Can you share your qualifications with us? That's been more or less de rigueur on this forum for those who are subject matter experts.
Rutgers Undergrad, Satan Hall Graduate both Chemistry,
First 14+ years spent in Pharmaceutical R&D. Promoted first 6 out of 7 years and into management while going to grad school at night. PhD's did not like reporting to someone with BA (from Rutgers College no less).
Last twenty years as Founder/CEO of a software product and consulting firm. Our customers are 7 of the top 15 in the world. Our software products are for R&D scientists and statisticians. I have no formal training in statistics or programming and yet two companies have globally standardized portions of their statistical evaluation process using the software I designed and created. One of them was paying SAS to explore this advanced concept and were told it is so complex it would take a team of 5 or more programmers along with a team of company R&D folks. The cost would be many millions (3.3 to be exact) and at least a year or more. I was done in less than 5 months with one other programmer and software validation analyst. I designed it to use a base SAS engine to boot. I threw in an interface system that allows users to connect it to almost any company data system and import their data directly into the desktop app for analysis. Both customers have been using it for 15 years. I get paid to help data scientists understand and solve problems.
So what are my qualifications? I put some stuff above because that is what people usually do. My view is I was given a gift, it is the 10 lb weight on my neck/shoulders. When I was young my school had me tested by a specialist and I guess was abnormally high on the distribution. On a personal note, I am not a fan of our education system as it is not designed well for those who are advanced. I went through undergrad and grad barely ever attending any classes as I found it boring. You want me to sit and write down what you say and then say it back to you a month later? I could train a baboon to do that. How about learning to critically think and solve problems. That is what our education should be doing. Life is a series of problems, some more difficult than others, that we are challenged to solve. My brain always seeks and prefers the elegant solution. Many people, even those highly educated, do not.
Here is a little anecdote: I don't need a calculator...never have. When I add or multiply numbers I work from left to right. I can get answers faster than you can write the problem down on a piece of paper. I can count upwards given any number meaning if you say 34, I can ssay 34, 68, 102, 136, 170, 204, 238, 272, 306, 240, 374, 408. I can go way faster than I can type or speak. When I add numbers I add them like this:
359
419
876
I see 163 sets of tens for 1630 with an additional 24 ones to give 1654. When I multiply I do it like this
52*68= 50*68=3400 and add 2 more 68's which is 136 for 3536. Sometimes for fun I will do 70 times 52 = 3640 and then subtract 2 52's to get 3536. My mind always seeks the simplest solution and then adjust to correct to solve the problem. I did not come up with that it just does it on it's own. So no credit to me.
When I was in third grade, I developed a routine to estimate square roots in my head to multiple decimal places and also created an error correction factor as the smaller the number the greater the error, once you approach trillions the estimate is accurate to more than 6 decimal places. My teachers understood from a young age that I never needed a piece a paper to answer a math question as I do it all in my head and it takes less than a second. Out of boredom one day as a 9 year old I decided to better understand the relationships betweeen numbers. I noticed that:
For squares: 2^2=4 3^2=9 4^2=16 and the relationship between them is to find the next in the series you add the two consecutive numbers together and add them to the previous square, i.e., 2^2=4 so 2+3 added to 4 is 9 or 3^2. 3+4 3^2=16 which is 4^2. So I decided how would I explain cubes, etc. I found:
For cubes: A*B x3+1 ---- 2^3=8 3^3=27 the difference between is 19 which is 2*3*3+1. So then the difference to the next number in the sequence is 3*4*3+1=37 so 37+27=64 which is 4^3. Next is 4*5*3+1=61 so 125 is the next solution which is 5^3
So my teacher asked me how would I explain ^4 series?
I said ^4: 1, 16, 81, 256, 625. What is the relationship? I looked at it and said to my teacher I see (A*B)*(A+B)*2+(A+B) as explaining the difference between each ^4. So (1*2)*(1+2)*2+3=15-------->1+15=16=2^4
(2*3)*(2+3)*2+5=65-------->16+65=81=3^4
(3*4)*(3+4)*2+7=175-------->81+175=256=4^4
Needless to say my teacher had me sent to a special program to be tested. I guess I see patterns where most people cannot. I was in 3rd grade.
For those interested, my thought process on how to estimate square roots in my head was this:
15^2=225
16^2=256
The difference between them is the two numbers added 15+16=31.
so sq root of say 243 is in between 225 and 256 and in fact (243-225=18) so 18/31 in between. So my initial hypothesis was the answer was 15.5806. But the actual answer is 15.588 so it underestimates it a fraction. What I also noticed was that as the number approaches infinity my result approaches the actual number. So sq root of 1,000,234 would be estimated to be 1000 + 234/2001 which is 1000.11694 and the actual answer is: 1000.11699. If you continue the estimate approaches the actual.