Exploring a Scientific Notation Curiosity -- What's Going On Here?

Inspired by a post on the calculators subreddit, this video explores why advanced scientific calculators in the $20 range were unable to solve what seemed to be a relatively simple problem.

[@cret859]

Hey! Scott ! Is there no LOG key on your cheap calculators ? At the time, this was a common problem with many limited range (typicaly -99 to +99 ) for ten's power calculators. All pocket calculators and first home computers suffer for this limitation in the 70's and 80's. Here is how to compute this out of ten's exponent range result on a limited or vintage scientific calculator (i.e. HP-15C): 9.1 EEX 31 CHS LOG 4 ENTER 1.6 EEX 19 CHS LOG * + display -105.224478. The integer part indicate me that the result is at 10^(-105) power. The fractional part FRAC 10^x give me the mantissa 0.5963776 So the final result is - 0.5963776*10^(-105)

[@scottcollins7513]

Very good thinking! I should have thought about using log, but it did not occur to me. Thanks for sharing this approach, C Ret -- it is a cool work-around.

[@cret859]

Yes, this is a cool work around. But you need at least a scientific calculator with LOG or LN function. There is a way to get the exact solution with a simpler calculator (as simple as a four banger + - × ÷ ): Start with ten’s exponent of the result : (-31) + 4 × (-19) = -107 note that the exponents add for multiplications or substrate for divisions such as logarithms. The result is something like X= x.xxxxx E-107 Next compute the mantissa : 9.1 × 1.6 × 1.6 × 1.6 × 1.6 = 59.63776 note that I repeat four times the multiplication by 1.6 since there is no exponentiation on my "calculette à quatre opérations” Finally, the result is 59.63776 E-107 we can also write it as 5.963776 E-106 or 596.3776 E-108 in engineer notation...

[@scottcollins7513]

@@cret859 The approach described here was the direction I went. Happy to have calculators that can handle the original problem without any gymnastics required.

[@metalcann]

What's that sick looking sketch pad your notes are on? Thanks

[@scottcollins7513]

Thanks -- it's called a "reMarkable". Mine is a first generation model. It works very well.

[@dooda1193]

on the more expensive ones,,, which one can go highest...pls check,,,10 to 10000 ?

[@scottcollins7513]

My hunch is the HP Prime would win. But, I'm not going to spend the time trying to confirm my hunch.

[@robjordan63]

HP Pioneer calculators 17bII (RPN) and 27S (Algebraic) both get it right.

[@scottcollins7513]

hp calculators were so great.

[@GalaxyFur]

Now calculating 11^6 ÷ 13 = 156158413/3600π exact answer on the Casio calculators. Say What? It does give the proper 10-digit decimal answer. "Matt Parker" covered this 2 years ago on his channel Stand-up Maths on KZbin. To be fair 156158413/3600π is accurate to 12 decimal places he said. But it's still not quite exact. (Perhaps due to the lack of a CAS capability to draw from?) I'm not very good at math. But I'm trying to teach myself. And watching all this math video stuff I'm finding interesting.🙂

[@scottcollins7513]

I think it is great that you are exploring math even if it is not the easiest thing for you. And, really cool that you are finding some of these nerdy videos helpful in your journey. Appreciate you letting me know. Your comment about my discovery being similar to the "Matt Parker" video is a very good observation. It seems both are directly related to the calculator's limited precision (having to round answers). And, you are right about both examples being due to the calculators lack of a CAS (Computer Algebra System). If a calculator has a CAS, it can provide exact answers and does not have to approximate anything. I think you are better at math than you realize! Your recognizing the parallels between Matt Parker's video and this one demonstrates some math chops!

[@AVL-Tree]

Do you know of any scientific calc that can do matrix with complex numbers? (Not using indirect method of var storing and then Cramer's Rule) For the FX-991EX, normal matrix for Ax + By = C equations could be found by using Simul Equation function. However, it will not allow complex number entries. Here's an example of a matrix with complex numbers to solve: (20 + 5i)x1 + (0 - 5i)x2 = (60∠0°) (0 - 5i)x1 + (0 - 15i)x2 = (90∠90°) Thank you.

[@scottcollins7513]

Hi, Giorno. I just tested your question on the HP Prime and the HP 48GX -- both handled it well. I entered the constants in a 2x1 matrix and the coefficients in a 2x2 matrix. I then divided the constants matrix by the coefficients matrix and got the answer. I plan to check the DM42 later.

[@scottcollins7513]

I just confirmed that the Free42 app for iPhone (and the HP42S and SwissMicros DM42, as well) also handles the complex matrices. It is not quite as easy as the Prime or 48GX, but still works well and gives the same x1=[2.25 -2.25i] and x2=[-6.75 0.750i] result. If you are used to algebraic-entry (infix notation), then the HP Prime would be the easiest to learn and most economical to buy (*edit* UNLESS you just go with the Free42 app which works great!).

[@scottcollins7513]

@@AVL-Tree I like the fx-991EX. If I was doing the FE, though, I would want the TI-36X Pro just b/c I think you can save a little bit of time by being able to grab previous results from the stack. That said, the fx-991EX is a great calculator, too. Yes, the Free42 by Thomas Okken is the right app. It works great. In fact, it is the code underlying the SwissMicros DM42 calculator. I am not aware of a strictly scientific calculator (non-programmable) that can handle complex matrices. The DM42 is the closest thing that I'm aware of, but I am by no means aware of all things calculator, so it's quite possible that there is something out there that can do it, but that is a relatively obscure calculator. I sure hope the FE does not throw complex matrices at you -- engineering is hard enough with that "complexity"!

[@scottcollins7513]

@@AVL-Tree 991EX is much faster in my experience, but I don’t know the specs.

[@Crazytesseract]

@@scottcollins7513 Yes, thanks to you, I realised 42S or Free42 does it, but it's complicated. Thankfully, I do not need to work on complex matrices.

[@gerardzi7930]

Use your HP-28S it's better calculator than Casio :-)

[@scottcollins7513]

It should be. It cost a mint back in the day!

[@saujanyapoudel7378]

Can you please explain exactly why is it. Does hp have some more exclusive features?🤔

[@gerardzi7930]

@@saujanyapoudel7378 HP-28S have the RPN (Reverse Polish Notation) When Do You Use a Bracket and a Parenthesis in Math?

[@scottcollins7513]

@@saujanyapoudel7378 HP28S was programmable, had a great solver, had RPN as Gerard mentions, was fluent with matrix and imaginary numbers math. Much, much more powerful than the 991EX even though we are talking about 1987. Amazing.

[@scottcollins7513]

But also had a steeper learning curve and was geared for engineers.