[…] Seeing complex numbers as an upgrade to our number system, just like zero, decimals and negatives were. […]
Not sure if anyone has pointed this out, but I think the sentence “Luckily, irrationals are at least algebraic” should be “luckily, some irrationals are algebraic” since the set of algebraic numbers is countable and thus has lebesgue measure zero in the reals, meaning “almost all” real numbers are not algebraic. Instead they are called trancendental. This is yet another bizarre property of our familiar real number system, I just figured the author would want to note.
I believe the Romans used a simpler method to multiply two numbers a and b: halve a and double b until a reaches 1; then add the values of b where a was odd. Using your example (a=IX, b=XXXIV):
odd IX XXXIV
even IV LXVIII
even II CXXXVI
odd I CCLXXII
then add XXXIV and CCLXXII to get CCCVI. Simples!
If you examine the details, it is a binary-shift method used by digital computers, and works in any base including decimal.
Hi Rohit, thanks for the note!
Very Very good.
@Jon: Thank you for the clarification! I’ll amend the post. As you mention, our number system can be really, really bizarre :).
I like that 1/3 paradox in base 10 but in base 3 it’s fine. But again you could use a similar argument against base three saying 0.1111111… in base three is 0.5 in base 10.
Loved the points you brought up. It’s really got me thinking about something I kinda always thought was set in stone.
Great post and website!!
[…] Explaining numbers intuitively […]
I would argue that our number system handles .9999999 quite well . . . at least if you’ve taken real analysis classes. We do have a definition of what an infinitely repeating decimal is - it’s equal to whatever the limit of the finite decimal approximations is. Since the limit of .9, .99, .999, .9999 etc. is 1, that means that .99999 repeating = 1.
Thanks Smo. I agree that we do have a well-defined definition of .999…, but most people are only accidentally using that definition. That is, kids who wonder what .999… means aren’t thinking “Gee, can someone compute the limit of the following sequence: …” but instead something like “What is the closest number I can get to 1 without (apparently) touching it?”
Intuitively, if you accept the idea of infinitesimals, you can say “There’s a possibility of getting infinitely close without touching. Standing on the line doesn’t mean you’re playing in the field.” and if you don’t, you’d say “There’s no way to represent an infinitely small gap, so you are either touching 1.0 or not, and .999… counts as touching. On the line is in the field.”.