
A mathematician's way of converting miles to kilometers - allthebest
https://twitter.com/TamasGorbe/status/1158348261683605504
======
rdiddly
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).

~~~
coldtea
> _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)

~~~
anongraddebt
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).

------
RcouF1uZ4gsC
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

~~~
mabbo
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).

~~~
crdrost
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.

------
yyyk
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.

~~~
lota-putty
What if we had just 4 digits on each limb or even 6 perhaps.

How does those numbers look in base 8 or 12.

~~~
wallace_f
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?..?

~~~
smcameron
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).

~~~
FabHK
There's an argument against intelligent design right there (4 or 6 fingers per
hand are obviously better).

~~~
alexanderdmitri
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...](https://www.scientificamerican.com/article/why-do-most-species-
have/)

~~~
FabHK
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,"

------
eps
In related news - pounds to kilos is "divide by 2, less 10%". Very precise
too.

160lbs = 80 - 8 = 72kg

~~~
xkgt
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.

~~~
cjbenedikt
Fahrenheit to Celsius: (100F -30)/2= 35C

~~~
achow
Not quite. It is 37.78.

In temperature couple of degrees means a lot.

~~~
chrisdhal
For outside temperatures it's close enough though. As an American who travels
to Central America quite a bit being about to do: (C*2) + 30 to get F is close
enough.

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avip
My way: x1.5 for speed limits, x2 for hikes.

~~~
lonelappde
Why x2?

~~~
_cereal
So it seems he went farther :D

~~~
klez
Or so they know it's still a long way.

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toolslive
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.

~~~
edflsafoiewq
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.

~~~
DerekL
By the way, those are called the Lucas numbers.

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_Microft
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.

~~~
_Microft
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 ;)

~~~
castratikron
Physicists have their own battles, like electrical engineers using Ohm's Law
as a definition of impedance.

~~~
rrss
What's the battle there? How do physicists define impedance?

~~~
castratikron
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.

------
learnstats2
It's both easier and more accurate to multiply by 1.6

~~~
mkagenius
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

~~~
phkahler
How many km is 4 miles? That's not on the list of numbers.

~~~
adtac
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.

~~~
OJFord
Or use them in combination:

9 is 8 + 1, so that's 8->13 + 1*1.6 = 14.6

------
magoon
I use a similar method of easy-to-remember numbers in conversion between
celsius to Fahrenheit:

    
    
      0 = 32
      10 = 50
      20 = 68
      30 = 86
    

Then roughly, subtract/add two F for every extra C. It’s easy to remember 32
and 50, while 68 and 86 are reversed.

~~~
gus_massa
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.

~~~
tzs
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℉.

------
Ragib_Zaman
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.

~~~
hackermailman
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.

~~~
OJFord
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?

~~~
lonelappde
10*X km is Y-base-16 miles, for single digit numbers.

------
joantune
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

------
js2
A runner's way of converting miles to kilometers:

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.

~~~
servercobra
Nice humblebrag. But agreed, the 5km = 3.1mi touchstone is helpful.

------
quickthrower2
I use π/2

Relevant xk: [https://xkcd.com/1047/](https://xkcd.com/1047/)

~~~
teddyh
I would think that [https://xkcd.com/526/](https://xkcd.com/526/) would be
more relevant.

------
robotosan
I generally just multiply miles * 1.5 for a rough approximation.

~~~
FartyMcFarter
Adding 10% of the original number to that is often easy as well, and makes the
estimate very close to the correct conversion.

~~~
hestipod
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.

------
aerophilic
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)
    

You could argue that it is a very rough estimation, but I find that most times
you are just trying to get a “rough speed” when doing this conversion anyway.

~~~
Phenomenit
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.

~~~
iiv
Well, that is literally how to convert between them. It's not an
approximation.

------
gingabriska
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.

~~~
etaoins
That converts miles to approximate hectometres.

~~~
gingabriska
It's within 10% error.

~~~
yholio
Yes, it's within 90%.

------
jsjohnst
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.

~~~
srl
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.

~~~
jsjohnst
> 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.

------
jonsen
If you know your powers of two

    
    
      1, 2, 4, 8, 16, 32, 64, 128, 256, 512
    

conversion from miles to kilometers is just two steps lexicographically

    
    
      1, 128, 16, 2, 256, 32, 4, 512, 64, 8
    

Ex.:

    
    
      32 miles is 51.2 kilometers
      80 miles is 128 kilometers (wrap around!)
    

See
[https://news.ycombinator.com/item?id=20546927](https://news.ycombinator.com/item?id=20546927)

------
dirkt
Nice. Makes it much easier to do the conversion in your head.

------
rihegher
Clever! And a good reason to learn the fibonacci sequence

------
whatever1
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.

------
youeseh
Come on guys, multiplying by 8 and dividing by 5 shouldn't be that difficult.
Worst case scenario, you use your phones to punch it in.

~~~
idlemind
Was hoping someone would say this. Times by 8 divide by 5 is super easy. And
easy to reverse when you’re doing km to miles.

------
ulfw
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.

------
miguelmota
For rough estimations, doubling it 4 times and dividing by 10 is usually good
enough

(300 << 4) / 10 = 480

300mi to km = 482.803

------
DrScump
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](https://en.wikipedia.org/wiki/Mars_Climate_Orbiter)

------
bloak
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.

------
yason
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.

~~~
jimmux
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.

------
kokey
An engineer's way: Pi = 3

~~~
rocqua
Pi is sqrt of 10 for all practical purposes.

~~~
mkl
Some practical purposes, maybe. Even 3.14 is a lot more accurate.

~~~
alaaalawi
or 355 / 113\. I memorized 113355. good enough for a lot. initially got it
from
[https://colorforth.github.io/pi.htm](https://colorforth.github.io/pi.htm)

------
prvc
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.

~~~
OscarCunningham
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.)

------
jarfil
The programmer's way would be to remember that 1.6 = (2^4)/10

------
dfeojm-zlib
What's interesting is that the absolute error converges to
~0.5399705049968784% (or a floating-point type or constant imprecision were
introduced).

~~~
mkl
.5*(1+sqrt(5))/1.609344 - 1 = 0.00539970867005111... actually, but that
doesn't seem interesting either. What do you mean?

------
usgroup
Wait, doesn’t everyone know there 1.6 times tables ?

~~~
rocqua
I only had to learn tables up to 10, 11 and 12 were useful to know, but not
mandatory.

~~~
taejo
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.

------
eggy
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/](https://frinklang.org/)

------
bhaak
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.

~~~
lonelappde
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)

------
servercobra
I made the mistake of reading some of the responses. This is your daily
reminder to never read the Twitter comments.

------
Causality1
I usually just do plus half plus a tenth.

------
wruza
And if it’s big enough, just prepend 0x.

~~~
joshca
How does it work? 1000 miles = 1609.34 km. But 0x1000 = 4096.

------
bena
Or you could just remember that a km is roughly 3/5 of a mile.

------
donpark
lazy me: cmd+L 5mi = ?km <enter>

~~~
joshca
On Mac: Cmd + Space 5 mi (and you have the answer already)

~~~
RolloTom
Same on KDE: ALT+SPACE
[https://i.postimg.cc/43ZSzJ2m/Schermata-2019-08-10-12-49-10....](https://i.postimg.cc/43ZSzJ2m/Schermata-2019-08-10-12-49-10.jpg)

------
dEnigma
Fascinating that the second reply (or second best reply?) is an unreasonably
angry response:

 _Who the hell needs all this .. Get lost [angry emoji]_

~~~
ghaff
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).

~~~
jedberg
> 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.

~~~
runarberg
I've heard that and I don't disagree with it. But things fail really badly.
E.g. 2 inches and 5 sixteenth

~~~
martin_a
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.

