equals the base of that number: The number 10 is stifled in base 4: 2x4^1 + 2x4^0 = 10 so 22 in base 4 is equal to ten in base 10 as 2+2=4.

StrangeAll strange numbers are prime. Every single digit prime number is strange. A number with two or more digits is strange if, and only if, the two numbers obtained from it, by removing either its first or its last digit, are also strange.{2, 3, 5, 7, 23, 37, 53, ...}

SurdWhen we can not simplify a number to remove a square root (or cube root, or nth root) then it is a surd. ie x^1/n.

T

TetrahedralA figurate number that represents a pyramid with a triangular base and three sides, called a tetrahedron. The nth tetrahedral number is the sum of the first n triangular numbers.{1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ...}

TranscendentalAny real or complex number that is not algebraic is transcendental. Examples include e and pi.

TriangularA triangular number is the sum of n numbers starting from zero. So some examples are: 0, 0+1=1, 0+1+2=3, 0+1+2+3=6, 0+1+2+3+4=10, ...{0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, ...}

U

UnhappyAn unhappy number or a sad number is a number that is ot happy. see happy numbers.{2,3,4,5,6,7,89,11,12,14,15,16,17,18,20, ...}

W

WierdThe smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70.{70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ...}

A for loop

document.write("n in base b")

b=2;

//n can be any number greater than zero and not greater than 1048575.//This program will run slow when n is entered as a larger number.n=367;

for (a = b; a

r7=Math.round(a*f7);

n8=d7/a;f8=n8-Math.floor(n8);d8=n8-f8;r8=Math.round(a*f8);











n19=d18/a;f19=n19-Math.floor(n19);d19=n19-f19;

r19=Math.round(a*f19);


document.write("base ten number n in base ",a," = ",r20,"-",r19,"-",r18,"-",r17,"-",r16,"-",r15,"-",r14,"-",r13,"-",r12,"-",r11,"-",r10,"-",r9,"-",r8,"-",r7,"-",r6,"-",r5,"-",r4,"-",r3,"-",r2,"-",r1,"");r0=r20+r19+r18+r17+r16+r15+r14+r13+r12+r11+r10+r9+r8+r7+r6+r5+r4+r3+r2+r1;document.write("digit summation = ",r20,"+",r19,"+",r18,"+",r17,"+",r16,"+",r15,"+",r14,"+",r13,"+",r12,"+",r11,"+",r10,"+",r9,"+",r8,"+",r7,"+",r6,"+",r5,"+",r4,"+",r3,"+",r2,"+",r1," = ",r0,"");}

_____________________________________________________________________

Output of the program below_____________________________________________________________________

n in base bbase b = 2base ten number n = 367base ten number n in base 2 = 0-0-0-0-0-0-0-0-0-0-0-1-0-1-1-0-1-1-1-1digit summation = 0+0+0+0+0+0+0+0+0+0+0+1+0+1+1+0+1+1+1+1 = 7

base b = 3base ten number n = 367base ten number n in base 3 = 0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-1-1-1-2-1digit summation = 0+0+0+0+0+0+0+0+0+0+0+0+0+0+1+1+1+1+2+1 = 7









base b = 216base ten number n = 367base ten number n in base 216 = 0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-0-1-151digit

Adi Cox Mathematics 25-6-15

Documents

x axis

checkered cube

stifled numbers14

angle x

w value of stifled numbers

list of null stifled

vertical z axis

wallpaper groups