A question for the Dremel afficionados

[Nicholas Farr]

On Nicholas Farr Said:

… however I think fractions often get a raw deal, and many only associate them with feet and inches, and consider then to be too complicated, but they are quite universal, e.g. 25% of 1M is 250mm and a 1/4M, 25% / 1/4 0f 1 Tonne is 250 Kg. / 1/4 Tonne, 25% of 1 Ton is 5 CWT / 1/4 Ton. The thing is they work with both imperial and metric,…

Regards Nick.

Don’t get me wrong about fractions, they’re fundamental, and very useful in their place.   Particularly good for maths involving ratios, especially common cases like sharing a quantity between ‘n’ people.  Before the industrial revolution, most commerce and building work was well served by fractions.  Then life got complicated!

Unfortunately it seemed convenient to our forefathers to build fractions into the weights and measures system by creating ad-hoc units.   Very often done by region, or trade, or both.  But, after a few centuries of aggro, pre-decimal Britain had simplified Imperial considerably:

12 inches make 1 foot  (¹⁄₁₂)
3 feet  make 1 yard (⅓)
5½ yards make 1 Rod, Pole or Perch (²⁄₁₁)
40 Poles make 1 Furlong (¹⁄₄₀)
8 Furlongs make 1 Mile (⅛)
3 Barleycorns make 1 inch (⅓)
12 lines make 1 inch (¹⁄₁₂)
4 inches make 1 hand (¼)
9 inches make 1 span (⅑)
18 inches make 1 cubit (¹⁄₁₈)
2½ feet make 1 pace (⅖)

21 shillings make 1 Guinea (1 pound + ¹⁄₂₀ ie 5%)
20 shillings make 1 Pound (¹⁄₂₀)
5 shillings make 1 crown (⅕)
8 half-crowns make 1 pound (⅛)
10 florins make 1 pound (⅒)
12 pence make 1 shilling (¹⁄₁₂)
4 farthings make 1 pence (¼)

16 Drams make 1 ounce (¹⁄₁₆)
16 ounces make 1 pound (¹⁄₁₆)
14 pounds make 1 stone (¹⁄₁₄)
28 pounds make 1 quarter (¹⁄₂₈)
4 quarters make 1 hundredweight (¼)
20 hundredweights make 1 ton (¹⁄₂₀)

Even though I’ve left out liquid measure, square measure, nautical measure or any of the trade measures, I hope it’s obvious that, taken as a whole, even after a decades of trying to clean it up, the Imperial system based on fractions is clumsy and obscure.   The maths starts well, nice and simple, but then grows into a monster!

Part of the problem is that Imperial relies heavily on fractions, which only work well inside their comfort zone.  Another is that Imperial is so simple and cuddly on the surface.  It’s the Nigerian Prince of Weights and Measures!  Seems great to traditionalists building an LBSC loco from a fractional dimensioned plan with their inch based Myfords, and then enjoying a friendly pint down the pub.  They’re not wrong, but their experience is of Imperial in it’s easiest basic form, where the system hasn’t shown it’s horns yet.

The real trouble kicks off when serious maths is attempted.   It’s not that Imperial and fractions don’t work, it’s that the system is internally incoherent – a mess.  In consequence, maths using it is full of conversion factors needed because the system encourages many different units for the same thing.   Miles per hour, feet per second, and many others, everyone an opportunity for mistakes.

Metric is a massive simplification, largely because it was designed to eliminate unnecessary conversions:

• Fractions are eliminated in favour of decimals, based on 10.
• Units are defined from single standards, one each for length, weight, time and amperes.  Sub-units are always scaled in powers of 10.
• As far as possible, metric units are related rationally.  Unlike Imperial, where relationships between units are an accident of history!

Back to fractions, consider this example:

⅐ + ⅛ + ⅑ = ?   (Hint: Lowest Common Denominator)

⅐ + ⅛ = ¹⁵⁄₅₆

¹⁵⁄₅₆ + ⅑ = ¹⁹¹⁄₅₀₄, or 0.37896825396825

Much easier to do the same sum in decimal:

0.142857143 +
0.125 +
0.1111111111 =

0.378968254

I look at decimal as a general purpose fraction.   0.125 is shorthand for ¹²⁵⁄₁₀₀₀, and 0.142857143 is shorthand for ¹⁴²⁸⁵⁷¹⁴³⁄₁₀₀₀₀₀₀₀₀₀₀  As decimal fractions all have factor 10 denominators, they follow the rules of ordinary arithmetic.  Compared with fractions, decimals are docile, and they facilitate mechanical aids like log tables, slide rules, and calculators.

Weird thing about the psychology of change is that although many want to retain Imperial Measure, no-one wants to go back to £sd.   I think because we all had to do mildly hard sums in £sd, and that it had sharp horns was painfully obvious.   Is two thirds of a pound 13/4d or not?

Now I’ve mentioned them, anyone got any experience with slide rules?  Not truthful devices by the standard Michael hoped of Bosch/Dremel, but still brilliant in my opinion.   Occasionally use my school Aristo to this day!

Dave

[duncan webster 1Participant @duncanwebster1]

SOD hasn’t scratched the surface. How about converting horsepower to foot pounds per minute, or btu per hour to horsepower, and the endless scope for confusion between mass and force. In 40 years doing engineering for a living I never came across anyone who preferred doing calcs in imperial.

And yes, I still have my slide rule, an easy way to get things wrong by factors of 10.

[Nicholas FarrParticipant @nicholasfarr14254]

Hi Dave, I started learning the basics of the slide rule in secondary school, and continued to use them at college, which we were allowed to use in college exams, but electronic calculators were banned in college exams back then, not that I could afford to buy one back then, but even my slide rule was a budget one from Boots, which I gave to my eldest grand daughter a while ago, which is a double sided one.

I do however, still have my late elder brother’s British Thornton P271 slide rule, which he would often use, working out quite complex equations, while remembering all the figures in his head, and that would also include fractions at times. He never ever revised before his exams, as he used to say, “you either know how to answer the questions, or you don’t” and he passed them all with good grades.

I still know how to use one, but nowhere near what my late brother’s level was.

As the old pound was 240 pennies, dividing it by three, will give you 80 pennies, that was 6s/8d, and twice that makes 13s/4d, which is two thirds of an old pound, but you can’t get two thirds of todays pound in loose change, and not even when the a half pence was in use.

Regards Nick.

 

[duncan webster 1Participant @duncanwebster1]

Why do you want exactly 2/3 of a pound? Won’t 67p do?

To put the counter argument, try 7 *£1 14s 6d in your head. Even on a calculator there are lots of ways to go wrong. £1.725 *7 is a lot easier, I did it by mental arithmetic, but on a calculator it’s a breeze

[HopperParticipant @hopper]

Loose change? Haven’t seen that since pre-Covid. Paying by card seems to be the modern way. Probably because today’s shop assistants can’t do even the most basic mental arithmetic.  Gone are the days when a bar tender could almost instantly work out in his head the total for a round of 8 drinks of various sizes and descriptions. Now they have to laboriously punch each specifically marked square for each drink on a touch screen till and let it do the tally. Makes no difference whether the currency is decimal or not when a machine does the thinking. Might as well have stayed with the old money!

[JasonBModerator @jasonb]

Most tills will tell you the amount of change if the amount tendered is entered so no need to work it out.

Punching in the items may take a little longer initially but that can also take care of stock control and even automatic reordering so it saves other peoples time.

Paying with an app on your phone has obviously not filtered down to the southern hemispheare yet if cards are the modern way😉

[not done it yetParticipant @notdoneityet]

Duncan,

It’s not that difficult!  7*6 is 42, so 3 shillings (to carry) and 6d. Count out your sixpence in one or several coins.

7*14 is 98.  Add in the 3s carried gives £5 and 1 shilling.  Find the one shilling from your purse, pocket or hand.

Adding £5 to £7 is not too difficult, either.  All in notes from your wallet – £5, £1 and 10s notes most probably (each of different size and colour).

Not actually too difficult to do that while counting out the change and notes back then.🙂

There were not that many calculators about, back then, so we were more adept at doing the mental ‘rithmatic in a systematic manner!

 

[Nicholas FarrParticipant @nicholasfarr14254]

Hi, I think my late elder brother got a calculator for his birthday or Christmas back in the early 60’s, similar to this type, Mechanical calculator

Regards Nick.

 

[JasonB]

On JasonB Said:

Most tills will tell you the amount of change if the amount tendered is entered so no need to work it out.

True. Now give the cashier the odd 27pence after she’s rung up the £10 note and watch her brain melt. Or convince them they’ve given you the wrong change because they entered the wrong amount and just dumped the cash in your hand without counting it out. This is bad enough, but you end up with the entire staff being unable to do basic paperwork like cashing-up 3 tills at the end of shift without errors – I would frequently see my fellow supervisors signing sheets that couldn’t be right. Some got every sheet back, but couldn’t understand why because they always used the calculator.

[noel shelleyParticipant @noelshelley55608]

Interestingly a lot of the checkout staff seem to be older folk in a big local supermarket – is this coincidence or have they realised that the standards of maths are so poor ?

the Thornton slide rule is in the loft. Noel

[Alan JacksonParticipant @alanjackson47790]

How about the massive further simplification for Pi, from 3.1419 etc to ( 3.00000) or just three point nought, that would be a good (woke) simplification. Make life much easier for anybody who is not deeply involved with high faluting science and the like.

[Michael GilliganParticipant @michaelgilligan61133]

Alan

🙂

MichaelG.

[Alan Jackson]

On Alan Jackson Said:

How about the massive further simplification for Pi, from 3.1419 etc to ( 3.00000) or just three point nought, that would be a good (woke) simplification. Make life much easier for anybody who is not deeply involved with high faluting science and the like.

Surely it depends on what you need? After all, proper mathematicians try very hard to not use actual numbers at all. The real world equivalent is realising there’s little point in making something to +/- 0.001″ when 1/8″ is more than good enough.

[SillyOldDufferModerator @sillyoldduffer]

With all respect to Nick, I fear his memory of pre-decimal Britain being good at mental arithmetic is rose-tinted!

Back when calculators first appeared, I showed one to my favourite uncle.  He was dead against the beastly thing for the exactly the same reasons as Nick, plus he saw no value in a device that relied on a battery!   I was surprised because uncle was a Tax Inspector (every family has a Black Sheep!), with lots of staff doing money sums.  But hey, he was an expert with years of experience!

A year later uncle mentioned he’d been commended by the Revenue for piloting a scheme whereby his team had all been issued pocket calculators.   Wonder where he got the idea?   Went well despite screams of pain from the old-school; productivity improved by about 20%, mainly by reducing errors caused by faulty mental arithmetic, but also by reducing the time it took new staff to get their mental arithmetic up to speed.  Mental arithmetic has to be practised, and most people aren’t good at it!

Mental arithmetic in a pre-decimalisation tax office was even harder : it needed practitioners to memorise the Rule of Practice (aliquot parts of a pound, shilling and penny.)  Uncle was also against dead against decimalised currency,  even wrote to his MP about it, but guess what?  Within a year of it’s demise he thought £sd was daft.

Some lucky folk are super-adept with numbers.   An accountant friend could total a column of numbers from top to bottom faster than I could read them.  His ability must hint at how the brain works because he couldn’t total from bottom to top, or total a row of numbers reliably.    When spreadsheets appeared, he switched to them immediately, because they could do all that and much more, and made it easier to spot and fix errors.

As a general rule, it’s best to replace humans with machines because humans can be redeployed to do work machines can’t.  Also better to replace inefficient systems like £sd with simpler equivalents.  The humans involved hate being forced to change, but we all want to be rich, right?

Dave

[Alan Jackson]

On Alan Jackson Said:

How about the massive further simplification for Pi, from 3.1419 etc to ( 3.00000) or just three point nought, that would be a good (woke) simplification. Make life much easier for anybody who is not deeply involved with high faluting science and the like.

I use 3 as a substitute for Pi when doing mental estimates on the job for things like how much material need to wrap around a round bar or pipe, or how long will be the walk around a lake of roughly x diameter etc. Or when working out camera and flash settings across the infield of a quarter mile circumference speedway track when I used to shoot it for newspaper reports. All low falutin stuff.

[not done it yet]

On not done it yet Said:

Duncan,

It’s not that difficult!  7*6 is 42, so 3 shillings (to carry) and 6d. Count out your sixpence in one or several coins.

7*14 is 98.  Add in the 3s carried gives £5 and 1 shilling.  Find the one shilling from your purse, pocket or hand.

Adding £5 to £7 is not too difficult, either.  All in notes from your wallet – £5, £1 and 10s notes most probably (each of different size and colour).

Not actually too difficult to do that while counting out the change and notes back then.🙂

There were not that many calculators about, back then, so we were more adept at doing the mental ‘rithmatic in a systematic manner!

 

I do know how to do it, you’ve just demonstrated why decimal is so much easier

[not done it yetParticipant @notdoneityet]

Yes Duncan, but life was so much more relaxed back then.  Few cheques, mostly cash, no ‘card transactions’, etc.

I knew a comptometer operator who could add up a list of numbers as she inputted the 4 or 5 figure data.

Sorry Nick, I meant pocket electronic calculators, which only became available in ‘65/‘66 with the Oxford model?  As I recall, only a four digit machine!?  Mechanical machines had been around for many a year.  Don’t forget the Abacus!

A fellow I worked with, back in the late ‘60s, could do multiple multiplications and divisions on a “mangle” (as we called them) faster, and more reliably, than most could input on the Busicom electronic calculator (with a neon read-out) that often gave incorrect results when it got hot.

I remember Dad often used a “Ready Reckoner” for the farm accounts (this was before red diesel came into use, so diesel duty/taxation for road vehicles and farm use had to be claimed/justified for tax purposes, for example).

At work we had enormously long slide rules available for better accuracy (approaching two feet long) or themore common cylindrical equivalent in both compact and larger sizes.

I still have my smallish ‘Ricoh’ slide rule.

At school, it was 4-figure log books (still have one).  At work there were 5-figure log books available, for yet another more-accurate calculations.  I believe there were 6-figure log books available, too?

At my first place of work (UKAEA), every calculated result had to be countersigned by a checker.  Woe betide anyone who signed off a mistake!

All memories from the 1960s.

 

 

[not done it yet]

On not done it yet Said:

 

Sorry Nick, I meant pocket electronic calculators, which only became available in ‘65/‘66 with the Oxford model?  As I recall, only a four digit machine!?

You mean like this:

 

which was one of the first electronic calculators advertised as “Pocket Size”. Good thing we all walked around in overcoats in those days.

In my recollection, the “pocket” versions started to appear around the end of that decade (this one certainly did). Basic (larger) electronic calculators  were a bit earlier (it wasn’t quite clear – to me – what you meant there).

[duncan webster 1Participant @duncanwebster1]

In any reputable organisation, all engineering calculations have to be checked and signed off. Project managers seem to think this is unnecessary bureaucracy, and don’t get around to it until the job is nearly finished. I was once called in to check some calcs done by the contractor who had installed some equipment, been paid and left site. Problem was the calcs were wrong, so I refused to sign. I was then pressured to either just sign them, or to put them right and then sign them. They couldn’t understand that if I put them right they would have had to get someone else to check my correction. Fortunately the error didn’t make the kit unsafe, or I would have gone to higher authority. As it is I’ve no idea how they got round it, but I wasn’t going to sign incorrect calcs. I’d have just gone back to the contractor and told him if he ever wanted any more work out of us he’d better have another look at his sums, but it wasn’t my project.

[duncan webster 1]

On duncan webster 1 Said:

… Project managers seem to think this is unnecessary bureaucracy,

 

My career included a few spells as a Project Manager, a job that straddles the sensible straightforward world of engineering and the much less obvious world of finance, office politics and organisational power struggles!

and don’t get around to it until the job is nearly finished. I was once called in to check some calcs done by the contractor who had installed some equipment, been paid and left site. Problem was the calcs were wrong, so I refused to sign. I was then pressured to either just sign them, or to put them right and then sign them. They couldn’t understand that if I put them right they would have had to get someone else to check my correction….

I bet the Project Manager understood perfectly what Duncan was saying! It stinks of cover up.  What follows is one of many possible scenarios:

That the contractor had been paid before the work was complete is a huge no-no, yet someone had authorised that.  Why? The accounting rules of large organisations often insist payments be made in the fiscal year in which they were budgeted, and managers who don’t balance the books lose bonuses etc.  When year end approaches, it’s tempting to pay for nearly finished work, even though that’s also against the rules.  Duncan arrives, innocent of the background, and his action threatens to expose someone senior.  And maybe the Project Manager is involved too!  Hence the pressure to just sign it off.

On a small-scale this kind of bad behaviour keeps things moving without doing much harm.  Problem is breaking the process to cut costs or meet political goals becomes institutionalised, the rot sets in, and them something nasty happens.  Like the Grenfell Tower fire, or Pearl Harbour…

[duncan webster 1Participant @duncanwebster1]

As I progressed up the greasy pole, I had to become more of a project manager. On my projects, the calcs got signed off before they started cutting metal! We has a system which said you had to spend the money in the fy it had been allocated for, but there was a means of ‘accrual’ whereby if the job wasn’t quite finished you could raise an accrual note and the money would be put into some kind of limbo account, so it looked on the books as if it had been spent, but the contractor didn’t get his hands on it. One fine April morning the phone goes

Finance: we’ve had an invoice from xyz and you didn’t raise an accrual, you’ll have to pay it out of one of this year’s projects

Me: you’ve had an accrual, I raised it myself

Finance: we’ve no record of that, so it didn’t happen

Me: well I’ve got a transmittal note signed by you, it’s your screwup, you find the money

Finance: but we don’t have a budget for that

Me: I don’t have a budget for your cockups, your problem, you deal with it.

I think it went up to senior staff, but I didn’t pay. It’s no wonder they let me retire 5 years early on full pension.

[Hopper]

On Hopper Said

I use 3 as a substitute for Pi when doing mental estimates on the job for things like how much material need to wrap around a round bar or pipe, or how long will be the walk around a lake of roughly x diameter etc. Or when working out camera and flash settings across the infield of a quarter mile circumference speedway track when I used to shoot it for newspaper reports. All low falutin stuff.

Perfectly reasonable – and it’s where this thread started! In the right context, taking pi=3 makes a lot of sense, and like Hopper, it is what any sensible engineer would do as a sanity check, maybe, on a calculation done in a different way. Or on a journalist’s figures in an article, some of whom seem to have only a loose grasp of arithmetic. In the Dremel case, 1/8″=3.2mm. Why not?

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