On striking a arc on the ro9 triangle. First arc. Given 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9= a line on y coordinate from y=0 to y=9. 0,0 1,0 2,0 3,0 4,0 5,0 6,0 7,0 8,0 9,0 =x. A line on the x coordinate from x=0 to x=9. Connecting the end points of x=9,0 and y=0,9 all in the Cartesian plane. So from 0,9 9,0 is a line with a negative slope dy/dx=-1. Further, the points are contained on this line, hypotenuse, of 0,9 1,8 2,7 3,6 4,5 5,4 6,3 7,2 8,1 9,0.. For x,y. This forms a triangle and these numbers are positioned on the hypotenuse of x,y coordinate lines. Taking x from x=0 to x=9 and rotating ccw to the point 1,8 to strike an arc. What can be derived from this motion? There are three well defined points on the line. dy/dx =m and are well defined. We know a arc. dy=9-8=1. dx=9-8=1. dy is a remainder and dx is a direct subtraction. dy=1-0 directly. Therefore dx,dy can be calculated directly and indirectly, much like s,d in n+1 of the counting sequence. Futher there is a new tangent, t=slope to the arc. 1.) From t=y=vertical to reciprocal of dy/dx1/8 to dt/dxdy=-8 the change in the slope of the tangent with respect to the change in x,y dydx for the arc with a fixed dydx for h and dh=c=0 with respect to dy/dx. 2.)The third will have dy/dx=2/7 with dt/dydx=-7/2. dh=0 3.) dy/dx=3/6 and dt/dydx=-6/3. and dt/dydx=-5/4. 5.) dy/dx=5/4 dt/dydx=-4/5. 6.) dy/dx=6/3, dt/dydx=-3/6. 7.) dy/dx=7/2, dt/dydx=-2/7. 8.) dy/dx=8/1, dt/dydx=-1/8. 9.) dy/dx=9/0, dt/dydx=-0/9. Not observing zero for infinity. What I'm looking to establish is a base line for mapping fractions from an interference pattern from consecutive circes and a base circle established by p. And consecutive curcles on h with the same curvature. Then apply ro9 for x,y. Ultimately resolving a bunch of things.

“No one shall cast us from the paradise that Cantor has created.”

On what is -1 1+1=2 1×1=1 1-2=-1 and all numbers satisfying this relation. Sqrt, look for squares In this sequence. 13,36 9+4=13 4×9=36->1+3+2+2+4=7+5=12. 7+6=13. 1+3=4. 3+4=7, 7+1=8. 13+34=47 4+7=11 47+11=58 5+8=13 58+13=71. 71+8=79 7+9=16 1+6=7 16+17=23 2+3=5 5+7=12 1+2=3 5,3,7. 5+3=8 8+7=15 1+5=6. 1+5+6=12 1+3+2+6=4+8=12 1+2=3 so 3,4,8. 3+4=7. 7,8. 7+8=15. 1+5=6. 6,7,8. 5+5+5+1+2+3 9+12=r3+4=31. Derive the multiplication factors from addition. Halving derives 2 equations. 4+6 and 5+6. So 10,11 -> 1+2+18. Sequence building. 1+8=9 12+18=r46 /3 /6=23 23+46=69 /3=23. 6+9=15 1+5=6. 3+2=5. Can cycle 11 12 13 14. 4,10 4,5,5 nc.9×4=36 36+4=40 40+1=41 40+41=81. Summands for 5=4,1 3,2 2,3 1,4 5,0. 36+1=37 =9×4+1. 13+1=14. 131+14=145 s=10 d=3+1 10+31=41 36+1=37. 3+6=9 3+7=10 9,0+9,1=18,1. 18+1=19 1+9=9,1 19+91=1010nc or110 100=99+1 or 99×1 2×991=18182nc 1+8+1+8+1=20. 1982 s=17+3=20 20=18+2. If 18=2 2+2=4 22+4=26=18+8=26 r13 so 8,4 r4,2 r2 1 12,6,3 /3= 421 or 1+2=3 363 -> 7,12. 421+363=784. 7+8+4=19 784->43444 43+44=87 87+4=91 9+1=10. Looking like complex number rational graph. 9×1+2=11 9×1+1=10 9×1+0=9. Thus 9,10,11,12,13->36,9 (55->36+9+9+1(19 1+9=10) 4×9=13 13+4=17, 31+20=51->4,5,6 av=5 15=3=5. 9,6 9,7 13+18=31. Closed form. On deriving p. 36-13=23. 1×3=3 1+3=4 43 3+4=7 7+4=11 4+13=17 30+17=47 40+17=57 20+17=37 40+13=53 30+13=43 20+11=31 30+11=41 40+11=51. Continously builds p. Rational interpretation of i,j... 49+14=63. 9+14=23 36+12=48 /2 =24 /3=19. 25+5=30 /2=15 /3=10. 64×8=72 /2=36 /3=26 26÷2=13, 36÷2=18, 36÷3=12. Rat circle universal div.

Will we be introducing the corresponding mset structure that represents these rational numbers?

at 40:17.......Can.....(pi/3) ......be replaced consistently....with (180/pi)......such that the curvature at vertice P = 401.070^o^degree.... or..... 1 period of rotation ...Yes or No...?

I'm curious how one explains Eulers identity, e^pi i + 1 = 0, without doing an infinite amount of work. Have you ever made a video about this identity?

10 11 12 13->17 18 19. 91 92 93 94 95 96 97 98 99 10+9+0=19 10²×9²=181 1+8+1=10 10+181=191 1+9+1=11 This second one is counting. This is a lot of work. I'll try to figure out the closed form for an equation. 10+9+1=20 11+9+2=22 18+4 4+1=5 4+8=12 so 17,8,9-> 25,26->7,8 7+8=15 1+5=6 where 6+15=21 21= 18+3=2×9+3 5+2=7 3+9=12 1+2=3 7=4+3 so 7+3+4 34+3=37 3+7=10 10=9+1 4+6=5+5 1+1+2+4+4++4+4=20 where 2+1+4=8 18=2×9 27=3×9=3³ 5+2=7 45+18=513nc 9n 9×5=45 45+13=58 5+8=13. Pass throgh numbers. 13+13=26 2+6=8 3+5=8 4+4=8 1+7=8 4×8=32 3+2=5 2+6+3+5+4+4+1+7=32. Why universal division works in the ro9. So a number 26354417 might not have any meaning in and of its self, does have meaning in magnitude and poisition in the ro9. 21 22 23 ->27 28 29 93 94 95 96 97 98 99 31 32 33 34 35 36 37 38 39 94 95 96 96 97 98 99 910 911 912. 31+95=396nc or 126 396+126=49266. 4986. Proven all number tend to order. Carries short circuit order.4+9=13 8+6=14 13+14=27=18+9 so 12+78=3+15 where 1+3=4 3+1=4 13 only has 1-9's while 31 has 3-9's 31+9=40 13+27=40 40+40=80 80-13=67 80-31=49. 67+49= 116 1+1+6=8 8+5=13 8+23=31 28+16=44 or 368 386 368+386=66886 or similiar r33443 2222222111 7,3 2,1 7+2=9 3+1=4 9+4=31 4+9=13 odd even decides order of pairs. 1 can be even or odd. So 94 is oe so 9+4=31 oe. Then 49=eo so 13. Proof 49+13=512 94+31=125 5+1=6 62 r31 1+2=3 35 oo. No way to derive odd odd from oo or oe except like this with addition and superposition. . 18+13=31 ro9 9+9+4=22=9,4. Pathological. 9+4=13

On the F(H)TM and the zeta function. Note: The three laws of motion should be framed differently compared to the proposed fundamental model of mathematics to be more comparable. 1.) The 3rd proposed law of mathematics, the real point and workhouse to achieve the highest understanding of our counting system. Expectations. Resolve; Cos, sin, exp, factorials, and zeta function, thoroughly explaining exponents, fractions, decimals, doubling, halving, averaging, multiplication, division, addition, subtraction, primes, rational circle, general geometry. Foundations for higher lever operations, physics both classical and quantum. Assuming complete uniformity across all of nature! Putting sin and cosine into context. Assuming the ro9 is true in all situations since the number system provides all the predictability in all mathematical operations.

It would be great if we had a curriculum and a book on Babylonian mathematics. It seems that there would be prerequisites in order to decipher all these tables more quickly. It sounds like a book to look forward to, with all of the Babylonian tables to make use of, perhaps with exercises and worksheets to work on for mastery.

It's really disrespectful how you name all Persians as Arabs. Khwarizmi, Khayyam, Biruni, Kashi

One is safe to assume space if infinite. therefor the only important definition of space would be localizing a space with some barrier to create a 123 situation. On compaction the number system its self is allready the most efficient and compact situation for numbers. If 9 can equal 1 then 18 can =2, 27 can equal 3... The average of 9 is 4,5. The average of 18 =9,9. The average of 27=13,14. Where 9+4=13 and 9+5=14. 4,5 then 4,4 5,5 s=18. 444555 s=12×15=27 where 2+2+2+7+7+7=6+21 and 6+3=9 in a sense n+1 for the rule of 9. Can construct, 2223333333 so 7-3's and 3-2's. 63+27=90 27+18=45. 90+45=135=9 or 9+27÷45=81. a+b+c=d. Can introduce n+1...n+n and n×n. N×n leave everthing unchanged with respect to the ro9. While introducing n+1in the addition in either a,b,c location hascs randomizing effect, however the ro9 can detect this change and the location at which n+1 was introduced. 1+2=3 1+5=6 => 12+15=27 3+6=9. abcd. b+1 variables all =1 except b. 1211. 9×12=102 102+11=113 102=9×11+3. 113=9×12+5. 23+8=31. 9=1 9999 9,18,99.27+18=36 486. 8,16,12. 8+7+3=18+18=1199 11+99=1010 nc1010+99=1109 9011. 102=90+11 3,2 45,90. 90=9×10 11=2×6 11=12 9+1 9+2 n+1. Something like that. udiv and stepfuction. Should lead to related triva to eventually bolster the final proofs. Acurate description of the problem. Answere still is svbit elusive. Quite some latitude. s=1 d=1 s+d= n+1 for s=1 d=1. 0+1=1=x. 0×1=0 multiplication isn't even a consideration at this point except for fraction. 1+2+1=4=x s+2d==2=s, where s=1 d=1 so 2×d=2=s. 2+1=3 3+1=4=x 2×s+3×d=3=s by parallel lines. Affine I guess. Gaussian = perpendicular? s,d=1,1 3×s+4×d=4×s s=1 d=1. 3+s+4+d=9 transpose equals 1? 7+2=9=x This is counting with dx. 2=av of 6 in 123. So we should be able to average 9. 7+1=8 7+3=10 8+9+10=27 2+7=9. 9 is the average. 27/3=9. Let's try the summands. 27+21=48. 4+8=12. 12÷2=6 . 468 halving 234=9. Halving again 2=11 3=21 4=44. 11+21+44=76 76/3= 25+1/3 25+4=29 29+13=42. 2+3+8=13 1+3=4 13+4=17 =9+8. Well the average was probably 22 of 11 21 44. Got stuck in a prime loop for odd numbers. 5,6 10,11 22 22 5+10+22 6+11+22 39,27 ÷3=13,9. 9,4 4,5. 13,9. Numbers pass through unchanged. 13×9=117 117+13=130. 9×13+13. Why carries work in multiplication and addition. x is the normal counting, completely disregarding dx. 3=s+4=d=>4=s. 1+2+3=6 core. 4+4+3=11 1+1=2=average. 222221 10+1=11. So 5-2's+1==>> 5×2+1=11 5+2+1==>>71,53. -> 124->having 62->31. 8+4=12. 1+3=4 31+12=43 4+7=11. A path in which multiplication cannot follow. The equations are obvious, bare. They can be further constructed. On Revisiting difinitive ro0. Of course this expands to the other addends of 9. 18 27 36 as well as 45 and the conjugates. 1×4=4 1+5=5 45 4+5=9 9+4=36 9×5=45 36+45=81 2×4=8 2×5=10 8+10=18 halving 4+5=9 9+18=27 obeys the odd even rules. 1+4=5 1+5=6 56 9n+y=45+6=51 9×5+6=51 4+5+6=16 10+5 so 9,1+23 1+×2+×3=6. 2+3=5 5+6=11 5+11=16. 9+7=16. Where 9×9+7=25. A rule for squares n+n+1. n=even or odd. n=1 1+1+1=3 9+11=20=18+2 3+1=4 2+2=4 n=2 2+2+1=5 5×9=45. 3×3=9 3+1+2=6=33 3×3+18=24=>27 2+7=9=>3,3,9 something there. 3×3×27=36 OK How to add in the rule of 9. Fin.

In base 60 the finite decimal points occur for division by powers of 2, powers of 3, and/or powers of 5? If we used a counting number base equal to n where n is the product of the first m primes. Division by the powers of the primes up to m would then result in a finite answer?

Please check the proof of Riemann hypothesis at: kzfaq.info/get/bejne/Y82ee5h-16ilZYE.html

On no derivation of the line that allows for connecting the line to the circle. Great instructions! Ty Let's see if we can put the line in sequence for a circle connectivity. I'm just jumping into the middle somewhere to conjugate(verbs) a number sequence.. 9=1 as in the ro9. Average =45 in (9×(n+1) for n+1 n+2 n+3... n+7 n+8 n+9. For n=1. 45+36=711=>81 7×9=63. 81+63=144=>12×12 1+4+4=9 9×1+4=13 9×1+4=13 13+13=26=>18+8. 8+8+1=17 1+7=8. 17+8=25 or 1+7+8=16. 25+16=36 sorts 5 4 6. S= 16 1+6=7 7+16=23 2+3=5. Collecting terms 4,5,6,7 s=22 d=3 2+2+3=7. 18+5=23 =>18+5 9=1 18=2 1+2=3 18+18+5=41. 4×9+1=37. 2+2+5=9. I am still working for the subcore, deriving the rest of the core. 789. 1+2+3=6 1×2×6=12 1+2+6=9 9+9+3=21=>> 3×7=21 3++7=10 10=2×5 9+1=10 10+7=17 1+7=8. 1×7=7 7+8=15 1+5=6. Multiplication can't derive 9. 9,6,6. 9×66=5454 nc or 594 9=1 514 =>10=9+1=> 6+3+1=>64,71,74 71+74=775 1414 154 145. 63+75=> 678. 145+154=299 18+99=117=13×9 where 4+=8 5+5=10 Note finally found . Halving 8,10 =4,5 doubling 16,20=>36. Even going to even all In the average of 1 in 4,5. Some proper order in n+1. The average 45 36.. for 1. Start looking at the decimals. 4,5,6,7 4+5=9 6+7=13. 4+7=11 5+6=11 4+6=10 5+7=12 10,12 11,11 9,13. 9,10,11,11,12,13. 9+10=19 11+11=22 12+13=25. 19,22,25. 19+22+25 both singularity and in pairs and triples. 16,66,317 n+1=1+6=7 or 9+7=16 9×1=9 7×1=7. 9×10=90 7×10=70. 90×70=160. 40×40=160. 9×16=144. 9×44 9+44=53 5+3=8 53+8=61 6×9+1=55. 53-> 45+3=48 4×9=36×8=44 48+44=92. 9+2=11 1+1=2 9+2=11, 9=1, 9+18=27. 2×9=18+7=25. 2×5+7=17. 13+9=22 Notes: Why do numbers become so unrecognizable? What is the step function in universal divisibility and those pesky decimals. Some work with tables: 21 18,3 983=20, 9=1. 23 18,5 985=22. 25 18,7 987=24. 27 18,9 989=26. 29 18,11, 98,11=19,28. Odd goes to even. 9×n, x,y in x=n 9×n+y=x,y. 9×n=>x,y-n if x=n. 31 18,13 8+3=11, 1111. 18+13=211 no carry 2×9=18 2+9=11. 11+18=29. 1+1=2 1+8=9. 9=1 9+1=10, 9+8=17. 10+17=27 9,1(10) 9,8(17). There are a lot of rotations. 10+90=100=10² 20+80=4²+8²+3=100 30+70=2²+4²+7²=100 4+6+9=19 1²+9²=82=>3+11=14 1+4=5. 1+4+5=10. 5 is half of 10. Etc. Conclusion, all numbers are cause-fully related. Halfing and doubling averages are important operation for predictability in using the numer system. More work will be required to fully bare this fact.

Thanks those videos are a treasure.

Omar Khayyam and Khwarizmi are Persians not Arabs

Sir,I could do differentiation without limits.It needs to understand Newton's method.Unfortunately we are all brought up in Leibnitzian education system,in which we learn by rote but not by thinking.We memorize formulas and rules without knowing or trying to know where they came from.

I really like constructive mathematics, therefore in some things I agree with your positions, for other I agree less. In this video, I think the problem is that for log and exp, one can approximate them, in the sense that one can prove, using formal arguments, that even adding more terms the value won't change too much (using the definition with epsilon and N, for example), while for the product of the P(n)'s, as you mentioned, as soon as P(N) = 0 for some N, then the whole product goes from 1 to zero, so a priori there is no clear argument to prove an approximation (in a formal sense with espilon and N), without knowing if such an N exists or not, i.e. having resolved the problem in the previous formulation. But I think this product could be thought more like a sorta of "Brouwerian (counter)example", that suggest how giving either true or false to a statement of the form "for all n ..." is only hiding the complexity of the problem instead of resolving it. Nevertheless a really interesting video! (Sorry for the numerous mistakes, English is not my main language)

To a common sense it appears that a quantity is less than,close to,bigger than is imprecise.It must be said 'how many times'.Limits cannot satisfy precision which is a must in logic.WHEN WE SAY DELTA DIFFERENCE GIVES EPSILON DIFFERENCE MAKES ONE TO PUZZLE AROUND WHAT ARE THE MAGNITUDES OF DELTA AND EPSILON .

On more, 789 2×2=4 and 3×3=9 2+2=4 3+3=6 2×4=8 3×9=27 2+4=6 3+6=9 2×8=17 3×27=81 2+6=8 9+3=12 1+7=8 8+8=16 Anyway the 2,4 6+2=8 8=cycle 6,9 =9 cycle. 7+8 cycles = 9 cycle. In 7+8=15 1+5=6. 7 cycle = 3+4=7 3+4=7 3×4=12 3+4=12 1234 in n+1 5 cycle =2+3 or 2×3=6. There is complete continuous map. Let the dust settle, i can prove this out. Therefor, we can triangulate all the numbers from counting to multiplication and addition properties and odd, even, p for a complete and comprehensive understanding of our number system by interpreting 7,8,9 and ultimately the rational circle.

Professor, I'm a student of logic and I've been quite interested in your proposals and the ultrafinitistic critique of mainstream "pure mathematics", and this series on mset/box arithmetic has been one of the most interesting projects I've seen for some time. As logic is my main interest, I would like to know if you have (or know) a ultrafinitist substitute for current classical logic (especially if it relates to mset arithmetic) you believe could be used for other forms of formal reasoning. I've seen Gajda's "Consistent Ultrafinitist Logic", "Model Theory of Ultrafinitism" by Mannucci and Cherubin and Terui's "Light Affine Set Theory"; what do you think of these proposals? Also, if you pardon me one more question, do you know other papers or materials using mset/box arithmetic for other areas of math such as group theory and topology? Thank you very much, professor, I've been loving your videos! Please continue with the great work! A warm salute from Brazil!!

Video Contents: 00:00 Introduction 3:15 Strongly Multiplicative Function 4:56 Euler’s Totient (fn) 8:25 ? 13:46 Applying Sum Operator 16:46 Taking Products Or Sum 20:12 More Multiplicative Functions 23:03 Helpful Tip

Gotta love the book 'Divine proportions Rational Trigonometry to Universal geometry N J Wildberger! 122/12=10.1666... Appling n+1. 10+1=11 1+1=2 11+6=17 1+7=8 17+6=23 2+3=5 23+6=29 3+9=11 1+1=2 29+6=35 3+5=8 35+6=41 4+1=5 41+6=47 4+7=11 1+1=2 47+6=53 5+3=8 63+6=59 5+9=14 1+4=5 59+6=65 6+5=11 1+1=2 65+6=71 7+1=8 71+6=77 7+7=14 1+4=5 77+6=83 8+3=11 1+1=2 83+6=89 8+9=17 1+7=8 89+6=95 95+6=101 10+1=11 1+1=2 101+6=107 10+7=17 1+7=8 107+6=113 1+1+3=5 113+6=119 1+1+9=11 1+1=2 119+6=125 1+2+5=8 125+6=131 1+3+1=5 2+8=10 1 11 2 1+2=3 10+5=15 6 21 3 6+3=9 15+2=17 8 25 7 8+7=15 1+5=6 17+8=25 7 32 5 7+5=12 1+3=3 25+5=30 3 33 6 3+6=9 30+2=32 5 37 1 5+1=6 32+8=40 4 44 8 4+8=12 1+2=3 40+5=45 9 54 9 9+9=18 1+8=8 45+2=47 2 49 4 2+4=6 47+8=55 1 56 2 1+2=3 55+5=60 6 66 3 6+3=9 60+2=62 8 70 7 8+7=15 1+5=6. 62+8=70 7 77 5 7+5=12 1+2=3 70+2=72 9 81 9 9+9=18 1+8=9 72+8=80 8 88 7 8+7=15 1+5=6 80+5=85 4 89 8 4+8=12 1+2=3 85+2=87 6 93 3 6+3=9 87+8=95 5 100 1 5+1=6 95+5=100 1 101 2 2+3=3 285285285. 2+8+5=15 2×8×5=80=8×9+8=72+8=80. 15÷40=.375. 3+7+5=15 3×7×5=105. 105=9×11+6. 15=3×3+6. 12+15=27=9×3. 120=93+27=9×13+3 25=2×9+7. Conclusion. Infinite decimals can't be added or even weaved together to make new decimals. How come there are no new videos? Have they stopped? The primitive core is detectable. Proved by universal divisibility. In this case with 3,6,9. ÷3=123. Or 3+6+9=18 1+8=9, ro9

On the question of 789, given there is a core, how to derive these 3 numbers? And is this fundamental? And given that the ro9 is not yet derived minutely. Will odd and even be fundamental? I see 3 kinds of derivations for 789. 1.) Purely in the form of addition or multiplication. 2.) Composite relation in the form of addition, multiplication. Especially since, now we know how they relate evenly and completely disjointedly. Core, p. Fundamental? 3.) Is odd and even fundamental? 4.) And in and of themselves, are the interconnectivity of these concepts mathematically decernable? 5.) Given the prospect of a rational circle these issues should be further delineated first.

On why the rule of 9 works in multiplication and addition. On retraction for the unit circle. That is a pretty strong statement that the disk and perimeter are impossible to relate. Since the number system has a core, the fundamental model of mathematics should reflect this fact. 1+2+3=6 1×2×3=6. We can derive more sequences. Ex. 1,6 2,6 3,6. Both in multiplication and addition. 1×6=6 2×6=12 3×6=18. 1+6=7 2+6=8 3+6=9. So, 6+7=13, 12×8=20, 18+8=27. Thus, 13,18,20.-> 4,9,2. Where 1×2=2 2×2=4 2×3=6. 2,4,6. In 122 224 236-> 5,8,2. 5+8+2=15 1+5=6. Returning the core. 1+2=3 2+2=4 2+3=5. 3,4,5. 123, 224, 235. 122,123 224,224 235 236. 247, 448, 4611-> 13,16,12. 13+16+12=41,14 The first conjugate in addition one at a time v. Pairs. This does flip oe to eo. Eo+eo=even,even. 44 11 16+1 8+2. 17,10->11,70->81. 8×1=8 8+1=9. 818, 819. -> 16 2 17 ->35 or 287=17 or 815 carry v no carry. Suming 8275=23=5. 8257-> 10,12 5,6 5+6=11=9+2 5+6=30=9×3+3. 10+12+56=78 7+8=15=9+6. Thus, 7896 or 15,15=30=9×3+3=15 1+5=6. 6 is the resultant core number. Conclusion, why, the rule of 9 works. This is the balancing of the two operations we know about, addition, multiplication. The most obvious manifestion of this truth is in the rule of 9. A closed form equation is possible. The number system itself is the highest level of proof. We'll see what kind of proof the impossible relating of the disk to the circle yields.

Professor, I was thinking like perhaps the three geometries actually fit into one rank-3 tensor g_{abc} where a, b, c run from 1 to 3. This acts as a metric and the quadrance would become a "cubance", and one works in the projective plane, therefore giving rise to 3 coordinates. Working in Riemannian geometry, the problem is how would curvature be defined from this rank-3 "metric"? And is there such thing as projective Riemannian geometry?

You Forgot Persians and especially Omar Khayyam

I solved the Riemann hypothesis. s=0 when zeta subzero=-.5=i=\sqrt{-1}=nontrivial zero.

On squares. 2->4 9,2 9,4 ->18,6 18+6=24 2×9=18 18+4=22 2+2=4 3->9 9,3 9,9->27,3 27+3=30 3×9=27 2+7=9. 4->16 94 91 96 -> 13,10,15. 13+31+10+1+15=16. Conclusion not very good at these yet. Had to use 2 conjugates for 16. Looked to 4 for help? I suppose the real question is what to do for unknows. And the general closed form for the equations. In time with more familiarity and practice.

Very interesting! Thank you. It is possible to accurately represent complex numbers by 2x2 matrices. For example, this can be used when your computer does not support the complex type. The key ingredient is the fact those 2x2 matrices commute. I understand the intrinsic difficulty for the formulation of electromagnetic theory (other theories too) is the fact there exist NO three-dimensional algebra that commute. That situation forced the physicists to insert "patches" into the theory like two different types of product etc... (The "symmetry" that makes any theory beautiful was lost!) The Quaternions theory does not make anything easier, it simply hides the multiple difficulties inside the definitions of the matrices. You still have to calculate the different products as if they were different operations. The matrix elements are not longer meaningful (intuitive), etc... I agree, that these new theories bring a multitude of possibilities to the young mathematicians. Not so much for the old engineers like me! LOL.! I really enjoyed the video. Thanks.

Until now in the series, he was drawing curves and circles in the plane; very "comfortable" are those complex numbers! Fun! But now he is mapping spheres and 3 D objects in the plane... I found this lesson very very difficult to follow.

Great topic! There is alot to learn and understand here! It looks like there are two et' on the slides or maybe a stray mark?. One for A and one for B. This is really a slippery set up from the first slide for me. Then r,s and s,r equals at least two different vectors. Then y=x for et' , ex' and for et,ex are the axis of symmetry for both coordinate systems. So they both have the same averages and the averages are 1:1 in either odd or even averages. The order of the relation going ccw ex ex' et' et all about y=x between ex' and et'. If the scaling on the slides are close then et'-et=ex'-ex=et'-ex' where et +et'=ex'+ex=ex'+et'. Difficult picking out the B world line =et'. A world line =et. That's a given then. Direction of realizing the vector et'=r s and ex'=s r. Have four possible assumptions for direction of vector ex et. r=et s=ex. Watch some more video to see the order of the dot product and start working with numbers. Yeah, this puts the hurt on my brain cells, jumbling of the units. x=etx. T=et×t. What if the order of the vectors are not the same as on the slide? Do the matrices become irrelevant? No clear diffinition if this is nomenclature or an actual multiplication problem or maybe just a magnitude, or position of some vector. Relating a coordinate with a vector is a fine a distinction. I dont think the actual number system works like this. I think the number was around before the dot and cross products and the predictability of the number ststem is not a result of these operations. Further thr relitivistic properties in nature are explained in odd and even numbers and their average summands. As simplistic of an approach as that seems, that approach is adequate to describe the predictability drivable from the counting system. I don't particularly care for matrices. Matrices are just a filing cabinet for certain orders of operations and to me are overrated. To put some kind of intuitive meaning to these numbers I would not take This approach. Onward. Overview. 5/4 3/4 5+4+3+4=16 16/4=4 average value 1+6=7. Middle summand, 3 4. 2 4 4 4, 9,12 3:2 =2/2+3/2=5/2 1/3 1/3 1/3=1 or .3×(33...)=1 = 1/2+1/2+1/2=3/2 3/6=1/2. 3+6+1+2=12=3. 3²+1=10. 5+1=6 6=3+3, 1:5.5, 10:1, 10+1:10×1/2 10:5 in 2:1 or in thirds 2/3:1/3. 10×1=10 10×1/2=5. A table in the ro9. 9+4=13. 1+3=4 18+4=22 2+2=4 27+4=31 3+1=4 36+4=40 4+0=4 45+4=49 4+9=13 !+3=4 22+31+40+49=25=18+7. 18+5=23 18+3=21. The extra step in 49 is transposing into a cycle. Think universal divisibility and the step function. 22+31+40+13=16=4². I see it's in the 9+4 cycle. Is there an independent cycle? 2+4+3+4+4+4+8+4=33. Summands 1+2+1+1=6. 4+2=6 18+4=22 2+2=4. At least one way to predictability return the identity in 4. 45+4=49 4+9=13. 54+4=57 4+8=13 63+4=66 6+7=13 72+4=76 13 4 81+4=85 13 4 90+4=94 13 4 99+4=103 4 One can merely keep summing 9 to the last total to find the same results. 103+9=112 4 112+9=121 4 121+9=129 4 129+9=138 13. Now these kinds of operations sum in and out of p. In addition the way these sum to the identity has a cycle also. And further these cycles depend on the number(location, magnitude) of origional digits and the relative location above or below the identity in the original digits. The rule of 9 in one digit, the last number in the sequence, for the whole sequence 0-9 is controlling! On why is there a core and subcore in our number system. We know the ro9 relatively well. What about the othe rules? 45 7+7=14 =9+5 9+5=14 7+6=13 =9+4 9+4=13 7+5=12 =9+3 6+5=11 =9+2 6+4=10 =9+1 6+3=9 =9+0 6+2=8 = 72+8=80 8+0 6+1=7 =63+7=70 7+0 6+0=6 =54+6=60 6+0 5+9=14 =1+4=5 9+4=13 14+13=27=3×9 3+9=12 so 12 13 14 or 3,4,5=12 12/3=4 so ±1, 141 or 345 so 6,12. The 3,6,9 or 486 falling back aspect of numbers, an undetectable cycle. 123445 4+4+5=13=1+3=4 9,4. 5+8=13 1+3 5+7=12 5+6=11 5+5=10 =9+1 5+4=9 5+3=8 5+2=7 5+1=6 5+0=5 4+3=7 4+2=6 4+1=5 4+0=4 3+1=4 4+3,01=71 4+1,0+3 53 summmands , need more distinctions anyway,p. 71+8=79 53+8=61. 79+8=87. 9n+y=x 9n drops x by n in multiplication and the summing. x+x≠2x sometimes. Ex 88+8=96 8×9+8=80. 9×7+6=69 80-69÷11. Ud 80-8=72 72+9=8 72/8=9 72+81=153. 153/9=17 153-17=136. 135/9=15. Something is here. 135/3=43 9×7=63 43+22,63+22=65,85 11,13 65+85=140 135/9=15 2+4+6=12 1+2+3=6 3+6+9=18 4+8+7=15=>6 34 87=1111 no carry. 109=11+98.109±20= 89,129 89×129=218 2+1+7. Returning 11 that means something, not just some hidden mystery. Gn 3+2=5 3+0=3 2+1=3 3+2+1=6 3+3=6 3+5+4=12 5+4=9 9+3=12 so 9,1 2,3 9123/9=1013r6. 2+0=2 2+1=3 2+2=4 ±111 123 345 ±222 012 456 ±333 -101 567 123+012+(-1)01=36 345+456+567=1368 9+18=27 1+2=3 10+20=30 11+22+33=66. 12+24+36=72 9×8=72. 9+3+18+6+27+9=>(3+2+1)×9=18+54=72 looking at steps. Somewhat undemonstrated.

Yes, that is how we see the phases of the moon, stereographic projection. On what should be the standard model in mathmatics. 1st by implementing the rule of 9 and proving again the rule of 9. Then building geometric shapes. Using position equals velocity multiplied by time as a non-confusing and established platform. For proving in different way the ro9. Then build the triangle which will allow extrapolations to other shapes. Physicist use units to check validity of their equations. So, lenght=(lenght/time)× time. Time cancels, lenght is the only unit left. So length=length. True statement,then. s=v×t. Asumming for now, constant speed in one unit of time. s=1, v=1. So, 1=1×1. Before we even do this I want to revisit the difinitive rule of 9 and make some more extrapolation. 1+4+1+5=11 11=2+9, where 14+15=29. Using 9n+y=x+1 in n=x. n=1, 9+y=1+1=9×1+1=2 in contrast for addition 9+n+y=x+n, n=1, so 10+y=1+1, y=1 x=1 xy=11. Modern math calls this multiplication. 1×1=1. 2+11=13. 211p. 2+1+1=4. First summand of 4=31=3×9+4. 39+31=70. 9n+y=9+7+0=63. 9n+y=n+x=Q. Q1=6×9+3=57. Q2=9×5+7=45+7=52. Q3=9×5+2=47. Q4=9×4+7=36+7=43. Q5=9×4+3=36+3=39. Q6=9×3=27. Q7=9×2+7=25. Q8=9×2+5=23. Q9=2×9+3=21 Q10=2×9+1=19. What this is yielding shows a cycle between squares, ro9,primes. The first postulant in the fundamental hypothesis of mathematics. Could there really be a fundamental theorem to mathematics? Quite conceivable, yes. This is very meritorious and deserves acknowledgment. Note 243=3⁵..

goatee Wildberger cannot be stopped, way to powerful!

8/11=7272... 9×8=72. 8+11=19. 1+9=10. 1+0=1. 7+2=9+1=10 1+0=1. So n+1 to every digit in the decimal. 7+1 2+2 7+3 2+4 7+5 2+6 7+7 2+8 7+9. 8 4 10 6 12 8 14 10 16. -> 8 4 1 6 3 8 5 1 7 8+4+1=13 6+3+8=17 5+1+7=13 -> (4 8 4 )/4=1,2,1,[or 9×12=108, 108+1=109, 10+9=19, 1+9=10 1+0=1. Anyway this is done there is n+1] 1+2+1=4. 4+9=13. 1+3=4. (.999...)/9=.111... .111/9=12345679. S=36 3+6=9. 11+8=19 1+9=10. 1+0=1 1+1+1+1+1+1+1+1=8. 8×9=72. 7+2=9. 19 9n+y=x. 9×1=9 9+9=18, 1+8=9. There are further expansions. Conclusion 8/11 is controlled as well as the whole number system, by the rule of 9. Maybe, not stop infinity, but surely infinity can be changed. Two rules of numbers: 1.) Numbers are infinite. 2.) Numbers can not be terminated only changed. 3.) I wonder what the third rule will be? All numbers are causality connected.

I feel like some of the problems in math are present even with Euclid’s definition of a point. I don’t know exactly what the Ancient Greek writing really says, but a common translation is “That which has no part or no magnitude.” It requires that a line segment of finite length, viewed as a collection of points, must contain an infinite number of points. It also leads to results where the perimeter inside a closed shape is the same perimeter on the outside of a shape. However when we construct geometric shapes with real materials the boundary always has a dimension to it. In many senses it seems that the rise of the continuum can stem from this definition of a point being without part or magnitude. However I am not sure if Euclid considered this to be equivalent to the number 0 or the number 1. It seems that the Greeks might not have considered the unit to have magnitude. Thus it seems possible to think that maybe the point and the unit are the same. If the point has a unit size then, a line segment wouldn’t contain an infinite number of them points… Part of the problem could be understanding what Euclid means by has no part, and by has no magnitude. It seems that the Greeks did not use zero conceptually and thus interpreting this to be zero might not accurately depict what Euclid was teaching.

Pr. Wildberger often mentions his interpretation of real numbers. He has many videos on the subject. Basically he sees it as a capitulation on the pursuit towards nicer mathematics. I use the opportunity provided by this message box to give my two cent on a related thought. For many many years, as a hobby, I have been writing a finite element program (my own!) in order to calculate electromagnetic fields in resonant cavities. In metrology, the accurate measurement of the resonance frequencies can be used to evaluate the mechanical dimensions of the cavity. Therefore my concern is focused on the precise determination of the eignevalues of the system. I assume that although the Finite Element Method is only an approximation, it could be improved with a higher order model. The final result is a (Lagrange like) polynomial with fractional coefficients. In one way or another, these coefficients always look like ratios of prime numbers. As expected, those numbers grow larger and larger as the order of the system increase. Fine with me as long as we have higher resolution. As opposed to "floating" (computer) numbers, it is not very easy to calculate on very large rational numbers. Truncating a large floating number result in an instant approximation, but rational number can not be modified. The only way to overcome this difficulty is to transform those beautiful fractions to floating numbers! What a pity... I have tried to find a solution using some kind of series of fraction where the high order terms could simply be dropped if the higher accuracy is not needed or the computer resources are limited. This way the fractional coefficients would be more 'universal' inserted in the model and then forgotten. I have never found what that theory! Any idea?

Thank You

Hello Respected Professor: At. 2:30 , to prove that m/n = p/q? My question is that, why do we brought a multiplication in the idea at the first place? Why did we started to multiply different entities at the first place. (ml)(kq) and (nk)(lp)? This is not clear, please could you expand on this?

thanks

On 9n+y=y-n, y+n 1+1=2 ->112->12×9+4 1×1=1 ->111->12×9+3 2+2=4 ->224->24×9+8 2×2=4 ->224->24×9+8 3+3=6 ->37×9+3 3×3=9 ->37×9+6 4+4=8 ->49×9+7 4×4=16->490×9+6 5+5=10->612×9+2 5×5=25->613×9+8 6+6=12->734×9+6 6×6=36->737×9+3 7+7=14->857×9+1 7×7=49->861×9+0 8+8=16->979×9+5 8×8=64->980×9+4 9+9=18->1102×9+0 9×9=81->1109×9+0: raw data. For all 'A' sequence. 1111->1,3 2,2 3,1. 2222->2,6 4,4 6,2 3333->3,9 6,6 9,3. 4444->4,12 8,8 12,4. 5. 1234 s=10. 369,12 . s=30 or 21 d=3+3+3+3=12. If including endpoints 3+2=5, d=12+5=17. 1×9=9, 7×9=63, deriving the conjugate from delta-d. 63+9=72. 72+27=99. Where 2×27=54 and 2×72=144. 54+144=198. 1+9+8=18=9+9 relates 22,11 20,10. 33,30 42,21 72,54 75,51 ->8,6 12,6 r4,3 2,1 4+2=6, 3+1=4->3,2 2,7,5 where 5+2=7 so 7,7. 7+7=14. 1+4=5 9+36=45. 22(2×9+4),23(2×9)+5.[19?+9=27, 2+1=3][24=18,6->9(1×9)+8=17][19+17=36][1+9=10 1+7=8, 10+8=18, so 2->4 with p on a path multiplication can't use directly],[17,18,19. 8+9+10=27 so 2->3->4. 4+3+2=9 then 1->3. 13+24=37,55. 37+55=92 9+2=11 1+1=2 11+2=13. 3+1=4 4×9=36 36+1=37. 3×9=27 27+7=34 3+4=7 27+14=41 4×9=36 36+1=37 3×9=27 2×9=18 18+7=25 9×2÷18 18+5=23 2+3=5 5×9=45 45+2=47... there is a rule and an equation here. This is getting circular. Beginning to talk about a diameter and a center. 11,13 is in here also.]. 4(2×9)+4 2,5(2×9)+5, etc. 1234 1111->6,6 9n+y=x for x,y 54(9×6)+6=60, 54(9×6)+0=54. 45(5×9)+4=49. 36(4×9)+9=45. 36(4×9)+5=41. 36(4×9)+1=37. 27(3×9)+7=34. 27(3×9)+4=31. 27(3×9)+1=28. 18(2×9)+8=26. 18(2×9)+6=24. 18(2×9)+4=22. 18(2×9)+2=20. 18(2×9)+0=18. One difficulty was increasing the sequence in 9n. Given 112,111 from above, can we make the sequence increase in 9n? ,12+111=123 to 132. Wasn't exactly the way I was trying to increase the sequence from 9n. That's just what the approach developed. Taking 11,21. 11+21=31(32?, 3+2=5,5×9=45, 45+1=46 4+4+5+5=19, there is also a 11 13 in here.), 3+1=4, 4×9=36. 31->27+1=28. 31+28=514(no carry) or 55. 55+64=(119). 64 from 514 5+1,4. 119+13=132. 13+2=15, 13+20=33, 32+10=42. So 33 42 15. Where 3+3=6 4+2=6, 1+5=6, so 6+6+6=18. =6+12=18. So 42+e48=90. 9+0=9. Ro9. The sequence increased from 123 . Conclusion, operations were not completed. The network is insufferablely slow.

A wonderful lesson, simply awesome. Thanks a ton and more

9+9=9 9×9=9 18 + 18=36, 3+6=9 8+10=18 1+8=9. 9+1=1+9. 3+6=9, thus 9+9=9. 9=9 however 9-9≠0 2+7=9 1+8=9 27-18=9 still equals 9. 9 can not be destroyed, only hidden. 9×9=81 (7+2)×(7+2) foil. 49 +14+14+4= 81. 49+4=53 53+14=67 67+14=81. Wow. 2 Sqrs, 2 primes, and a quadratic equation!!

Excellent!! Thank you!!!

If Zeno had been a buddhist then Norman Wildberger is his present day reincarnation (IMHO) and a master of the phrase "But, . . .".

What I find stranger are the Lorentz transformations, especially the gamma factor “1/sqr(1-v^2/c^2)”. This factor goes to infinity at v = c. The light quantum should have no expansion.

I understand the argument here. But what if we just say that it converges to "some number" but we will never truly know what that number is? We can only specify it up to a certain precision. At least I'm comfortable with that level of doubt.

Regarding the diagonalization method, what happens when the length of a number in the number of digits, does not grow with the same rate as the possible permutations of its digits? If we write out all 2 digit numbers what is the diagonal of this set? 00 01 02 03 04 … 99 What happens when the set is not a square set n x n? It would seem that the diagonal implies n x n. It also seems obvious that you could make any new m digit number where m=n+1 was not in the list of n digit numbers. Could you not also use this same method to show that any finite list of integers is also not a complete list of integers?

Regarding the diagonalization method, what happens when the length of a number in the number of digits, does not grow with the same rate as the possible permutations of its digits? If we write out all 2 digit numbers what is the diagonal of this set? 00 01 02 03 04 … 99 What happens when the set is not a square set n x n? It would seem that the diagonal implies n x n. It also seems obvious that you could make any new m digit number where m=n+1 was not in the list of n digit numbers. Could you not also use this same method to show that any finite list of integers is also not a complete list of integers?

You state without proof or motivation that perpendicular lines have spread of 1 and that supplementary angles have the same spread. Not intuitive at all, based on the provided definition.