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).
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.
55mi -> 55 + (25 + 2.5) -> 82 (let's say around 85)
80km -> 40miles (let's say around 45 miles)
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).
For whatever reason, six tenths and add is mentally quicker for me than division.
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
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 for example 55 mph,
double 4 times:
110, 220, 440, 880
divide by 10:
So 55 mph is approximately 88 km/h
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).
135, half is 67, a tenth is 13, add those two together to get 80, 135+80=215.
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.
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.
How does those numbers look in base 8 or 12.
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 fact, every number base is base "10" when you interpret the "10" in that base.
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.
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.
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.
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.
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:
> 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,"
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...]
160lbs = 80 - 8 = 72kg
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.
Fₙ in ≈ Fₙ₊₂ cm
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!
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!
2 inch ≈ 5 cm
In temperature couple of degrees means a lot.
Anyway, I can't remember where I learned it.
Disclaimer: I am physicist myself.
Let a = distance in km;
b = distance in mi; and
φ = the conversion factor from mi to km.
Then, a = φb.
I'm fine with that - on the condition that you never, ever treat a differential operator like a fraction again ;)
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.
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.
9 is 8 + 1, so that's 8->13 + 1*1.6 = 14.6
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).
Because knowing a km is 3/5 of a mile is more useful.
It follows that a mile is 5/3 of a km. From there it's basic math. 4 * 5 is 20. 20 / 3 is 6.666...
3 miles ~=5km
1 mile = 1.6km
So 5km + 1.6km ~= 6.6km
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.
0 = 32
10 = 50
20 = 68
30 = 86
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.
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℉.
-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
I did say I was imprecise, though (;
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.
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.
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.
Relevant xk: https://xkcd.com/1047/
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.
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)
To convert from m/s to km/h you need to divide by 1000 (there are 1000 meters in 1 km) and multiply by 3600 (there are 3600 seconds in 1 hour).
So 3600/1000 = 3.6
Brain is surprisingly good at doubling, tripling and halving operations.
And visually added zero is quite easy too.
*3, *10, /2 == *15
I'm surprised this is faster than adding half of the original number.
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.
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.
150/5 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.
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!)
(300 << 4) / 10 = 480
300mi to km = 482.803
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.
16 would have also been useful for mi/km conversion and hex/dec conversion.
5 mi * 0.6 + 5 = 8 km
8 km * 0.6 = 4.8 mi
Or just use Frink !
The Golden Ratio squared is 2.618 which is pretty close to 2.54.
Who the hell needs all this .. Get lost [angry emoji]
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).
We can't really know to what category this one belongs to.
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.
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.
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.
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.
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.
Now, in Europe (excluding the UK) they also still use some weird units from the past. Calories for example, and horse (!!) power.
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.)
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.
They're given in N/m^2, but all the petrol station machines are in PSI, probably because the numbers just fall better.
Standard units are much more useful in carpentry. It's a lot easier when you can divide by 2,3,4, and 6.
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
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.
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.
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.
Can anyone provide any reason to use Fahrenheit scale, apart from historical ones? Legit question.
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.