A mathematician's way of converting miles to kilometers(twitter.com)
twitter.com
A mathematician's way of converting miles to kilometers
https://twitter.com/TamasGorbe/status/1158348261683605504
207 comments
Nice humblebrag. But agreed, the 5km = 3.1mi touchstone is helpful.
I made the mistake of reading some of the responses. This is your daily reminder to never read the Twitter comments.
Clever! And a good reason to learn the fibonacci sequence
Nice. Makes it much easier to do the conversion in your head.
In related news - pounds to kilos is "divide by 2, less 10%". Very precise too.
160lbs = 80 - 8 = 72kg
160lbs = 80 - 8 = 72kg
You saved my life :)
So n * (1-0.5) * (1-0.1)... This could also work for miles to km conversion in a similar way: n * (1+0.5) * (1+0.1). Not as accurate as golden ratio but still acceptable. :D
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Any similar shortcut for feet inches to cm? I find this conversion to be slower to compute than miles to km or lbs to kg.
@sigfpe in a reply to the tweet mentioned
Fₙ in ≈ Fₙ₊₂ cm
https://twitter.com/sigfpe/status/1158465136891269121
Fₙ in ≈ Fₙ₊₂ cm
https://twitter.com/sigfpe/status/1158465136891269121
Fahrenheit to Celsius: (100F -30)/2= 35C
Just remember 100F was Fahrenheit's wife's body temperature when she had a low grade fever. Normal body temperature is 36.8C, and with a low fever it's between 37 and 38C. It's so easy! haha.
this was the size of my ruler: 12 inch ≈ 30 cm. therefore,
2 inch ≈ 5 cmA metre is about 40 inches, and an inch is about 2.5 cm. So, first into total inches, then take out as many 40s as you can, then multiply what's left.
5' 7" -> 5 * 12 + 7 inches -> 67 inches -> 1 metre 27 inches -> 1 metre 67 centimetres
Which isn't quite right, but is close-ish! And not really easy enough to call a shortcut either!
5' 7" -> 5 * 12 + 7 inches -> 67 inches -> 1 metre 27 inches -> 1 metre 67 centimetres
Which isn't quite right, but is close-ish! And not really easy enough to call a shortcut either!
A yard is 0.9m.
1.1 yard per m or 1yd and 4in.
A foot is 30cm.
An inch is 2.5cm.
4in per 10cm.
2m is 6ft 8in =180+20=2m
5f 4in is 150+10=1.6m
1.6m is 3ft 4in + 24in=5ft 4in
Even converting to and back isn't bad... work in a machine shop for a while and it will be second nature. Also metric/standard tools do not cross the line on the floor, only you and the work piece on different pieces of equipment. 25um looks like a mil, but it's not!
1.1 yard per m or 1yd and 4in.
A foot is 30cm.
An inch is 2.5cm.
4in per 10cm.
2m is 6ft 8in =180+20=2m
5f 4in is 150+10=1.6m
1.6m is 3ft 4in + 24in=5ft 4in
Even converting to and back isn't bad... work in a machine shop for a while and it will be second nature. Also metric/standard tools do not cross the line on the floor, only you and the work piece on different pieces of equipment. 25um looks like a mil, but it's not!
Yeah it's not actually that bad, the base-12 system helps you out. You multiply the feet by 3, divide the inches by 4, add, multiply by 10. Optionally add 1.5% to really get that extra precision.
It's 30.5•(a+b/12) understood as about 1.015•10•(3a+b/4).
So I am 6’4”, that becomes 18+1=19, so I am about 190cm. Adding between 1-2% gives me that I am between 192cm and 194cm. I know I am on the lower end of 6’4” (maybe 6’ 3.75”) so I usually report 192cm.
It's 30.5•(a+b/12) understood as about 1.015•10•(3a+b/4).
So I am 6’4”, that becomes 18+1=19, so I am about 190cm. Adding between 1-2% gives me that I am between 192cm and 194cm. I know I am on the lower end of 6’4” (maybe 6’ 3.75”) so I usually report 192cm.
It took me a second to parse what you were saying. Essentially since it’s 2.2LBS per kg the first divide gets you the number on both sides of the decimal. Very cool
I have similar way of calculating square meters from square feet. "Subtract 10% divide by 10". So for example 900 square feet: 900 - 90 = 810/10 = 81. Real answer is 83.6, close enough for me.
I generally just multiply miles * 1.5 for a rough approximation.
Adding 10% of the original number to that is often easy as well, and makes the estimate very close to the correct conversion.
Works great for kg/lb conversions too.
kg x2 + 10% = lbs
lbs /2 - 10% = kg
I've yet to find a fast one for C to F temp conversions though. It takes a bit longer to do the 9/5-5/9 + or - equation in your head.
kg x2 + 10% = lbs
lbs /2 - 10% = kg
I've yet to find a fast one for C to F temp conversions though. It takes a bit longer to do the 9/5-5/9 + or - equation in your head.
Triple the initial number, add a zero at the end, now halve the resulting number is good enough for my purpose.
Brain is surprisingly good at doubling, tripling and halving operations.
And visually added zero is quite easy too.
Brain is surprisingly good at doubling, tripling and halving operations.
And visually added zero is quite easy too.
That converts miles to approximate hectometres.
*3, *10, /2 == *15
I'm surprised this is faster than adding half of the original number.
So 1 mile equals 15km? and then you wonder why all these units conversion and up with probe crashing on mars.
If you are going that route, why not multiply by 8 and divide by 5 for a much better approximation. But to be honest for me it has always been easier to directly multiply by 1.6 (or 1.5 for your case)
I usually just do plus half plus a tenth.
The pendulum has now swung the other way on the misconception by the general public of mathematicians being great at mental arithmetic, and it's become a meme for some mathematicians to take some pride in being average or poor at mental arithmetic. In my experience, most mathematicians (or anyone working in a quantitative field) don't have issues quickly approximating 1.6 * X.
It seems to have swung from the old joke of "von Neumann summed the infinite series" to exactly the same place.
Anybody who's taken CS 101 can estimate 1.6 * n too since they spend half the year in base2 or base16.
It's 1(n) + 6(n)/10 since we're using base 10. For example, 5mi to ~8km: 1(5) + 6(5)/10 = 5 + 30/10 or 5 + 3.
It's 1(n) + 6(n)/10 since we're using base 10. For example, 5mi to ~8km: 1(5) + 6(5)/10 = 5 + 30/10 or 5 + 3.
That's how I was taught to do it in primary school, yes it's simple, but I don't see what bases 2 or 16 have to do with it?
10*X km is Y-base-16 miles, for single digit numbers.
Fascinating that the second reply (or second best reply?) is an unreasonably angry response:
Who the hell needs all this .. Get lost [angry emoji]
Who the hell needs all this .. Get lost [angry emoji]
There are a fair number of people who seem to take personal offense that the US (and to some degree the UK) don't use SI for everyday types of things and add it to their laundry list of grievances about the US generally.
Never mind the fact that SI is generally used for engineering and other areas where it has legitimate advantages (which miles vs. kilometers doesn't really in day-to-day life).
Never mind the fact that SI is generally used for engineering and other areas where it has legitimate advantages (which miles vs. kilometers doesn't really in day-to-day life).
> that SI is generally used for engineering
Except when it isn't? Are you telling me SI is used in engineering everywhere in the US? I've heard too many stories (especially accidents) to believe that in any way.
Except when it isn't? Are you telling me SI is used in engineering everywhere in the US? I've heard too many stories (especially accidents) to believe that in any way.
don't they still use pounds per square inch ?
Now, in Europe (excluding the UK) they also still use some weird units from the past. Calories for example, and horse (!!) power.
Now, in Europe (excluding the UK) they also still use some weird units from the past. Calories for example, and horse (!!) power.
PSI is very commonplace though I'm not in a field where I use it in an engineering way. (I actually had to think for a second to come up with the SI equivalent.)
Pounds are one of the real bugbears of imperial in mechanical engineering with lb-f and slugs (i.e. "pounds" conflates mass and weight). Always hated that. I'd convert things to metric and then convert back when I had to work in Imperial.
And I still hear stones from time to time in the UK if you really want archaic.
Pounds are one of the real bugbears of imperial in mechanical engineering with lb-f and slugs (i.e. "pounds" conflates mass and weight). Always hated that. I'd convert things to metric and then convert back when I had to work in Imperial.
And I still hear stones from time to time in the UK if you really want archaic.
Yeah, a lot of people in the UK use stones for weight. Even young people in my experience.
We still use PSI for tyre pressures in UK, probably some other things too.
They're given in N/m^2, but all the petrol station machines are in PSI, probably because the numbers just fall better.
They're given in N/m^2, but all the petrol station machines are in PSI, probably because the numbers just fall better.
A calorie was originally defined as as the amount of heat required at a pressure of 1 standard atmosphere to raise the temperature of 1 gram of water 1° Celsius.
But you're right that is sort of a weird unit from the past insofar as it's related to the SI system but isn't formally part of it. (It's basically now defined by its ratio to joules.)
But you're right that is sort of a weird unit from the past insofar as it's related to the SI system but isn't formally part of it. (It's basically now defined by its ratio to joules.)
Adding: There are a number of measurements that aren't necessarily obviously related but are actually the same thing in terms of units. Heat, work, and energy are all newton meters or kg m^2/sec^2 in SI units.
Nope. Hence generally. I haven't made a study of it but I assume Imperial units are more common in civil engineering and other areas that converge with local construction and building trades. Metric is pretty standard in my experience more broadly. (Note also that the better known examples of unit conversion accidents are pretty old as far as I know.)
On the UK railways, I think I'm right in saying that the only place imperial units are used is in measuring the curvature of track, which involves chains.
> which miles vs. kilometers doesn't really in day-to-day life
Standard units are much more useful in carpentry. It's a lot easier when you can divide by 2,3,4, and 6.
Standard units are much more useful in carpentry. It's a lot easier when you can divide by 2,3,4, and 6.
Yep. Factoring is useful in a lot of cases.
Also, although not strictly a debate around metric, the Fahrenheit scale gives you more granularity without going to decimals and requires less use of negative numbers on a day to day basis. And, if one is really concerned about a scientifically relevant temperature scale, we'd be using Kelvin, not Celsius.
Also, although not strictly a debate around metric, the Fahrenheit scale gives you more granularity without going to decimals and requires less use of negative numbers on a day to day basis. And, if one is really concerned about a scientifically relevant temperature scale, we'd be using Kelvin, not Celsius.
Kelvin IS Celsius, only starts at absolute zero. 0 C is 273 Kelvin, and 0 Kelvin is -273 Celsius. You add 273 to any Celsius measurement, and voila, it's Kelvin.
No. That's actually not quite correct. You add 273.15... The size of the degree is the same of course.
My point is that if you want to use a "correct" scientific measurement on a day to day basis you'd use Kelvin. As soon as you're converting, you're converting whether from Celsius or Fahrenheit.
ADDED: There's an argument for Celsius vs. Fahrenheit of course in so far as the size of the degree is baked into some other SI measurements. But there's no particular other reason that Celsius is superior on a day to day basis other than familiarity for some. There's some logic to basing easy to remember numerical points around water properties but it's not clear that actually has a lot of advantages for day-to-day questions about how hot or how cold it is.
My point is that if you want to use a "correct" scientific measurement on a day to day basis you'd use Kelvin. As soon as you're converting, you're converting whether from Celsius or Fahrenheit.
ADDED: There's an argument for Celsius vs. Fahrenheit of course in so far as the size of the degree is baked into some other SI measurements. But there's no particular other reason that Celsius is superior on a day to day basis other than familiarity for some. There's some logic to basing easy to remember numerical points around water properties but it's not clear that actually has a lot of advantages for day-to-day questions about how hot or how cold it is.
The fact that degree is the same means that energetically 1 degree C is equal to 1K.
So when doing scientific calculations, for example to calculate energy needed to change the temp by X, they are the same.
The fact that the scale is based on water properties seem to be just so values are easier to relate to. We are water based, after all.
Can anyone provide any reason to use Fahrenheit scale, apart from historical ones? Legit question.
Can anyone provide any reason to use Fahrenheit scale, apart from historical ones? Legit question.
Sure. Where I live:
1. Fahrenheit is far more likely to cover the range of temperatures I encounter without having to use negative numbers. 0 is really cold. Anything below zero is really freaking cold. The temperature of the boiling point of water is mostly an academic point in my day to day life.
2. Fahrenheit provides about 2x the granularity of Celsius without having to use decimals
I'm not going to argue that if Fahrenheit didn't exist, we'd invent it. But it does have some advantages as an existing system.
ADDED: For engineering using SI units, of course Celsius and Kelvin make a lot more sense.
1. Fahrenheit is far more likely to cover the range of temperatures I encounter without having to use negative numbers. 0 is really cold. Anything below zero is really freaking cold. The temperature of the boiling point of water is mostly an academic point in my day to day life.
2. Fahrenheit provides about 2x the granularity of Celsius without having to use decimals
I'm not going to argue that if Fahrenheit didn't exist, we'd invent it. But it does have some advantages as an existing system.
ADDED: For engineering using SI units, of course Celsius and Kelvin make a lot more sense.
I've heard that and I don't disagree with it. But things fail really badly. E.g. 2 inches and 5 sixteenth
I find this hilarious when watching DIY videos on YouTube.
All the makers are talking about "5/16 of an inch" and I'm trying to convert this to a metric value and it doesn't make any sense at all, because "complex" fractions like this are not used in every day life.
All the makers are talking about "5/16 of an inch" and I'm trying to convert this to a metric value and it doesn't make any sense at all, because "complex" fractions like this are not used in every day life.
Unless you are working with wold and the measurements are all nonlinear lies.
https://en.m.wikipedia.org/wiki/Lumber#Dimensional_lumber
Oh yes, then you need to figure out if 7/16 is bigger or smaller than 3/4. No, thanks.
You can easily divide by 2 and 4 in metric, because guess what, you just add a decimal point to your measure and 10cm/4 is 2.5cm and not some crazy fraction or weird unit and the math is not harder if you're dealing with 10m as opposed to 10cm.
10cm/3 is 3.3cm or whatever the precision of your tools is. It's the same "problem" (it isn't) with dividing 1ft in 5 for example. You go by the tolerance of your tools
You can easily divide by 2 and 4 in metric, because guess what, you just add a decimal point to your measure and 10cm/4 is 2.5cm and not some crazy fraction or weird unit and the math is not harder if you're dealing with 10m as opposed to 10cm.
10cm/3 is 3.3cm or whatever the precision of your tools is. It's the same "problem" (it isn't) with dividing 1ft in 5 for example. You go by the tolerance of your tools
It's a lot easier to do things when you don't have to go "okay, what's the conversion between these two units again?". The problem isn't base 10 vs base 12, it's the fact that in SI, every unit scales in the same way (the prefixes). With imperial units, every unit scales differently. How many fl.oz to a tablespoon? No idea. How many teaspoons to a tablespoon? No idea.
I agree it's not worth the hate, but in my opinion SI units are much more convenient.
Using your own example, miles vs kilometers may not feel like a big difference, but the fact that you have proportional units (mm, cm, m, km) for any possible distance has the potential to save you a lot of trouble. Having completely different units for the same thing (land miles / nautical miles for distance), usually not proportional between them, is less practical than the SI at every level. Not only for an engineer but also for home owners and almost everyone of us.
I understand why there are some places in the world where they still use systems other than the SI, mainly for legacy reasons. But I think the practical benefits of a more coherent (and standard) unit system exist not only for the engineer.
Using your own example, miles vs kilometers may not feel like a big difference, but the fact that you have proportional units (mm, cm, m, km) for any possible distance has the potential to save you a lot of trouble. Having completely different units for the same thing (land miles / nautical miles for distance), usually not proportional between them, is less practical than the SI at every level. Not only for an engineer but also for home owners and almost everyone of us.
I understand why there are some places in the world where they still use systems other than the SI, mainly for legacy reasons. But I think the practical benefits of a more coherent (and standard) unit system exist not only for the engineer.
That's fair enough. Whatever the advantages of Imperial in some narrow contexts, I'm not sure very many would argue for them absent legacy.
On the other hand, it's pretty much an academic debate because a switch isn't happening for mainstream use. The desire and political will just aren't there. The little push that once existed is essentially gone whatever some tech types might want.
On the other hand, it's pretty much an academic debate because a switch isn't happening for mainstream use. The desire and political will just aren't there. The little push that once existed is essentially gone whatever some tech types might want.
I like arguing for arcane standards, and there are aspects of the imperial system I like.
Specifically, I tend to advocate for a base 12 system based arround the inch. Base 12 because the very structured multiplication table, which also makes for easy dividing. The inch because I find my lengtg estimation accuracy to be better captured by inches.
Specifically, I tend to advocate for a base 12 system based arround the inch. Base 12 because the very structured multiplication table, which also makes for easy dividing. The inch because I find my lengtg estimation accuracy to be better captured by inches.
I assume that 12 inches to a foot is the reason that, at least when I was growing up, we were taught multiplication tables up to 12x12 rather than 10x10.
My favourite use of units is still the „2mm Scale Association“, a British model railroad club that builds their trains at a scale or two millimeteres to the foot (which is slightly larger than the standard N-track). It always blows my mind when I try to think how people come up with a scale like that.
Kinda a crazy ratio, but not -really-.
If you have modelmaking tools measured in mm, and a bunch of legacy dimensions of trains and buildings that are often round numbers of feet, it becomes tempting to use a scale like this. After all, you tend to get round numbers of mm to make things...
It might be convenient to make a starmap where e.g. 10 light years is 1 cm. Yes, the actual dilation factor is strange, but it's pretty easy to plot a set of cartesian coordinates measured in light years on a 1 cm grid.
If you have modelmaking tools measured in mm, and a bunch of legacy dimensions of trains and buildings that are often round numbers of feet, it becomes tempting to use a scale like this. After all, you tend to get round numbers of mm to make things...
It might be convenient to make a starmap where e.g. 10 light years is 1 cm. Yes, the actual dilation factor is strange, but it's pretty easy to plot a set of cartesian coordinates measured in light years on a 1 cm grid.
The programmer's way would be to remember that 1.6 = (2^4)/10
It's both easier and more accurate to multiply by 1.6
Not if you just remember 3,5,8,13 . You do not need to remember all fibonacci numbers, just the above and rest you can multiple by 2,5,10 and get them
How many km is 4 miles? That's not on the list of numbers.
4 is the average of 3 and 5, so its km value is the average of 5 and 8 (assuming the Fibonacci series to be a geometric progression as described in the tweet).
While I can do this for 4km, I can't for different values like 9km. I've done the multiply-by-1.6 thing often enough by now to be fast enough at it, so I'll likely be sticking to it. This is a cool trick nevertheless.
While I can do this for 4km, I can't for different values like 9km. I've done the multiply-by-1.6 thing often enough by now to be fast enough at it, so I'll likely be sticking to it. This is a cool trick nevertheless.
Or use them in combination:
9 is 8 + 1, so that's 8->13 + 1*1.6 = 14.6
9 is 8 + 1, so that's 8->13 + 1*1.6 = 14.6
8 -> 13 and halve it for 6.5?
Or any other combination, but using a higher Fibonacci number is going to be more accurate than combining smaller ones.
Or in this case, as it's only 1 off a Fibonacci number you can convert that one and add 1.6 without multiplying it by anything (multiplying by the 1 off).
Or any other combination, but using a higher Fibonacci number is going to be more accurate than combining smaller ones.
Or in this case, as it's only 1 off a Fibonacci number you can convert that one and add 1.6 without multiplying it by anything (multiplying by the 1 off).
I would solve it mentally as 4 miles = 3 miles + 1 mile.
3 miles ~=5km
1 mile = 1.6km
So 5km + 1.6km ~= 6.6km
3 miles ~=5km
1 mile = 1.6km
So 5km + 1.6km ~= 6.6km
For me, I think it’s pretty close for the numbers that are right on, and multiplying is way easier for intermediate numbers.
50->80 is one beat in my brain, it’s basically immediate because I know the first many Fibonacci numbers.
But multiplying by 1.6 is only two beats, it’s “add a half” and then “add a tenth”, each of which come just as automatically as recalling a two Fibonacci numbers. 50->75->80.
For the in-between numbers, 1.6 seems way easier. 40->60->64 is much quicker for me than averaging 50 and 80.
50->80 is one beat in my brain, it’s basically immediate because I know the first many Fibonacci numbers.
But multiplying by 1.6 is only two beats, it’s “add a half” and then “add a tenth”, each of which come just as automatically as recalling a two Fibonacci numbers. 50->75->80.
For the in-between numbers, 1.6 seems way easier. 40->60->64 is much quicker for me than averaging 50 and 80.
Yeah times 1.6 isn't too hard right? Just add 50% and 10% in succession, very easy to do in your head.
The other way around is trickier. I tend to divide by 8 (or /2/2/2) and multiply by 5, which is harder but still consists only of steps that are clearly defined in my head.
The other way around is trickier. I tend to divide by 8 (or /2/2/2) and multiply by 5, which is harder but still consists only of steps that are clearly defined in my head.
Yeah, converting via the 5/8 ratio is how I've always done it. I traveled to Florida a number of times as a kid, and for whatever reason I have a permanent memory of the speed limit signs showing "55 MPH / 88 KMH". It's weird what one's brain seems to fixate on sometimes...
I thought the same. As Australians in the US for a few months recently, I'd regularly ask my 6-7yo son to convert miles to kilometres as a test - say, 53 miles into km. Multiply by eight and divide by five (or divide by five first in cases where that was quicker). He found it straightforward enough.
Nice. Wish I'd thought of that. FWIW, I played points based games like cribbage with my youngsters to better encourage mental maths.
5 furlongs in a kilometer, 8 furlongs in a mile.
Easy peasy.
Easy peasy.
But that is not the mathematicians way...
Yes. 1.6 = (1 + 0.5 + 0.1).
So just halve the number and add it to itself. Need more accuracy add 10%.
lazy me: cmd+L 5mi = ?km <enter>
On Mac: Cmd + Space 5 mi (and you have the answer already)
Same on KDE: ALT+SPACE
https://i.postimg.cc/43ZSzJ2m/Schermata-2019-08-10-12-49-10....
I've been doing it like that as well for years. Most people you explain it (replace the number with a sum of fibonacci numbers, and for each one, take the next)
to come back with "but it becomes less accurate for larger numbers, right?". After you say "hm, no!" there's a pause, and then the penny drops: Golden ratio!
Anyway, I can't remember where I learned it.
Anyway, I can't remember where I learned it.
It's better for large numbers since the asymptotic property dominates. It's questionable for small numbers since then the effect of the initial condition dominates. For example, the tweet's argument works the same way for the Fibonacci sequence that goes 1,3,4,7,11,... but obviously that gives different numbers.
By the way, those are called the Lucas numbers.
I use π/2
Relevant xk: https://xkcd.com/1047/
Relevant xk: https://xkcd.com/1047/
I would think that https://xkcd.com/526/ would be more relevant.
[deleted]
it's good for a gag - but it's much simpler to just go and add half of the number + a tenth to convert from miles to kms
An engineer's way: Pi = 3
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Pi is sqrt of 10 for all practical purposes.
Some practical purposes, maybe. Even 3.14 is a lot more accurate.
or 355 / 113.
I memorized 113355. good enough for a lot.
initially got it from
https://colorforth.github.io/pi.htm
Wait, doesn’t everyone know there 1.6 times tables ?
I only had to learn tables up to 10, 11 and 12 were useful to know, but not mandatory.
I had one slightly eccentric teacher who decided that the 25 and 125 times table were also necessary. While I don't know about necessary, I've definitely used that knowledge (possibly more useful to think of them as the one-quarter and one-eighth times tables with a shift).
16 would have also been useful for mi/km conversion and hex/dec conversion.
16 would have also been useful for mi/km conversion and hex/dec conversion.
The Mars Climate Orbiter[0] was lost due to errors in converting between Newton seconds (SI) and pound-force seconds.
[0] https://en.wikipedia.org/wiki/Mars_Climate_Orbiter
[0] https://en.wikipedia.org/wiki/Mars_Climate_Orbiter
The even more useful conversion for me is inch to centimetre: F(n)in ~ F(n+2)cm
The Golden Ratio squared is 2.618 which is pretty close to 2.54.
The Golden Ratio squared is 2.618 which is pretty close to 2.54.
That's also in the thread, posted by Oscar winner and Project Loon developer Dan Piponi, who also has a fantastic technical blog at
blog.sigfpe.com
https://mobile.twitter.com/sigfpe/status/1158465136891269121
https://mobile.twitter.com/sigfpe/status/1158465136891269121
2.5 is closer and easier.
What's interesting is that the absolute error converges to ~0.5399705049968784% (or a floating-point type or constant imprecision were introduced).
What's interesting about that number?
.5*(1+sqrt(5))/1.609344 - 1 = 0.00539970867005111... actually, but that doesn't seem interesting either. What do you mean?
While using only additions and shifts (if we generalize) without floating point or multiplication should appeal to HN, this method uses a lookup table and calculating values needs a lot of memory accesses e.g. f(7) => f(4) + f(3) => (f(3) + f(5)) >> 2 + f(3) .
The Human Mk1 processing units are also capable of small multiplication/divisions, especially on bases 2 and 10, but bad at lookups - who thought manufacturing units with such slow memory access was a good idea??
I'd rather we play to their strengths and multiply by 1.6 (f(X) = X + X/2 + X/10 for all X) requiring only a few memory accesses. This is already as accurate as the other method. We could make it 1.61 (+ X/100) if we must be more accurate. Any floating point error should be too small to matter.
The Human Mk1 processing units are also capable of small multiplication/divisions, especially on bases 2 and 10, but bad at lookups - who thought manufacturing units with such slow memory access was a good idea??
I'd rather we play to their strengths and multiply by 1.6 (f(X) = X + X/2 + X/10 for all X) requiring only a few memory accesses. This is already as accurate as the other method. We could make it 1.61 (+ X/100) if we must be more accurate. Any floating point error should be too small to matter.
What if we had just 4 digits on each limb or even 6 perhaps.
How does those numbers look in base 8 or 12.
How does those numbers look in base 8 or 12.
I never bought this argument, but I'm not confident about it. Isn't base 10 inherently intuitive because of the obvious reasons? IE an order of magnitude is just another 0?
Since I learned about base 2, etc, long ago, I always thought there was something magically elegant about base10 and never understood this? The explanation I've always heard, being 10 fingere, doesn't seem to explain all the elegance with base 10 being easy to work with?..?
Since I learned about base 2, etc, long ago, I always thought there was something magically elegant about base10 and never understood this? The explanation I've always heard, being 10 fingere, doesn't seem to explain all the elegance with base 10 being easy to work with?..?
In base 8 (if we'd had 8 fingers), an "order of magnitude" would have been defined as "times 8" instead of "times 10", so it would also be adding another 0. Same with base 12. Base 16 would have the further advantage that we could easily halve, quarter, eighth, or 16th any number ending in 0 to a whole integer (in base 10, we can only halve, fifth, or tenth).
There's an argument against intelligent design right there (4 or 6 fingers per hand are obviously better).
Sure, if you assume the only purpose of fingers is for counting.
Eh, evolution is a pretty nifty mix of oo class extensions, recursion, brute-force and bias weightings.
I'd wager Gawsh made the best system S/He could given product constraints (completely unfocused if you ask me [which I know no one did]) and the real need to deliver (take it easy over there Leibniz, the world is still crap as evidenced everywhere).
Anyway, can't knock it 'til you've built it.
This is an interesting article: https://www.scientificamerican.com/article/why-do-most-speci...
I'd wager Gawsh made the best system S/He could given product constraints (completely unfocused if you ask me [which I know no one did]) and the real need to deliver (take it easy over there Leibniz, the world is still crap as evidenced everywhere).
Anyway, can't knock it 'til you've built it.
This is an interesting article: https://www.scientificamerican.com/article/why-do-most-speci...
Well, my comment was partly in jest (though I do think it's by no means clear that 5 is a local optimum, thus it's quite possible that 4 or 6 would be better, and twice either would give us a better base for counting), but I'm amazed that there's actual scientific discussion of the issue. I wish to quote the most pertinent part of the article though:
> Is there really any good evidence that five, rather than, say, four or six, digits was biomechanically preferable for the common ancestor of modern tetrapods? The answer has to be "No,"
> Is there really any good evidence that five, rather than, say, four or six, digits was biomechanically preferable for the common ancestor of modern tetrapods? The answer has to be "No,"
Yea you are obviously right now that I think about it, and I still have such a strong willingness to think there is something special about the number 10.
Meanwhile, there is something special about base-12, namely that the log base 2 of 3 has a really good rational approximation as 17/12, the log base 2 of 5 has a pretty good approximation as 7/3 (you can do better with 28ths).
This is the basis for the 12-tone equal temperament scale in music, and it only works if you use base-12. So if we used base-12 for our numbers then someone would have the bright idea to name all of our musical notes with numbers and we could just do a key change (or chord formation) by addition.
This is the basis for the 12-tone equal temperament scale in music, and it only works if you use base-12. So if we used base-12 for our numbers then someone would have the bright idea to name all of our musical notes with numbers and we could just do a key change (or chord formation) by addition.
> In base 8 (if we'd had 8 fingers)
Seven fingers.
Base 11 is the natural base for a ten fingered person: Base 11 has a distinct symbol for ten, base 10 does not.
[A prime base has quite a few practical disadvantages... and their advantages are fairly esoteric...]
Seven fingers.
Base 11 is the natural base for a ten fingered person: Base 11 has a distinct symbol for ten, base 10 does not.
[A prime base has quite a few practical disadvantages... and their advantages are fairly esoteric...]
I mean it's not so bad. If we meet aliens there is a decent chance that they will have a base-9 counting system in balanced ternary: so their digits would be -4, -3, -2, -1, 0, 1, 2, 3, 4. Rather than prefixing negatives with a minute sign, maybe negation would flip a number top-to-bottom.
Base 10 is intuitive because we are taught to work in base 10. If we worked in base 7, then multiplication by seven would be just another 0. (And if we worked in base 7, we would probably have defined “an order of magnitude” to be a multiplication by 7, rather than 10).
Though probably a base with a couple convenient small factors is useful. Especially 2, since parity (even/odd) is so useful.
Past cultures thought even more factors were good, e.g. sexagesimal with 2, 2, 3, and 5. It means that e.g. the expansion of 1/3rd and 1/6th don't form a repeating fraction in sexagesimal notation.
Past cultures thought even more factors were good, e.g. sexagesimal with 2, 2, 3, and 5. It means that e.g. the expansion of 1/3rd and 1/6th don't form a repeating fraction in sexagesimal notation.
I’m convinced base 12 would be far superior to base 10. It has four common factors: 6, 4, 3 and 2, rather than just one. This would make handling common whole number fractions in place value form much easier. It’s also easy to count to 12 on one hand - just point to your finger bones with your thumb. That way with two hands you can count all the way up to 24 (in base 10 equivalent).
Me too! And telling if a large number was a multiple of 2, 3, 4, 6 would be trivial - just check the last digit!
Still only two prime factors. I really wanted bass thirty - negative powers of 2 3 and 5 all terminate.
You are correct that "10" is a very special number, as long as you don't assume that it can only mean "ten".
In fact, every number base is base "10" when you interpret the "10" in that base.
Try it:
10 binary is two.
10 octal is eight.
10 hexadecimal is sixteen.
This is the very definition of a number base: it is the multiplier that you represent by appending "0" to a string of numeric characters in that base.
So that is where the fundamental and special nature of "10" comes from; it's not because it happens to mean "ten" in our customary number base.
Ten is nothing special, "10" is. "10" is simply the way you write N where N is whatever number base you're working in. It's just as special in every number base!
p.s. I'm sorry you were downvoted so heavily for asking an honest question. You can't be the only one who has wondered about this, and your question led to an interesting discussion.
In fact, every number base is base "10" when you interpret the "10" in that base.
Try it:
10 binary is two.
10 octal is eight.
10 hexadecimal is sixteen.
This is the very definition of a number base: it is the multiplier that you represent by appending "0" to a string of numeric characters in that base.
So that is where the fundamental and special nature of "10" comes from; it's not because it happens to mean "ten" in our customary number base.
Ten is nothing special, "10" is. "10" is simply the way you write N where N is whatever number base you're working in. It's just as special in every number base!
p.s. I'm sorry you were downvoted so heavily for asking an honest question. You can't be the only one who has wondered about this, and your question led to an interesting discussion.
This illustrates the idea well: https://i.imgur.com/II5W6Pl.png
Amusingly, the alien and astronaut would be referring to the same thing when they say 'Base 3'
That is nice of you to provide such a well-written answer, and a human response at the end.
Your reasoning seems quite obvious now with hindsight after having given it some thought; and yet, even still I have such a strong inclination that ".........." is a special number? Why? I guess it really is entirely my cognitive bias, because I can't find a reason for it.
But I should have known better as I've--probably obviously being a user here--encounteted binary more than just a few times. And even still, it never occurred to me that 10 in binary is just ".." And then 100 in binary is 1 order of magnitude of "..," 4. And 1,000 is just two orders of magntiude, 8. But still, intuitively this does not seem as natural as 10, and I guess that is completely cognitive bias.
Am I retarded to not have realized this? Maybe, but I actually was so curious about this that I tried to quiz some colleagues by asking what 1000 and 1001 is in binary and only one person got it right immediately, probably by understanding orders of magnitude and not by rote memorization. All the others got it by counting in binary, and one final person was annoyed and questioned why I was asking about binary (oops, sometimes being inquisitive is not socially acceptable). By the way, I work with app developers, most of whom do not have backgrounds in computer science, same as myself.
Your reasoning seems quite obvious now with hindsight after having given it some thought; and yet, even still I have such a strong inclination that ".........." is a special number? Why? I guess it really is entirely my cognitive bias, because I can't find a reason for it.
But I should have known better as I've--probably obviously being a user here--encounteted binary more than just a few times. And even still, it never occurred to me that 10 in binary is just ".." And then 100 in binary is 1 order of magnitude of "..," 4. And 1,000 is just two orders of magntiude, 8. But still, intuitively this does not seem as natural as 10, and I guess that is completely cognitive bias.
Am I retarded to not have realized this? Maybe, but I actually was so curious about this that I tried to quiz some colleagues by asking what 1000 and 1001 is in binary and only one person got it right immediately, probably by understanding orders of magnitude and not by rote memorization. All the others got it by counting in binary, and one final person was annoyed and questioned why I was asking about binary (oops, sometimes being inquisitive is not socially acceptable). By the way, I work with app developers, most of whom do not have backgrounds in computer science, same as myself.
It seems you are inutitively always converting everything to the decimal system and taking that as "the way" to think about numbers. That wouldn't be surprising, because we are brought up this way and even our language focusses on the decimal system. Not having good words to speak about the binary number 1101001 makes it difficult to think about it without converting it first. Two (I'm decimal again!) isn't a good base for human communication because there is a lot of repitition of simple symbols. Maybe a new way to pronounce hex numbers like a17c03 would be able to replace the decimal system.
I'd say we're way past MK1 units... 7 million years of evolution.
I use a similar method of easy-to-remember numbers in conversion between celsius to Fahrenheit:
0 = 32
10 = 50
20 = 68
30 = 86
Meh. Every 10F difference is a bit under 5C difference (since the ratio is 9:5). So you can estimate that 40F~5C, 50F=10C, 60F~15C, etc. All I remember is 2:1 and 32F=0C.
I find it helpful to do similar, but mine is fuzzier and tied to day-to-day activities. I am a gross farenheit user, so this is my mental table for cross reference
-42|-42: 9th layer of hell
0|32: freezing point
10-15: maybe think about a jacket
20-25: room temp
37|98: body temp
50: death valley
100|212: boiling
I use "double and add 30" (or "minus 30 and halve" for F->C). The accuracy isn't that great, but it's within single digits between freezing and boiling. More than good enough to parse "oh my god it was like 40 degrees outside" as being a meaningful statement about the weather.
and it has never failed me! +1
If you want to get fancy, you can call it the Taylor approximation.
I'm a bit lazy so I try to use the F(c)= 2 * c + 32 approximation described in a sibling comment, but for the range of human-friendly temperatures the error is too big. The problem is not the absolute difference, but how each temperature feels. So I have to resort to making the exact calculation or using Google for the conversion. I'll try your method in the future.
I'm a bit lazy so I try to use the F(c)= 2 * c + 32 approximation described in a sibling comment, but for the range of human-friendly temperatures the error is too big. The problem is not the absolute difference, but how each temperature feels. So I have to resort to making the exact calculation or using Google for the conversion. I'll try your method in the future.
Just go back 10% after computing 2C and the result is exact. If you don't want to deal with non-integers, round to the nearest integer when taking 10%. The final result will be the exact result rounded to the nearest integer.
Example: 22℃ -> 44 - 4 + 32 -> 72℉. (Exact is 71.6℉)
Example: 23℃ -> 46 - 5 + 32 -> 73℉. (Exact is 73.2℉)
I've seen people do the going back 10% before the doubling. That's fine if you are not going to round. If you are going to round, take off the 10% after the doubling or you could end up off by up to 1℉ for the final rounded amount.
For example, 26℃ with rounding after -> 52 - 5 + 32 = 79℉ (78.8℉ exact). With rounding before it goes -> (26 - 3) + 32 = 78℉.
Example: 22℃ -> 44 - 4 + 32 -> 72℉. (Exact is 71.6℉)
Example: 23℃ -> 46 - 5 + 32 -> 73℉. (Exact is 73.2℉)
I've seen people do the going back 10% before the doubling. That's fine if you are not going to round. If you are going to round, take off the 10% after the doubling or you could end up off by up to 1℉ for the final rounded amount.
For example, 26℃ with rounding after -> 52 - 5 + 32 = 79℉ (78.8℉ exact). With rounding before it goes -> (26 - 3) + 32 = 78℉.
> ... while 68 and 86 are reversed.
Just had a flashback - my mum giving me a comparably helpful mnemonic about four decades ago: 16 (c) ~= 61 (f).
That, along with human body temperature (38C ~= 98F) have stuck, and between them I can guesstimate most numbers in between on the foreign (F) scale.
Just had a flashback - my mum giving me a comparably helpful mnemonic about four decades ago: 16 (c) ~= 61 (f).
That, along with human body temperature (38C ~= 98F) have stuck, and between them I can guesstimate most numbers in between on the foreign (F) scale.
37°C (not 38): 37 is a prime, and a centered hexagonal number, if you need some facts to hang your memory on. It's only approximate (same for °F where 98.6 is now seen as "normal"). "Normal" temperature depends on where on the body you measure anyway.
Since we are talking about “useful approximations”, one I have always found useful in robotics is doubling m/s to get miles per hour. While a bit “rough” usually “good enough” for when thinking about normal driving speeds. Here are some examples:
1 m/s ~ 2 mph (2.2 mph)
5 m/s ~ 10 mph (11.2 mph)
10 m/s ~ 20 mph (22.4 mph)
20 m/s ~ 40 mph (44.7 mph)
30 m/s ~ 60 mph (67.1 mph)
In physics class we used 3,6*m/s to get km/h all the time. It's not superclean but it 3,6 is still pretty easy to multiply or divide with.
Isn't that rather the physicists way? Works well enough, used in a way not quite as intended by mathematicians, ...
Disclaimer: I am physicist myself.
Disclaimer: I am physicist myself.
Ha, I saw a downvote there!
I'm fine with that - on the condition that you never, ever treat a differential operator like a fraction again ;)
I'm fine with that - on the condition that you never, ever treat a differential operator like a fraction again ;)
Physicists have their own battles, like electrical engineers using Ohm's Law as a definition of impedance.
What's the battle there? How do physicists define impedance?
Ohm's law is an empirical law that only holds in certain circumstances. A classic exercise is measuring the current and voltage across a lightbulb, plotting it, and measuring the slope of the line. The slope is the impedance. Then you turn up the voltage and watch the line turn into a curve, which is where the law breaks down and doesn't apply anymore. The engineers treat it like a definition and assume linearity over all voltage.
Mathematicians' way:
Let a = distance in km;
b = distance in mi; and
φ = the conversion factor from mi to km.
Then, a = φb.Does not seem to be very convenient for figures which are not the product of one of the first few members of the Fibonacci sequence and a power of 10, at least compared to multiplying by 1.6.
You can sometimes use it by breaking the number down into smaller Fibonacci numbers. For example 29km = 21km + 8km = 13 miles + 5 miles = 18 miles. (Correct result is 18.02 miles.)
Or you could just remember that a km is roughly 3/5 of a mile.
It's all about 6 for me:
5 mi * 0.6 + 5 = 8 km
8 km * 0.6 = 4.8 mi
Or just use Frink [1]!
[1] https://frinklang.org/
5 mi * 0.6 + 5 = 8 km
8 km * 0.6 = 4.8 mi
Or just use Frink [1]!
[1] https://frinklang.org/
I multiply by 1.5 (+50%) and then by 1.1 (+10%). Both are easy and fast to do by head and the total (1.5*1.1=1.65) is close enough.
As a programmer, I am much more facile with powers of 2, than Fibonacci. My way is to double the number 4 times (multiplying by 16) and divide by 10.
So for example 55 mph,
double 4 times: 110, 220, 440, 880
divide by 10: 88
So 55 mph is approximately 88 km/h
So for example 55 mph,
double 4 times: 110, 220, 440, 880
divide by 10: 88
So 55 mph is approximately 88 km/h
This is the method I'm using :D
Yes! I'm a Canadian whose brain works in km but I spent a year and a bit working a job that had me flying to (and subsequently driving around) places across the USA. All the time I'd be finding myself going "Okay, 30 mph... what the heck is that?".
Double four times, and divide by ten as soon as possible (ie: the first time you see a zero on the end, drop it).
So 30 mph => 3 (drop the zero) => 6 => 12 => 24 => 48 km/h
The nice thing about this method is that you know that the kph is going to be more than mph, but less than double it, so you don't have to count the doublings very well- when you're in the right range, you're at the right number.
Example: 135 miles to destination is how many km? Okay, double to 270; drop the zero to 27; double to 54; double to 108;... we're still less than the mph, so we must need to double again to get 216 km. Now we're between 135 and 2*135, so we must be at the answer. 216 km. (actual answer is 217.3).
Double four times, and divide by ten as soon as possible (ie: the first time you see a zero on the end, drop it).
So 30 mph => 3 (drop the zero) => 6 => 12 => 24 => 48 km/h
The nice thing about this method is that you know that the kph is going to be more than mph, but less than double it, so you don't have to count the doublings very well- when you're in the right range, you're at the right number.
Example: 135 miles to destination is how many km? Okay, double to 270; drop the zero to 27; double to 54; double to 108;... we're still less than the mph, so we must need to double again to get 216 km. Now we're between 135 and 2*135, so we must be at the answer. 216 km. (actual answer is 217.3).
You can also add a half and a tenth.
135, half is 67, a tenth is 13, add those two together to get 80, 135+80=215.
135, half is 67, a tenth is 13, add those two together to get 80, 135+80=215.
This is so brilliant! As a runner (most distances in running are measures in meters) and a programmer, this is just perfect! Thanks for sharing!
Good trick. A little easier: divide by 10 first and then double four times.
so in other words, you multiply by 1.6. How is having facility with powers of 2 helping you here other than knowing that 2^4 = 16?
I mean, I get what you're saying in that doubling a value several times is relatively easy mental math, and at the same time, this whole thread strikes me as overcomplicating something that doesn't need this much complexity and abstraction.
I mean, I get what you're saying in that doubling a value several times is relatively easy mental math, and at the same time, this whole thread strikes me as overcomplicating something that doesn't need this much complexity and abstraction.
Division and multiplication by 5 and 8 should be fairly easy for most folks, imho. Lot easier than trying to remember the closest Fibonacci number to me.
Quick, what’s the closest Fibonacci number to 150? Can you do that faster than 150/5*8 in your head? What about 500?
The reverse is almost as easy. Even with numbers not as evenly divisible, say 490km, most will know 490/8 is about 61 quickly. Multiple that by 5 and you 305mi.
Maybe I’m just better at basic speed math than average, but I still feel it’s easier for most people.
Quick, what’s the closest Fibonacci number to 150? Can you do that faster than 150/5*8 in your head? What about 500?
The reverse is almost as easy. Even with numbers not as evenly divisible, say 490km, most will know 490/8 is about 61 quickly. Multiple that by 5 and you 305mi.
Maybe I’m just better at basic speed math than average, but I still feel it’s easier for most people.
I'm admittedly pretty bad at basic speed math (even dividing by 5 takes me a bit -- and I know it shouldn't). I use the same fibonacci trick, and at least for me it really is much easier.
It's important to remember that all arithemtic tricks are made more useful when combined with others. I don't know what fibonacci number is close to 500, but I don't need to: 5 -> 8 means 500 -> 800. Really, the only fibonaccis I have memorized are 2,3,5,8,13,21.
150 is harder, but I would use the same trick. 13->21, so 150->230 plus a bit. Maybe 240.
It's important to remember that all arithemtic tricks are made more useful when combined with others. I don't know what fibonacci number is close to 500, but I don't need to: 5 -> 8 means 500 -> 800. Really, the only fibonaccis I have memorized are 2,3,5,8,13,21.
150 is harder, but I would use the same trick. 13->21, so 150->230 plus a bit. Maybe 240.
> 150 is harder, but I would use the same trick.
150/5 is something literally any adult should be able to do instantly. I realize that’s a bit hyperbolic, but still seriously easy. 30*8 is also very simple too.
So yeah, I get what you’re saying, but seriously, practice a little and I swear you’ll be able to learn it.
150/5 is something literally any adult should be able to do instantly. I realize that’s a bit hyperbolic, but still seriously easy. 30*8 is also very simple too.
So yeah, I get what you’re saying, but seriously, practice a little and I swear you’ll be able to learn it.
I basically just use 6.
1.6 is the factor everyone talks about (approximation of 1.609), but it has 2 significant figures. To make mental calculations easier/quicker I use its reciprocal, 0.621, which I approximate with 0.6, which has only one significant figure, 6.
Instead of multiplying by 1.6 you would divide by 0.6, which basically amounts to dividing by 6 and then moving the decimal point to someplace plausible.
55 mi / 6 = about 9, so 90 km (actual answer is 88.5)
Going the other way:
80 km * 6 = 480, so 48 mi (actual answer is 49.7)
If you can't remember whether to multiply or divide, just remember multiplying a number by 0.6 makes it smaller, which you would do if going to miles (which are bigger so there are fewer of them). And dividing by 0.6 makes a number bigger, so you must be going to km (which are smaller so there are more of them).
1.6 is the factor everyone talks about (approximation of 1.609), but it has 2 significant figures. To make mental calculations easier/quicker I use its reciprocal, 0.621, which I approximate with 0.6, which has only one significant figure, 6.
Instead of multiplying by 1.6 you would divide by 0.6, which basically amounts to dividing by 6 and then moving the decimal point to someplace plausible.
55 mi / 6 = about 9, so 90 km (actual answer is 88.5)
Going the other way:
80 km * 6 = 480, so 48 mi (actual answer is 49.7)
If you can't remember whether to multiply or divide, just remember multiplying a number by 0.6 makes it smaller, which you would do if going to miles (which are bigger so there are fewer of them). And dividing by 0.6 makes a number bigger, so you must be going to km (which are smaller so there are more of them).
Multiplying by 1.6 is the same as adding 0.6, so the calculation going to KM is about the same either way.
For whatever reason, six tenths and add is mentally quicker for me than division.
For whatever reason, six tenths and add is mentally quicker for me than division.
>1.6 is the factor everyone talks about (approximation of 1.609), but it has 2 significant figures. To make mental calculations easier/quicker I use its reciprocal, 0.621, which I approximate with 0.6, which has only one significant figure, 6.
How about just 1.5 + 0.1? I.e. the same amount plus half, plus a tenth?
So, 55 mi -> 55 + (25 + 2.5) + 5.5 -> 88km
And inversely, 0.5 + 0.1, so half plus a tenth:
80km -> 40 + 8 -> 48 miles
And for most purposes, just (one and a half = 1.5) and (half = 0.5) is close enough. You can mentally add a little more.
So
55mi -> 55 + (25 + 2.5) -> 82 (let's say around 85)
80km -> 40miles (let's say around 45 miles)
How about just 1.5 + 0.1? I.e. the same amount plus half, plus a tenth?
So, 55 mi -> 55 + (25 + 2.5) + 5.5 -> 88km
And inversely, 0.5 + 0.1, so half plus a tenth:
80km -> 40 + 8 -> 48 miles
And for most purposes, just (one and a half = 1.5) and (half = 0.5) is close enough. You can mentally add a little more.
So
55mi -> 55 + (25 + 2.5) -> 82 (let's say around 85)
80km -> 40miles (let's say around 45 miles)
This is exactly the way I do math in my head.
I often do mental math like this. However, sometimes it is faster to not split up calculations like this. For instance, in the case of converting 55 mi, it's faster (at least, for me) to just divide 54 by 6, move the decimal, and call it a day.
I've found that calculations are faster when you know when and when not to split calculations like the way you describe (even though at least 80% of the time it's faster to split them up).
I've found that calculations are faster when you know when and when not to split calculations like the way you describe (even though at least 80% of the time it's faster to split them up).
[deleted]
I had a Canadian car in the US. My rule was simple. I knew the first few multiples of 16 up to 80. And I knew half of 16 is 8. So to convert 35 miles an hour it's just 48+8. And it's very convenient to know that 50 mi is 80km. Beyond that it's just adding 8s or 16s.
For going the other way is just find the nearest multiple of 5 in miles and do a quick approximation. So 70 km. I know that 40 mi is 64 km. Thus 45 mi is 72km. So I'll guess a little lower than 44mi
For going the other way is just find the nearest multiple of 5 in miles and do a quick approximation. So 70 km. I know that 40 mi is 64 km. Thus 45 mi is 72km. So I'll guess a little lower than 44mi
1.6 = 8/5
So n miles = 8n / 5 km
And m km = 5m / 8 miles
Fraction multiplication is actually much easier than decimal multiplication imo. I'm surprised so many people seem to avoid it. Keeping track of where the decimal point should be is a significant cognitive load.
So n miles = 8n / 5 km
And m km = 5m / 8 miles
Fraction multiplication is actually much easier than decimal multiplication imo. I'm surprised so many people seem to avoid it. Keeping track of where the decimal point should be is a significant cognitive load.
[deleted]
I just remember that an inch is 2.54 cm, exactly (unless it's a "survey inch"), and there are 12 inches in a foot, and 3 feet in a yard ... but then I get stuck remembering how many yards there are in a chain and how many chains in a furlong.
If you know your powers of two
1, 2, 4, 8, 16, 32, 64, 128, 256, 512
1, 128, 16, 2, 256, 32, 4, 512, 64, 8
32 miles is 51.2 kilometers
80 miles is 128 kilometers (wrap around!)
For rough estimations, doubling it 4 times and dividing by 10 is usually good enough
(300 << 4) / 10 = 480
300mi to km = 482.803
(300 << 4) / 10 = 480
300mi to km = 482.803
I had arrived to the same approach by accident, basically by looking at the car analog speedometer and noticing which denoted mile increments match almost exactly the km increment. You can see for example that the 80kph "tick" is located exactly where the 50mph is. Then if you observe all the ones that match, you can see a sequence emerging.
Surely you can find various ways to reason or calculate these unit conversions.
But for practical purposes adding a half goes a long way. And it's even easier to add the missing 0.1, if you really need to.
But for practical purposes adding a half goes a long way. And it's even easier to add the missing 0.1, if you really need to.
When I'm in the US I find that with a little practice I can guess it with surprising accuracy. It might help that I have mild synaesthesia so visualising a number line makes it possible to almost see the conversion.
1 mile = 1609 meters.
3.1 miles = 5 kilometers.
6.2 miles = 10 kilometers.
...
26 miles, 385 yards = 42.195 kilometers.
100 miles = 160.9 kilometers.
Sorry, that's as far as I've run.