Scientific Notation for Numbers?

[MrPeabody]

Question: At what size (both huge and tiny) does a number become more easily understandable in scientific notation? I'm pondering this question for myself. I bet it's different for different people.

The answer to this question is absolutely different for different people. I became intimately familiar with scientific notation a very long time ago. One of the important, fundamental advantages is that the true accuracy of the number is conveyed by the number of significant digits. Of course this requires that it be used properly. If you see a number like 5.000x10^3, you know that the number has four significant digits, because those zeros wouldn't be there otherwise. If you see 5,000 instead, there's no way to know. Most people would implicitly assume that all three of the zeros in 5,000 are significant, but in general this is not a valid assumption.

Engineering notation seems very similar to scientific notation, but the twist added in engineering notation is not without a subtle complication. In scientific notation, the radix point is always located to the immediate right of the first non-zero digit, and the power-of-ten exponent can be any integer. In engineering notation, the exponent is restricted to integer multiples of +/-3; the radix point is shifted as needed to accommodate this requirement. The scientific number 5.7x10^-4 is written .57x10^-3. Or is it written 570x10^-6? Hmmm. How many significant digits are there in this number?

In scientific notation, the rule is that the least significant digit does not need to be fully accurate but must contain some accuracy. Occasionally, there will be a number with just one significant digit. In scientific notation, 6.x10^-5 is a proper number, notwithstanding that there are no digits following the decimal point. In engineering notation, is .06x10^-3 proper, or is 60x10^-6 proper? If it happens that 60x10^-6 is deemed proper or acceptable by whatever governing body is authoritative for the use of engineering notation, then to me it seems that engineering notation has lost a fundamentally important attribute of scientific notation. If 60x10^-6 is deemed properly equivalent to 6.x10^-5, then it would seem to me that every number written in engineering notation should be accompanied by a tolerance value.

 

[paulraphael]

How do you change your preference to Roman numerals?

Someone once wrote a replacement calculator for the Mac that used them. It was called Calculatorium.

 

[Wes]

Engineering notation is nice because you can make quick comparisons across a small scale

6.5e9 vs. 67.5e9

Scientific notation is for bigg scales - where the exponent gives you the scaling

I was not aware of Eng. notation until I got an hp calculator that had it

 

[egellings]

Why is scientific notation needed for what gets discussed in hobbyist audio? You could also use engineering notation where the exponent is always in multiples of 3, I 'spoze.

 

[MrPeabody]

Are you joking?? Why on earth do yo want to see 3.28 x 10^2 instead of 328?
or maybe you are too used to the HP 41c calculator using 1 ENTER 2 ENTER + =2 instead of 1+1=2

I have no idea whether the OP was joking. I expect you were joking about HP 41c input syntax, i.e., RPN (Reverse Polish Notation). But whether you were purely joking or maybe being a little serious, a cursory look at the real advantages of RPN won't cause any long-term damage.

In RPN, you do not press ENTER after the second operand (unless you want to add 2 to itself and leave the 1 remaining on the stack), and you do not press = after pressing the key for the operation you want to perform. The number of keystrokes needed to add two single-digit integers is the same for both RPN and traditional algebraic entry: four. To appreciate the advantage of RPN, it is necessary to consider more complex algebraic expressions where, when you use traditional algebraic syntax, you will encounter multiple, nested pairs of parentheses. The only well-defined advantage of the use of RPN syntax in the routine interaction with a calculator is the avoidance of those nested parentheses. Many people obviously believe that this advantage doesn't compensate for the lack of intuitiveness of RPN, but it isn't clear why they think that RPN isn't intuitive or why they have any particular difficulty with it.

The less clearly identifiable advantages of RPN are with the synergy between RPN and the run-time stack. These two constructs are joined at the hip, and they are the key features of the user interface in the RPN calculator. For a good many people, the runtime stack and RPN syntax are so completely intuitive that they are at a loss to understand why all calculators aren't clones of the HP calculators that were based on RPN and on RPL (the programming language that evolved naturally from RPN). There are lots of people who fully grasped the stack concept literally within a minute or so of being exposed to it, and who fully adapted to RPN with no difficulty whatsoever. The truth of the matter is that neither the stack construct nor RPN is particularly difficult. In fact, if object-oriented programming is placed at one end of the spectrum of difficulty for programming constructs, the runtime stack and RPN/RPL belong very near the opposite end of that spectrum. The only downside that I've ever noted with RPN or RPL is the difficulty of keeping track of the state of the stack when writing a program. This is undeniably difficult, and I would say that this is the only reason that RPL isn't easy. For anyone who isn't trying to write an RPL program, though, this difficulty doesn't apply so much to immediate use where you typically don't need to have more than two or three numbers on the stack at any given time, and where the calculator display continuously shows you what's where on the stack.

Not wanting to overlook something that shouldn't be overlooked, I'll note that many modern, serious calculators have symbolic capabilities, e.g., the ability to solve and rearrange expressions symbolically, to factor, etc. When the reason for inputting an algebraic expression is for this sort of symbolic use, the equation editor that takes advantage of the multi-line LCD display is the right tool for the job, and the only alternative is to input the expression as a command line expression but using traditional algebraic syntax complete with the nested parentheses. With the premium HP calculators that are based on RPN, sadly no longer in production, the user has the ability to input an algebraic expression using traditional algebraic syntax, simply by placing the expression within quotes.

 

[PaulD]

I have no idea whether the OP was joking. I expect you were joking about HP 41c input syntax, i.e., RPN (Reverse Polish Notation). But whether you were purely joking or maybe being a little serious, a cursory look at the real advantages of RPN won't cause any long-term damage.

In RPN, you do not press ENTER after the second operand (unless you want to add 2 to itself and leave the 1 remaining on the stack), and you do not press = after pressing the key for the operation you want to perform. The number of keystrokes needed to add two single-digit integers is the same for both RPN and traditional algebraic entry: four. To appreciate the advantage of RPN, it is necessary to consider more complex algebraic expressions where, when you use traditional algebraic syntax, you will encounter multiple, nested pairs of parentheses. The only well-defined advantage of the use of RPN syntax in the routine interaction with a calculator is the avoidance of those nested parentheses. Many people obviously believe that this advantage doesn't compensate for the lack of intuitiveness of RPN, but it isn't clear why they think that RPN isn't intuitive or why they have any particular difficulty with it.

The less clearly identifiable advantages of RPN are with the synergy between RPN and the run-time stack. These two constructs are joined at the hip, and they are the key features of the user interface in the RPN calculator. For a good many people, the runtime stack and RPN syntax are so completely intuitive that they are at a loss to understand why all calculators aren't clones of the HP calculators that were based on RPN and on RPL (the programming language that evolved naturally from RPN). There are lots of people who fully grasped the stack concept literally within a minute or so of being exposed to it, and who fully adapted to RPN with no difficulty whatsoever. The truth of the matter is that neither the stack construct nor RPN is particularly difficult. In fact, if object-oriented programming is placed at one end of the spectrum of difficulty for programming constructs, the runtime stack and RPN/RPL belong very near the opposite end of that spectrum. The only downside that I've ever noted with RPN or RPL is the difficulty of keeping track of the state of the stack when writing a program. This is undeniably difficult, and I would say that this is the only reason that RPL isn't easy. For anyone who isn't trying to write an RPL program, though, this difficulty doesn't apply so much to immediate use where you typically don't need to have more than two or three numbers on the stack at any given time, and where the calculator display continuously shows you what's where on the stack.

Not wanting to overlook something that shouldn't be overlooked, I'll note that many modern, serious calculators have symbolic capabilities, e.g., the ability to solve and rearrange expressions symbolically, to factor, etc. When the reason for inputting an algebraic expression is for this sort of symbolic use, the equation editor that takes advantage of the multi-line LCD display is the right tool for the job, and the only alternative is to input the expression as a command line expression but using traditional algebraic syntax complete with the nested parentheses. With the premium HP calculators that are based on RPN, sadly no longer in production, the user has the ability to input an algebraic expression using traditional algebraic syntax, simply by placing the expression within quotes.

While I am a useless Forth programmer, I cannot use a normal calculator because I grew up with RPN calculators. I have a 50th anniversary HP 32S RPN that was only available to employees - I love it, and still use the RPN mode on the Mac calculator on my computer and phone apps.

 

[Blumlein 88]

I hate RPN. If my only calculator were RPN, I'd be using pencil, paper and a slide rule.

 

[restorer-john]

I hate RPN. If my only calculator were RPN, I'd be using pencil, paper and a slide rule.

I think it's what you get used to.
Scientific calculators through the 1980s were totally different to the DAL (direct algebraic logic or whatever) calculators the kids get now. I still use old scientifics where the operator is after the value.

 

[Blumlein 88]

I think it's what you get used to.
Scientific calculators through the 1980s were totally different to the DAL (direct algebraic logic or whatever) calculators the kids get now. I still use old scientifics where the operator is after the value.

I was around when calculators first became available. I worked part time as an assistant to a university physics professor. He loved those first HP's with the RPN. I eventually saved up and purchased my very own TI30. I had used his HP for a while. It never set well with me. When I finally had my own TI30 I was done with that goofy HP. I had used a slide rule for years already and it was second nature to me.

The TI30 had the algebraic entry in 1976. I purchased one in 1977. They were $25 when the cheapest HP was over $100 still. It looked like this, and I still have mine. Last time I put a battery in it the thing still worked though many keys were sticky. That was their nature to get sticky keys pretty quickly. I later switched to solar powered Casios which as far as I can tell you can't wear out. Of course use my cell phone these days.

Here is the prof's calculator which was the first HP35. I remember the blue keys on it. He had the HP45 which he carried in his suit pocket which is why he let me use his HP35.

 

[restorer-john]

The TI30 had the algebraic entry in 1976. I purchased one in 1977.

No, the TI-30 has the same entry as all the 1980s ones, but not the hardcore full RPN of the early HPs. I have one right here actually and it still works of course!

For log 100, you type 100, log. for 4.7x10^-6, you type 4.7 EXP 6 then the +/- key to make it negative.

Casio have VPAM/S-VPAM (visually perfect algebraic method) and Sharp use DAL (direct algebraic logic), not sure what TI call theirs but it's the same.
For log 100, you type log 100. Sin45 for Sin45. For 4.7x10^-6 you type 4.7 EXP -6 (note, there is no +/- key anymore you use the minus operator key, just like you'd write it)

Basically, you have to think ahead (like we all did) with the old scientific calculators. Just like doing all the denominator calculations first and then using the 1/x key or dumping it in memory and performing the functions only to use it later. The old X-M key got a hammering. The modern ones you just key it in like you were writing it. You never need more than the orignal expression's parentheses, whereas older scientifics you could end up nesting quite a number to overcome the 'logic' of the 'half' RPN.

PS. I picked up a lovely Casio graphing scientific the other day for $2. Three colour LCD with controls for each colour. I can't resist scientific calculators. I still have the one my parents bought for my senior school years.

 

[Blumlein 88]

No, the TI-30 has the same entry as all the 1980s ones, but not the hardcore full RPN of the early HPs. I have one right here actually and it still works of course!

For log 100, you type 100, log. for 4.7x10^-6, you type 4.7 EXP 6 then the +/- key to make it negative.

Casio have VPAM/S-VPAM (visually perfect algebraic method) and Sharp use DAL (direct algebraic logic), not sure what TI call theirs but it's the same.
For log 100, you type log 100. Sin45 for Sin45. For 4.7x10^-6 you type 4.7 EXP -6 (note, there is no +/- key anymore you use the minus operator key, just like you'd write it)

Basically, you have to think ahead (like we all did) with the old scientific calculators. Just like doing all the denominator calculations first and then using the 1/x key or dumping it in memory and performing the functions only to use it later. The old X-M key got a hammering. The modern ones you just key it in like you were writing it. You never need more than the orignal expression's parentheses, whereas older scientifics you could end up nesting quite a number to overcome the 'logic' of the 'half' RPN.

View attachment 120593

PS. I picked up a lovely Casio graphing scientific the other day for $2. Three colour LCD with controls for each colour. I can't resist scientific calculators. I still have the one my parents bought for my senior school years.
View attachment 120594

View attachment 120595

True, I forgot how nice the first Casio was when I purchased on of those (for something around $10 I think). The half RPN was way better than full RPN even so. It took like almost zero time to get used to the Casio VPAM method.

I had two of the old TI30s, one SR40, one solar TI30 and then I purchased the exact Casio you have in your picture. Had the little 4 way button on it. It was used, and used and used. It is still kicking around here in a kitchen drawer somewhere. Also have a Casio FX260 solar here somewhere. Amazingly they still make and sale that exact FX260. And they make the one with the 4 way button or one very nearly identical to it.

Back in the early days I would loved to have had one of those graphing calculators. By the time they were cheap however I would do any such things on a computer so never replaced my two Casio calc's for portable calculating.

 

[Wombat]

I got my HP33E out of its box recently, and it was dead - not PS or batteries.. It happens it seems.

I purchased it after I left my HP45, in my briefcase, on a train. Heartbroken.

I noticed the current HP35s, with RPN, is around $AU108. I must get one.

 

[restorer-john]

I got my HP33E out of its box recently, and it was dead - not PS or batteries.. It happens it seems.

Did you dismantle it and attempt a repair? Don't throw it out, it may be able to be rescued.

 

[KSTR]

I love RPN calculators. I had HP32E's when I was a school kid. It had two advantages, (I) nobody would ask me if they could borrow it, (II) I was much faster in doing longer calcuations with lots of parenthesis levels where regular notation is just f*cking up. The stack principle is so natural and elegant (that's why I also quickly fell in love with FORTH later).
The early ones didn't have soldered ICs, there were only pressed onto a soft flex PCB with a plastic frame. When you twisted the case you could make some pins lift and sometimes this produced nice endless loops with funny numbers running through the display.

As for topic, I'm all in for dB's (or any other log based measure) for everything, with no more than two significant digits plus a 1-digit tolerance value.

 

[Wombat]

Did you dismantle it and attempt a repair? Don't throw it out, it may be able to be rescued.

No. I am pretty well past working on stuff. I will miss the old red led display on the new one. I won't miss the short battery life between charges, though.

 

[Wombat]

No. I am pretty well past working on stuff. I will miss the old red led display on the new one. I won't miss the short battery life between charges, though.

P.S. I'll give it to you, all original packaging and contents.

 

[restorer-john]

P.S. I'll give it to you, all original packaging and contents, if postage is reasonable.

Just post it to me and if I can fix it, you can have it back- on me. (or not whatever). I have a soft spot for HPs and all the time and head scratching that it produced or helped solve.

Postage shouldn't be more than 10-15. let me know and I'll pay for it.

 

[q3cpma]

Might be the right place to post https://knightos.org/ to amaze people.

 

[MZKM]

I took two years of Latin in high school so I don't mind, but yes I don't get the continuance of this. They were scheduled to end it but went on.

54 or 55:

 

[rdenney]

I used a Keufel & Esser calculator starting out. Picket calculators, popular with my mates, were so pedestrian.

I had an SR-something by TI (and being in Texas made me partial), and the keys stopped working reliably. I’m stubborn, though, and I kept buying TIs up through the TI-59. I even had the print cradle for it, and wrote transportation planning software for it as a student worker for Texas Transportation Institute. Software was stored on magnetic cards that the calculator would feed through a slot. The keys got flaky on that one also, after a while, as did the NiCad battery packs.

In grad school, I bought an HP11C, using the new LCD technology. I still have it on my desk, and I still use it. I bought my first iPhone 3 when I discovered an HP11C app for it:

The keys still work, and it uses common button cells. (Low power draw from the LCD makes that work.) HPs were more expensive, but they used far more reliable key switches.

I have never used a handheld graphing calculator—computers came into my life before I ever visualized a need.

Rick “need to change calculator batteries—third time since 1983” Denney