A mathematician's way of converting miles to kilometers • Hacker News Board

Re: A mathematician's way of converting miles to kilometers

#161

Post by nullc » Sat, Aug 10, 2019, 4:53 PM UTC

smcameron wrote: post #109

wallace_f wrote: post #102

Earlier quoted context omitted.

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…

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).

> 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...]

Re: A mathematician's way of converting miles to kilometers

#162

Post by Retra » Sat, Aug 10, 2019, 4:54 PM UTC

FabHK wrote: post #137

smcameron wrote: post #109

Earlier quoted context omitted.

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.

Re: A mathematician's way of converting miles to kilometers

#163

Post by anamexis » Sat, Aug 10, 2019, 5:00 PM UTC

greggyb wrote: post #87

magoon wrote: post #73

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.

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 It's not precise, except for some intersection points, but it sticks well for me and allows me to be conve…

Note: the 9th layer of hell intersection point is -40.

Re: A mathematician's way of converting miles to kilometers

#164

Post by skykooler » Sat, Aug 10, 2019, 5:06 PM UTC

xkgt wrote: post #101

eps wrote: post #4

In related news - pounds to kilos is "divide by 2, less 10%". Very precise too. 160lbs = 80 - 8 = 72kg

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.

For inches to cm in my head, I multiply by 10 and then divide by 2 twice. E.g. 7 inches -> 70 -> 35 -> ~17.5 cm.

Re: A mathematician's way of converting miles to kilometers

#165

Post by rdiddly » Sat, Aug 10, 2019, 5:14 PM UTC

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).

Re: A mathematician's way of converting miles to kilometers

#166

Post by crdrost » Sat, Aug 10, 2019, 5:17 PM UTC

xkgt wrote: post #101

eps wrote: post #4

In related news - pounds to kilos is "divide by 2, less 10%". Very precise too. 160lbs = 80 - 8 = 72kg

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.

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.

Re: A mathematician's way of converting miles to kilometers

#167

Post by lanstin » Sat, Aug 10, 2019, 5:25 PM UTC

JoeAltmaier wrote: post #144

simonh wrote: post #141

Earlier quoted context omitted.

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.

Re: A mathematician's way of converting miles to kilometers

#168

Post by [deleted] » Sat, Aug 10, 2019, 5:31 PM UTC

[deleted]

Re: A mathematician's way of converting miles to kilometers

#169

Post by hestipod » Sat, Aug 10, 2019, 5:42 PM UTC

FartyMcFarter wrote: post #9

robotosan wrote: post #5

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.

Re: A mathematician's way of converting miles to kilometers

#170

Post by crdrost » Sat, Aug 10, 2019, 5:46 PM UTC

wallace_f wrote: post #146

smcameron wrote: post #109

Earlier quoted context omitted.

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).

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.