Can any Number be a Base?

There are many different ways to express numbers. The most popular is definitely the decimal system, or in other words base 10. Base 2 and base 16 are also used in computers. But did you know that we can make number bases not only from integers?

Chapters:
00:00 Introduction
02:22 Base 1
03:12 Negative bases
04:34 Fractional bases
10:06 Irrational bases
15:10 Imaginary bases

[brunnomenxa]

16:27 Small error here. You say "21 + 2i", but it is written "21i + 2".

[ryan]

Interestingly, tally marks or even just counting with your fingers are an example of base 1, and probably the oldest number system we have. Roman numerals were also derived from tally marks, and they could be considered an example of a system with multiple bases, where the auxillary bases are 5 and 10

[TheArtOfBeingANerd]

I can see base pi being useful for trig. imagine cos(10)=-1 and sin(10/2)=1, etc. Also sum of reciprocal squares would be 100/(whatever 6 would be)

[RandyKing]

and before this i didn’t think my number universe could get any bigger…. thanks!

[landsgevaer]

For a number base a/b, you use digits 0..a-1, but that allows numbers to be written many different ways. In fact, you only need 0..floor(a/b).
Let's say for your example of 265, converted to base 7/3. You write it as 64366. But you could also write it as 1110020.001...
Similar for other non-integer bases.

[yunogasai]

This man put so much work effort to show us the beauty of math. I’m highly appreciating your videos dude. I hope u get a good job and good life mate

[pierreabbat]

There's also base φ, with digits 0 and 1, no two 1s in a row. 2 is represented as 10.01 in this base.
You can use base 2-i with the digits 0, 1, i, -i, and -1. Similarly, you can use base 2.5-√-0.75 with digits 0, 1, -.5+√.75, .5+√.75, -.5-√.75, .5-√.75, and -1.

[matroqueta]

Mind = blown

Respect for explaining such far out concepts in a way that is so easy to follow

[DoctorIknowWho]

Funnily enough, base 1 has a fun application where you can represent a string of numbers by having the “length” of the number represent a number in some other base, like base 9 for example, using 9 as a separator. This allows you to write any number of numbers in a string in base 1. Fun thought experiment.

[leave-a-comment-at-the-door]

my favorite type of number system that wasn't brought up here is factoradic, where instead of having one radix that you keep squaring, you take each digit as the next factorial, so each position can range from 0 up to the position number. to give you a feel for the system, here's some numbers counting from 0 to 24:
0, 10, 100, 110, 200, 210, 1000, 1010, 1100, 1110, 1200, 1210, 2000, 2010, 2100, 2110, 2200, 2210, 3000, 3010, 3100, 3110, 3200, 3210, 10000, and so on
to go back and forth it's very similar to a normal base; for example to render 5835241010(!) into base 10 you would do:
5*9! + 8*8! + 3*7! + 5*6! + 2*5! + 4*4! + 1*3! + 0*2! + 1*1! + 0*0!
=1814400 + 322560 + 15120 + 3600 + 240 + 96 + 6 + 0 + 1 = 2156023

[nbspWhitespaceJS]

really cool video but i dont think you covered about the golden ratio base?

whats interesting about this is that if the base is the golden ratio, you get an interesting phenomenon.

(btw base golden ratio only needs 2 digits, 0 and 1)

let the golden ratio = phi

we know that phi = 1 + 1/phi

multiply both sides by phi.

we get phi^2 = phi + 1, (a(b+c) = ab + ac)

rewrite this as phi^x because we are in base phi

phi^2 = phi^1 + phi^0. (x^0 = 1)

remember that we can always multiply both sides by phi to increment all of the exponents.

its really cool cause we get 100 = 11 in base golden ratio. just something to note.

if you found this comment interesting, consider checking the combo class, another channel covering this topic and is the source of all these equations.

[chrisengland]

I remember reading several research papers in the 70's about unusual number bases. It was a long time ago, so my memory has faded, but I do remember being intrigued by negative number bases. Their main attraction is that no sign is needed to handle negative numbers. Nowadays, of course, two's complement arithmetic is so entrenched in all computers that nobody ever uses anything else.

[blim]

Wait wait wait! I want to point out a lot of things:
First we want to state what a "generalized base b" should be. L'll start with just real numbers.
I think we should ask that, with a finite set of digits (natural numbers), we want to be able to write any real number as a sequence like: ±an[...]a0.b1[...]bn[...] (where we have a finite number of "a" digits on the left of the dot and an unlimited sequence of "b" digits on the right).
This is what the "traditional" base b allows ud to do.
We trivially notice that base 1 des not work, since we are not alloed to represent 1/2 in any way (just like any other fraction).
Moreover base -1 does not work even to represent just integers with just one kind of digit...
Using a negative integer just like you said causes no problem and could be done just fine (except for -1, of course).
Up to now we should point out that each number can be written in base b in just one possible way (with the exception of "b-1" periodic, where for exmple, in base 10 we can write and and they are the same number).
With non interger numebers we have to renounce to this property, but we'll be ok with that.
Now think how to write 1/3 in base 2, it should be 0, 0101010101... (and that's the only way to write it).
Now how can we write 3 in base 1/2?
It should be the reversed of the previous writing, namely: ...0101010.000...
Here we have an unlimited sequence of digits in the left of the dot, and this cotraddicts our defiinition (see above).
We could stretch that definition to include unliited sequence of digits on the left. Quite strange but ok, let's do it.
We could now prove that, using any real number b (besides 0, 1 and -1) the amount of digit we need will be the maximum between b and 1/b, rounded up. Witch is much better than your proposal, since fof 3/7 we will just need 3 digit instead of 7.
Talking now about complex numebers: it's not clear which digits are you allowed to use in case of complex numbers like 2+2i.
Since it's a fourth root of -64 I suppose you will want to allow us to use all the integer digits between 0 and 63.
If so 1 can obviously be written as 1, while i will be -0.08 (you can easily check it).
Since you can wrote any number in base -64 using those digits and you can represent those numbers in base 2+2i by using just fourth powers (like 3000500020003.00040007... instead of 3523.47... in base -64), you can write any complex number like a+ib doing the operations: x-0.08y.
I'm not sure about which complex numbers cannot be used as a base b, but I'm pretty sure that the n-th roots of 1 cannot be use, regardless of the amount of digits we will allow us to use.
I don't know if other numbers with modulus 1 can be used (and, if I have to guess, I think they could work, with the proper, finite, amount of digits) nor I can extimate the amount of necessay digits for generic complex numbers, like e+iπ.
I hope I was understandable (I'm sorry but I'm not a native English speaker) and please answer me noticing my possible mistakes.

[mikerickson]

Yes, any number can be expressed as SUM(i=-∞ to ∞) ai*r^i (a are the digits, r is the base) where r≠0 and 0≤ai<r, but the digit may not be an integer. Expressing the number 3.5 in base 3.7, would require a 3.5 in the "units".

[mcrow]

Very well expressed and executed video. I never thought of this before. Thank you.

[Naniblocks]

this is a beautiful video. the topic is so absurd but you explained it in the most understandable way possible

[legygax]

Great content. I never thought bases could be something else than integers, but it actually makes sense.
I just spot a very little mistake at 16:25 it shows 21i+2 (which is absolutely correct) but the voice says "21+2i".

[lloydbotway]

As I worked my way through grad school as a teaching assistant, one of the full professors sat in on a class I was teaching, where I explained negative bases. He was shocked and amazed and thrilled, because he'd never considered the possibility. No minus sign required! Cool!

[gustavojacobina]

This is so mind blowing and really well explained. I barely can believe what I see

[yanntal]

12:55 But this problem should also happen for some Algebraic numbers. There are Algebraic numbers that you can't write in terms of radicals, for example a solution to some general quintic equation.