# Base 4,000 Mathematics?

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Would we be better off using a higher base in our maths?

Humans mostly use base-10 arithmetic. Probably because we have 10 digits on our hands.

But of course other bases are possible: computers use binary base 2, old computer programmers in machine-code used "Hex", base 16, and the ancient Babylonians calculated in base 60 (at least in their astronomical tables).

For such astronomical calculations, base 60 is supposed to have been particularly convenient. Because 60 could be divided by so many smaller numbers: 2,3,4,5,6,10,12,15,20,30.

This convenience brought the disadvantage, of having to remember 59 different numerical symbols. However, modern Chinese manage to remember far more symbols than that: at least 4,000 (ideograms or logographs, as you prefer).

So suppose we adopted a base 4,000 arithmetic. Or better still - base 4,096. As that would fit in nicely with computer binary code.

Our numbers would obviously be shorter. But would there be any other advantages? Would mathematicians delight in this new notation - maybe discover new theorems?

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I doubt changing base would lead to new theorems. changing the base does not affect the properties of numbers, just the way you write them.

I don't even really see the practical application, we don't really have anything that would be made simpler by base 4000.

computers use base two because it is easy to distinguish between on-off and easier to create such a signal.

hex was used because it was 16-bit which a lot of early processors used for everything.

put simply we only use bases other than ten when it provides an advantage(smaller numbers isn't an advantage).

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Would we be better off using a higher base in our maths?

Humans mostly use base-10 arithmetic. Probably because we have 10 digits on our hands.

But of course other bases are possible: computers use binary base 2, old computer programmers in machine-code used "Hex", base 16, and the ancient Babylonians calculated in base 60 (at least in their astronomical tables).

For such astronomical calculations, base 60 is supposed to have been particularly convenient. Because 60 could be divided by so many smaller numbers: 2,3,4,5,6,10,12,15,20,30.

This convenience brought the disadvantage, of having to remember 59 different numerical symbols. However, modern Chinese manage to remember far more symbols than that: at least 4,000 (ideograms or logographs, as you prefer).

So suppose we adopted a base 4,000 arithmetic. Or better still - base 4,096. As that would fit in nicely with computer binary code.

Our numbers would obviously be shorter. But would there be any other advantages? Would mathematicians delight in this new notation - maybe discover new theorems?

Mathematicians would be nearly unaffected. Most theorems have nothing to do with any number base (there are a few exceptions).

Balancing your checkbook would be a trick -- you would need to memorize addition tables for numbers up to 4000.

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Shorter numbers aren't a huge advantage, either -- we developed scientific notation because it makes large numbers easy to handle in our base-ten system.

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Mathematicians would be nearly unaffected. Most theorems have nothing to do with any number base (there are a few exceptions).

Balancing your checkbook would be a trick -- you would need to memorize addition tables for numbers up to 4000.

Thanks insane_alien and DrRocket.

Of course I accept that the the way we write down numbers, can't affect the properties of numbers themselves.

But doesn't the way we write them, influence our perception of their properties.

For example, in base 10 arithmetic, the constant "Pi" starts 3.14159265..... and goes on endlessly.

And the endless stream of digits displays no perceptible pattern.

I think many people are puzzled by this. The ratio of the radius of a circle, to its circumference, seems such a basic thing. It ought to be something simple like plain 3. Or at least, something with some kind of pattern to it. Not just a stream of apparently random numbers.

Maybe if we wrote down Pi in base 2, we'd see some pattern in the stream of 0's and 1's?

Or if we wrote it in base 4,000, or base 4,000,000, or whatever, might we perceive a pattern of some kind?

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$\pi$ would still have a pattern of repeating "random" decimal places. Irrational numbers are irrational numbers and this can be seen geometrically.

I'm not sure, but maybe a base system could be set up that sets $\pi = 1$ but I don't know if it can be done and I doubt it would offer much advantage if any.

We can already represent irrational numbers symbolically. $2 \cdot \pi = 2\pi$

Edited by mississippichem
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I doubt changing base would lead to new theorems. changing the base does not affect the properties of numbers, just the way you write them.

I don't even really see the practical application, we don't really have anything that would be made simpler by base 4000.

computers use base two because it is easy to distinguish between on-off and easier to create such a signal.

hex was used because it was 16-bit which a lot of early processors used for everything.

put simply we only use bases other than ten when it provides an advantage(smaller numbers isn't an advantage).

hex is also easily interchangeable with binary, i think time is the most interesting number sequence we use because it incorporates many base systems. If you could elaborate on why you think base 4000 has any advantage over base 10? i dont think we use base 10 because of our fingers to the contrary it is the most efficient base to use for general purpose mathematics.

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Random aside:

Computers sometimes use large bases for calculations. Especially those involving division or multiplication of numbers with 100s of digits.

The idea is that you have to store each digit in a different place in memory in its own place in some form of data structure.

As they have to be dealt with separately, and there are all sorts of optimizations and conveniences for dealing with things that are one word (frequently 64, 32, or 16 bits) long then it can be useful to use a base close to your word size. Ie. 65536 for 16 bit.

It gets a bit more complicated than this and bigger isn't always better, but this outlines the basic idea.

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Random aside:

Computers sometimes use large bases for calculations. Especially those involving division or multiplication of numbers with 100s of digits.

The idea is that you have to store each digit in a different place in memory in its own place in some form of data structure.

As they have to be dealt with separately, and there are all sorts of optimizations and conveniences for dealing with things that are one word (frequently 64, 32, or 16 bits) long then it can be useful to use a base close to your word size. Ie. 65536 for 16 bit.

It gets a bit more complicated than this and bigger isn't always better, but this outlines the basic idea.

when your talking about 64/32/16 bit you are only referring to FSB size and essentially the amount of tracks that buss holds, not the numbering system as that is still binary. In Higher level programming you can start manipulating number systems but still only through manipulation of binary sowwy

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hex is also easily interchangeable with binary, i think time is the most interesting number sequence we use because it incorporates many base systems. If you could elaborate on why you think base 4000 has any advantage over base 10? i dont think we use base 10 because of our fingers to the contrary it is the most efficient base to use for general purpose mathematics.

theoretically there is no advantage, thats the point. The only advantage there is is that all our systems are set up to use base 10, we are all used to using base ten and so on. Any change to this must have a SIGNIFICANT advantage, and i'm talking world changing, from big business banking to street stalls.

we use ten because we have ten digits, it's also a low number. you can think about ten. base 4000 will have difficulties in memorising the numbers, calculators will need a few thousand extra keys etc. etc.

there is a significant disadvantage in moving up to 4000.

infact, there would be a significant disadvantage in moving (not a property of the base your moving to, just the act of switching itself).

now, I'm thinking of this in terms of industrial equipment(something i'm very familiar with).

now, lets say i'm purchasing some stirrers for my process tanks. My plant is in the UK so we've got 50Hz electricity. I'm looking at two suppliers who supply stirrers that are the same in every specification and price except one uses 50Hz and the other uses 60Hz.

now, if i had access to both 50 and 60Hz it would matter one iota which one i bought. but all the existing infrastructure means i have an advantage with the 50Hz stirrer as it interfaces with existing infrastructure. If i buy the 60Hz one i'll either have to change the infrastructure of the whole UK (not gonna happen) or I pay extra for a converter device so i can interface with the infrastructure.

The converter would add cost, complexity and failure points.

the same with base 4000.

If a company decided to use some base other than 10 then they would need to convert EVERY number they need to share with the outside world or recieve from the outside world between bases. a modern company(even a small one) will exchange hundreds of thousands of numbers with existing infrastructure.

It's just not worth it.

Its like saying "lets all speak binary from tomorrow" its not going to happen.

until you can prove a world changing advantage to b4000 then there IS NO ADVANTAGE. certainly no more than moving to 11, 12, 5, 42 or even staying at 10. the only thing about 10 is that there is zero cost to move as thats where we are.

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Also, how do I make a base-4000 calculator keyboard?

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Also, how do I make a base-4000 calculator keyboard?

not sure, i think it may be fun to make up 3991 new characters though, i can just see all the squiggles right now

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Have fun memorizing the 4,000 X 4,000 multiplication table so you can do multiplication in your head. Or the 4,000 digits themselves. Or designing the new number pad. Or convincing everyone to switch. And all just to write the numbers more compactly.

On the other hand, there might be good cause to switch to base 8, or 16, because it will make it easier to work with computers and some math that works better in binary.

There are a few funny things that work only in one base or another. For example, in base 10, if you add the digits together the result will be divisible by 3 only if the original number was divisible by 3.

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Maybe if we wrote down Pi in base 2, we'd see some pattern in the stream of 0's and 1's?

Why not write pi in base pi? 10. There's a pattern for ya. Now, writing 2 in base 10.... that's the tricky one now.

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Dekan , Base 4096 would be simple to use and simple to represent symbolically because these would be the pre-requisites for it's global use .

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A base gives a representation of a number system. Any theorems that are of real deep importance are not going to case about the representation used. As DrRocket has said, there are some theorems in number theory that do depend on the base, however my knowledge of number theory is poor and other may be able to elaborate here.

There maybe some practical advantages of picking uncommon bases. Then, the disadvantage is that other people may find it hard to understand. Trying to keep things as clear as possible must be a goal in communicating mathematical theorems etc.

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Also, how do I make a base-4000 calculator keyboard?

Cap, your objection is so plausible, that at first I was at a loss for a reply.

My pocket calculator's keyboard uses an area of about 2 square inches to accommodate the 10 decimal numeral keys.

So if each Base-4000 numeral had its own key, the area would be 800 square inches, ie over 2 feet by 2 feet.

Such a large size would obviously rule out pocket calculators.

However, if I can refer again to the Chinese and their thousands of ideograms. The Chinese seem to be able to send text messages, using the tiny keyboards on their mobile phones. Do they use "predictive text" exclusively to generate the ideograms?

Or are the keyboards on their phones programmed so that each ideogram can be built up, by a series of keystrokes.

If the latter, the Base 4,000 calculator could use a similar method to build up the numerals.

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However, if I can refer again to the Chinese and their thousands of ideograms. The Chinese seem to be able to send text messages, using the tiny keyboards on their mobile phones. Do they use "predictive text" exclusively to generate the ideograms?

I really doubt that all the characters are necessary for sms communication. they may use a menu system.

Or are the keyboards on their phones programmed so that each ideogram can be built up, by a series of keystrokes.

i don't think so

If the latter, the Base 4,000 calculator could use a similar method to build up the numerals.

so essentially you'd want to use a smaller number base to construct your larger number base, kind of defeats the purpose.

and also you see yourself the complexity this would add to even simple addition and subtraction.

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A base gives a representation of a number system. Any theorems that are of real deep importance are not going to case about the representation used. As DrRocket has said, there are some theorems in number theory that do depend on the base, however my knowledge of number theory is poor and other may be able to elaborate here.

There maybe some practical advantages of picking uncommon bases. Then, the disadvantage is that other people may find it hard to understand. Trying to keep things as clear as possible must be a goal in communicating mathematical theorems etc.

Yes, DrRocket's remark was intriguing and I'd be grateful any elaboration on the point.

What I was really thinking about, is that different bases might reveal hidden patterns in numbers. "Hidden", in the sense that we can't easily perceive them in our customary base-10.

For example, suppose we write this series of numbers in base 10:

10

204

3,640

Do these numbers show any easily perceived progression, that would enable us to predict, with confidence, what the next number in the series should be? Probably not.

However if we write the same numbers in base 2 binary, we see:

1010

11001100

111000111000

And then it's easy to see that the next number would be:

1111000011110000 (= 61,680 in decimal)

You may be quite right in saying that theorems of deep importance, don't care about the method of representation used.

Still, mightn't playing around with patterns of numbers, lead to useful insights?

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You may be quite right in saying that theorems of deep importance, don't care about the method of representation used.

Still, mightn't playing around with patterns of numbers, lead to useful insights?

Maybe, but I do not know enough number theory to point at examples.

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You may be quite right in saying that theorems of deep importance, don't care about the method of representation used.

Still, mightn't playing around with patterns of numbers, lead to useful insights?

Working in different bases does at times help show some interesting patterns. A few examples that come to mind are:

-Defining the Cantor Set as all numbers in the interval [0,1] such that there base-3 representations do not consist of only 0's and 2's.

-When working with the Collatz Conjecture you can notice certain classes of numbers that will iterate to 1 by looking at binary.

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Yes, DrRocket's remark was intriguing and I'd be grateful any elaboration on the point.

What I was really thinking about, is that different bases might reveal hidden patterns in numbers. "Hidden", in the sense that we can't easily perceive them in our customary base-10.

For example, suppose we write this series of numbers in base 10:

10

204

3,640

Do these numbers show any easily perceived progression, that would enable us to predict, with confidence, what the next number in the series should be? Probably not.

However if we write the same numbers in base 2 binary, we see:

1010

11001100

111000111000

And then it's easy to see that the next number would be:

1111000011110000 (= 61,680 in decimal)

You may be quite right in saying that theorems of deep importance, don't care about the method of representation used.

Still, mightn't playing around with patterns of numbers, lead to useful insights?

The certain usefull insight would be to show how useless patterns can be found everywhere.

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when your talking about 64/32/16 bit you are only referring to FSB size and essentially the amount of tracks that buss holds, not the numbering system as that is still binary. In Higher level programming you can start manipulating number systems but still only through manipulation of binary sowwy

Well you can use an addition table if you really want, and you don't even have to make the symbols match their binary representation.

Ie. base 4 could be 0= 10, 1=11, 2=00,3=01

then you'd save a table of what the addition operation did and implement carry giving you the ability to do arithmetic.

This doesn't involve binary arithmetic at all, at no point do you perform a base 2 carry.

Realistically you use binary arithmetic, but you store the 'digits' of a massive number in something like a linked list or an array.

let's take base 256 multiplication as an example

2 30 45 (equivalent to 138797 base 10), and;

3 200 120 (248928 base 10)

First you calculate 45*120, you need 16 bits to store this

gives you 24 with a carry of 21. and so on

     	1  23  35
14  21
2  30  45
* 	3 200 120
___________________
254  37  24
1 167 147  40  00
+6  90 135  00  00
___________________
8   3  24  77  24


Edit: can't get it to align properly, should have looked up whatever tags make a table :/

Or 34411662616 base 10.

This can be useful when dealing with numbers much larger than your word size. You will have to store these in some kind of structure regardless, so you take advantage of the fact that many cpus are optimized to do multiplications in one cycle. Using base 2 or base 10 in this case would be very wasteful.

In this case you are using the base 2 optimized multiplication in lieu of memorizing a times table and the slower higher order operation as your

actual multiplication algorithm. This could be achieved by making a 1 megabyte (256*256*16) (and thus no base 2 multiplication would be involved) hash table and looking up the value each step, but this would be no faster and you'd use up all your L1 cache.

Naively using the square root of your largest int value as your radix seems like it would work best, but I think there is some reason to prefer smaller numbers (perhaps something to do with the addition step to avoid additional carries before starting to translate your answer to base 2 or 10).

I think there is also a hack where you can take advantage of the way the 32 and 16 bit registers on a 32 bit x86 cpu overlap to optimize this a bit further, but I don't know much about assembly and can't recall where I saw this so I can't say for sure how it works.

Edited by Schrödinger's hat
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• 3 weeks later...
This convenience brought the disadvantage, of having to remember 59 different numerical symbols. However, modern Chinese manage to remember far more symbols than that: at least 4,000 (ideograms or logographs, as you prefer).

The Babylonians did not use base-60 for their digits, but a modified base-10 system. Each "digit" was represented by two symbols (representing 10s and 1s) and the digits had a slight space between them.

Why not write pi in base pi? 10. There's a pattern for ya. Now, writing 2 in base 10.... that's the tricky one now.

PI is an transcendental number, making it one of the two worst possible bases you could possibly use for a counting system. You would not be able to represent any non-PI multiple with exact precision.

Edited by baric

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