Simple math, ancient Egypt:

Originally posted by @eladar
Is that taught in basic math?

I don't think so.

Finding the inverse for functions is taught to anyone going to college in all math classes.

Logarithms and exponentials are also taught to all, but few understand them. I believe it is because of how it is taught. My kids seem to get it pretty quickly once I changed how I taught it.

So you are not creative, just educated, right?

Originally posted by @sonhouse
So you are not creative, just educated, right?

It depends on what you are talking about.

If you mean creative as in improving mathematical explanations so the average non math major type understands college algebra and calc one and two concepts, then I am pretty creative.

I would expect every educated person to be familiar with at least college algebra level stuff.

Speciality areas I would not expect most educated people to be familiar.

Originally posted by @eladar
It depends on what you are talking about.

If you mean creative as in improving mathematical explanations so the average non math major type understands college algebra and calc one and two concepts, then I am pretty creative.

I would expect every educated person to be familiar with at least college algebra level stuff.

Speciality areas I would not expect most educated people to be familiar.

How far up the scholastic ladder did you climb in math? MS? Phd?

Originally posted by @sonhouse
How far up the scholastic ladder did you climb in math? MS? Phd?

Basically a math minor. The math department head at ACU said that they offered a BA in math which then would allow me automatic entrance into Texas A&M's engineering master's program, but I went into education instead. I should have taken him up on it, but I wanted out of school at the time.

So have been teaching Algebra 2, Trig, AP Calculus and now AP Stats for the past 20 years or so.

The AP stuff has helped solid up some Calc stuff as well as learning how to use the ti 84. It is really a very powerful math tool.

Originally posted by @eladar
Basically a math minor. The math department head at ACU said that they offered a BA in math which then would allow me automatic entrance into Texas A&M's engineering master's program, but I went into education instead. I should have taken him up on it, but I wanted out of school at the time.

So have been teaching Algebra 2, Trig, AP Calculus and now AP ...[text shortened]... me Calc stuff as well as learning how to use the ti 84. It is really a very powerful math tool.

Yes it is, I have a lot of calculators, I kind of collect them sporatically, I use the casi0 115 series because of the hour minute second key which I need to do to calculate how long a run will take on our 4 sputtering machines. The problem is the productivity, that is to say the depth rate, how many angstroms per pass of the platen in front of the target, one machine might clock in at 70 angstroms per pass while another comes in at 50 for a certain RF power and backgroound argon gas pressure (a few millitorr, 5 to 10 or so) and the field strength of the magnets inside the RF magnetron target, some strong magnets in there, long story, but the stronger the field strength of the magnets, the more angstroms per pass all other things being equal. A lot of variables so I take the time of each pass, multiply that times the total passes programmed in, but that section of the program doesn't start for about a half hour from the time of parts loading to the pertinent coating, in our case, either aluminum, silcon carbide, silicon chromium and silicon dioxide (glass) and so I have to input the time of each pass, which varies from machine to machine and from run to run in a single machine, calculate from that the total run time so the operators downstream knows when to expect the parts to be taken out of the machine.

So I only need a scientific for that, one with the hours minutes and second buttons and not needing programmability.

For my hobby work, I use the TI 84 and Casio 9850 for calculations involving my study of gravitational lensing. I worked up quite a bit of self taught knowledge about that subject.

I also have an HP11C and an HP12C when I get in the mood to do RPN, which is somewhat more efficient by eliminating the need for parentheses. Also have the big guy the HP48GX which has the plug in's for math like the 'math pro' and another slot for more memory. Still pretty slow but it has a very steep learning curve.

I built my own four banger back in the 60's, a heathkit, add subtract, multiply, divide, and not much else, was stolen when I got back from employment in Thailand. No huge loss and the thing would have been dead in a few days since it had rechargable batteries. The idiot who stole it probably thought he had a thousand dollar piece of elelctronics when in fact I think it cost about 30 bucks as a kit🙂

Most of what you are talking about is something outside my expertise, but you did mention a need to avoid parenthesis. If you are working with compositions of functions and you don't need to do this too many times, you can enter your equations into the y-editor of the ti-84 and then use the vars to pull of the equation you want. It functions just like function notation, so if you want to keep the same variable you just do something like this y2(y1(x)).

If you want to simply avoid parenthesis you can use the ti-84's fraction button and put a 1 in your denominator.

Then if you want the ti-84 to solve near specific values, you can use the ti 84 solver, input the two Y's you are using to trying to solve.

An example, if you input one side of your equation into y5 and the other into y6, then in the solver you just put in y5-y6 or if you have a new one just type y5 into one field and y6 into the other. Putting your stuff into the y-editor would then allow you to see the graphs or tables so you can see what is going on.

After you input the equations, it should give you a guess. It will start trying to find it from there. Teaching AP stats this year has opened my eyes to the vars menu. I hope it will be very useful for kids who struggle with math so that they can program their calculators accurately to find solutions. I have older ti 84's so they do not allow the new math script in the solving area. Y1- Y2 not only makes it easier to see what is going on, but it also allows easier use of the solver. You don't have to input a new function anymore, nor do you have to use parenthesis. More user friendly math is good for people who struggle with math.

User friendly math is good for everyone.

Originally posted by @eladar
Most of what you are talking about is something outside my expertise, but you did mention a need to avoid parenthesis. If you are working with compositions of functions and you don't need to do this too many times, you can enter your equations into the y-editor of the ti-84 and then use the vars to pull of the equation you want. It functions just like func ...[text shortened]... ndly math is good for people who struggle with math.

User friendly math is good for everyone.

Have you ever used RPN? Stack arithemetic? They used to have just 3 or 4 stacks to pile in numbers upon which functions were done but the later ones like the HP48 or 28 has unlimited stacks, as much as your memory can handle but most work doesn't need more than 4 stacks. like 4 (in the display) enter, now 4 is in display plus 4 in stack 1. So simple stuff, go (-), it pumps down 4 and 4 and returns zero. Or 4, ent, 5, ent, 6, ent, 7 ent
So in a four stack 4 is on top, 5 below, 6 below that, 7 below that. So if you just hit + each time you end up with 4+5+6+7 with 22 in the display. But of course you are not limited to just that kind of simple arithmetic, a couple of numbers in stacks can be operated on by a whole program, such as the one I figured out (not original, I missed that by 2 years), where given a radius number and a mass, it outputs where the focal point is for that particular radius. (for gravitational lenses). The Schwartzchild radius is 2GM/C^2R where G is gravitational constant, C ^2 is speed of light squared, M is mass in Kilograms, R is radius in meters gives the Schwartzchild radius of a black hole.

But Einstein worked out the angle of deflection of light (in radians) is twice the Schwartzchild radius, so his went 4GM/C^2 R gives the angle of deflection of light passing by a star.

I simplified that by considering 4G/C^2 as a constant, which I called Z. Then I worked up the focal point formula which was just R^2/ZM where R is radius in meters, M in Kg, Z, my new constant is very close to 3E-27. That is what 4G/C^2 as a constant comes out to.

So that gives a focal length in meters of any mass input being mass and radius.

So you can do radius numbers of anything positive. For instance, I also figured the approximate focal point for neutrino's which would pass right through the sun so it would be focused by the mass below the surface, and giving a somewhat closer focal point than any electromagetic radiation like light or RF or x rays.

Einstein figured out there would be a focus about 80 billion km past the sun for light coming in from a distant star, say Sirius so all stars start to focus at that point, about 1000 AU away from the sun. The trick would be to figure out how far away that focal line goes. I figured that one out also.

Originally posted by @sonhouse
Have you ever used RPN? Stack arithemetic? They used to have just 3 or 4 stacks to pile in numbers upon which functions were done but the later ones like the HP48 or 28 has unlimited stacks, as much as your memory can handle but most work doesn't need more than 4 stacks. like 4 (in the display) enter, now 4 is in display plus 4 in stack 1. So simple stuff, ...[text shortened]... The trick would be to figure out how far away that focal line goes. I figured that one out also.

Nah, I've only worked with ti's.

Would what you are doing work well with matrices?

Originally posted by @eladar
Nah, I've only worked with ti's.

Would what you are doing work well with matrices?

No, it's just basic algebra mainly. My work with the focal point of gravitational lenses is just a simple algegraic. R^2/ZM = F (focal point past the object for light coming from a distant source) Earth has a focal point/line so does Jupiter, or Luna or Mars, I did them all very quickly and easily knowing the radius and mass of the objects.

Originally posted by @sonhouse
No, it's just basic algebra mainly. My work with the focal point of gravitational lenses is just a simple algegraic. R^2/ZM = F (focal point past the object for light coming from a distant source) Earth has a focal point/line so does Jupiter, or Luna or Mars, I did them all very quickly and easily knowing the radius and mass of the objects.

So f and z are the only true variables?

Originally posted by @eladar
So f and z are the only true variables?

Z, as I said, is a constant, ~=3 E-27. F is a variable, the focal length, M is a variable, mass in Kg, R is a variable, radius number in meters. F, focal point in meters. So take example of the sun and light coming in from way far, Sirius, say, 8 light years away, light skimming the surface of the sun would be using R as 695500000 (meters) squared= 4.83E17 Divided by Z times 2E30, (3E-27*2E30=6000 so 4.83E17/6000= ~8E13 meters, or 8E10 Km. or about 51 billion miles out. But say you want to know where the focal point is for R=2, it's 4 times out (2^2) so at radius 2 the focal point goes out to about 200 billion miles and so forth.

My general solution works for any mass, at least large mass like the sun, Earth, Jupiter and so forth. There is a lot more to be said about that but that is just my little derivation of the focal point. 2E30 is close to the mass of the sun in kilograms.

Originally posted by @sonhouse
Z, as I said, is a constant, ~=3 E-27. F is a variable, the focal length, M is a variable, mass in Kg, R is a variable, radius number in meters. F, focal point in meters. So take example of the sun and light coming in from way far, Sirius, say, 8 light years away, light skimming the surface of the sun would be using R as 695500000 (meters) squared= 4.83E1 ...[text shortened]... just my little derivation of the focal point. 2E30 is close to the mass of the sun in kilograms.

How does R=2 for the sun? The radius is constant, 695 700 000 meters.

Originally posted by @eladar
How does R=2 for the sun? The radius is constant, 695 700 000 meters.

No it is not. R can be any number. If R=2 that represents an altitude 695 million meters above the 'surface' of the sun . If you imagine the Einstein ring of light coming from a distant source, there can be a huge number of such rings all denoted by some R number. So a ring could be denoted by R=1 which would be light skimming the 'surface' of the sun, going past but now deflected inwards by the famous 1.75 arc seconds so the energy in that ring will converge to a focus at about 80 billion Km away from the sun.

But there are other rings, say a ring at R=1.1, an altitude of 765 million meters above the sun, and puts that at about 9.75 E13 meters out, or 97 billion Km or about 60 billion miles away from the sun, that is where the ring of light around the sun focuses.

Or R=2. In that case, 4 times the skimming light, or about 200 billion miles out.

Get the picture now? R is the NUMBER of radius numbers. So R can be 3 too which would put the focal point for that ring at 9 times the distance of the first focus, so about 735 billion km away from the sun.

And so forth, till the end of focusability, which is another story.

Originally posted by @sonhouse
No it is not. R can be any number. If R=2 that represents an altitude 695 million meters above the 'surface' of the sun . If you imagine the Einstein ring of light coming from a distant source, there can be a huge number of such rings all denoted by some R number. So a ring could be denoted by R=1 which would be light skimming the 'surface' of the sun, goi ...[text shortened]... ion km away from the sun.

And so forth, till the end of focusability, which is another story.

So R is your variable here where R is actually the ratio of the light's radius to planetary body's radius.

Do you find different values for consistantly growing R's? If so the Ti 84's table could be helpful.

Originally posted by @eladar
So R is your variable here where R is actually the ratio of the light's radius to planetary body's radius.

Do you find different values for consistantly growing R's? If so the Ti 84's table could be helpful.

It's basically a square function, 3XR, F=9times that of 1 R, 6R, 36X 1R and so forth. 6R is 4.17 million Km above the 'surface' of the sun and the focal point is 36 times 80 billion Km or 2.88E12 Km out.

That reaches a limit but that is another story.

The thing about a ring that high up is, visualize an ice cream cone with a small ball in the end. So the apex of the cone represents the distant star. The angle of the cone indicates where the 'Einstein ring' will be above the surface of the sun,

So there are two competing effects going on here. If you raise the angle of the cone, the opening as it flys by the sun, being higher, will be deflected less than an Einstein ring just skimming the surface of the sun. The funny part about that is that part of the story is just linear.

That is, at 1 R you get 1.75 seconds of arc deflection, at 2 R you get 1.75/2 NOT 1.75/2^2. So our last example, the R at 6 is 1.75/7 or about 0.3 arc seconds of deflection.

At some point making the angle of the cone bigger and bigger, you reach a point where the light from the distant source is going away from the surface, the Einstein ring is diverging. So at some R number, the amount of bending will only be enough to 'straighten out the Einstein ring, like an expanding smoke ring forced to go away parallel never converging.

So that is the last of the focal line. It turns out, analysing it geometrically like that, the maximum length of the focal line is approx. equal to the distance between the stars. So for Alpha Centauri, light from that star passing by the sun, the focal line will end at about 4 light years, so there is a 4 light year long spike of energy from AC and that's it.

For Sirius, the line goes on longer, the disance between Sirius and the sun is about 8 light years so the Serius focal line would extend 8 light years so there would be a spike 8 light years long from Serius.

The same goes for every star sending light past the sun. There will be varying amounts of energy focused since each star puts out is own level of radiation so Sirius would have more energy in its Einstein ring than Alpha Centauri at the same distance up from the sun, the same R number since Sirius puts out a lot more light than AC.

So the grand vision of all that is invisible energy spikes coming off ALL the stars in our galaxy, a grand vision at least to me, the energy lines a god would see if it saw all the radiation from all the stars. Spikes of radiation in lines all around our sun for instance, all starting at around 80 billion Km out and going on the length of the distance between the star being plotted.

I figured that one out by just drawing the two angles and matching them up for the resultant angle of deflection. Two competing effects limiting how far out into space a line of concentrated energy will be deposited and after that, not much of anything.