Construction of number system

1. Mar 9, 2005

C0nfused

"Construction" of "number system"

Hi everybody,
How is the number system created and defined? I am talking about the natural numbers, but I am not actually asking about the definition and properties of N (Peano) but for the way that we have agreed to produce numbers. I mean, we have defined ten digits:0,1,2,3,4,5,6,7,8,9 and all the natural numbers can be written as a series of these digits. I would like to know if there's any logic behind these ways we have for writing numbers.

For example, we have decided that the successor of 9 is 10, of 19 is 20, of 99 is 100 etc. Why we have chosen this way to create any number, as big as we want? A number with k digits n(1),n(2),...,n(k) ,beginning from left to right with n(k) and ending with n(1) is actually defined like this:
=n(k)*(10^(k-1))+n(k-1)*(10^(k-2))+...+n(2)*10+n(1) ? (I am referring to the decimal system) Is this just a convenient way that also is in agreement with the axioms and properties of naturals?

I hope you understood what I am asking (i know it's not very clearly stated )

Thanks

2. Mar 9, 2005

honestrosewater

The symbols (0, 1, 653) used to represent numbers are called numerals so as not to confuse them with numbers. Systems of numerals are called- surprisingly enough- numeral systems. Using different numeral systems doesn't effect the rules of the number system- they aren't tied together in that way. You can use whatever symbols your heart desires (as long as you use them consistently, of course). The binary system (with 0, 1) is used with computers. I imagine there are several reasons why the decimal system is standard. You can start with wikipedia's entry.

Edit: Someone can probably explain this better than I can, but I should add that the number system and numeral system are of course tied together in some way- an interpretation. For instance, in the decimal system, "1 + 1 = 2" is true while, in the binary system, "1 + 1 = 10" is true. "1 + 1 = 2" isn't even false in the binary system since 2 isn't a symbol of the binary system. The link should explain this or link to somewhere that does.

Last edited: Mar 9, 2005
3. Mar 9, 2005

BobG

A base 10 numbering system is just anatomically convenient. It's not the best mathematically or practically. It's a compromise - it keeps the number of symbols to remember down to a reasonable amount (only 10), plus you can express fairly high numbers without too many digits.

Base 12 is a much more practical base. It's divisible by 2, 3, 4, and 6 and only has two more symbols to remember than base 10.

Mathematicians would probably prefer a prime number base, such as base 11. Using a prime number, every fraction would be irreducible and represent a unique number.

Unfortunately, I don't think any civilization thought to try either of those two bases.

Early navigators preferred a base 60 system, in spite of the fact that 60 is such a large number of symbols to remember that they had to add an auxiliary base to their numbering system. 60 is divisible by lots of numbers, plus an even multiple of 60, 360, is approximately equal to one year, making it easy to track how the stars vary in their position each night.

Base 5 would be a natural human base. Humans can discriminate up to four objects before they have to resort to counting. In fact, the first human numbering systems were base 5, similar to how people keep tallies of a score by marking 4 'ones' and then designating '5' by drawing a diagonal line through the four 'ones'.

Personally, I like binary. In binary, I can count up to 1,048,575 on my fingers and toes.

4. Mar 9, 2005

jcsd

The use of base-10 number systems is just an histoirical accident and a matter of convenience. This system can be used to represent the naturals as that is exactly what it was designed to do, but there's no compelling reasons to use it other than convenience and convention.

5. Mar 9, 2005

C0nfused

Thanks for your answers. I checked the link honestrosewater mentioned. My main question is how do we produce the numbers that follow the digits we choose in any system. I have found this in the wikipedia site:
"In general, numbers in the base b system are of the form:"
$$((a_na_{n-1}...a_1a_0.c_1c_2c_3...)_b=\sum_{k=0}^{n} \ a_kb^k + \sum_{k=1}^{\infty} \ c_kb^{-k}$$
If this is the "definition" of numeral notation in a system with a base b, then
1)how b is written?
2)how $$b^k$$ is calculated and written?I feel this leads in a circle

Thanks again

6. Mar 10, 2005

honestrosewater

Did you understand the few paragraphs above this? Do you understand what a positional system is?

7. Mar 10, 2005

HallsofIvy

Staff Emeritus
In a "base b" numeration system, b itself is always "10". bk is a 1 followed by k 0s. b-k is a decimal point followed by k-1 0s and then a 1.

Is that what you are asking? I don't see any thing "circular" about that.

8. Mar 10, 2005

DaveC426913

Actually, I'd call this 'Base 1'. Every time you use up all your symbols, you start a new column. In base 1, there's only 1 symbol - so every time, you start a new column. (The horizontal cross for a 5 is merely flourish.)

Last edited: Mar 10, 2005
9. Mar 10, 2005

DaveC426913

Perhaps - but that's not an advantage of binary, that's a limitation of your counting system. The problem lies in the fact that your fingers and toes can only count two symbols (open and closed).

If you were to allow the same 10 digits (digits - fingers & toes, heh) to the other bases, you would get much larger numbers. Using ten digits in decimal, you can count up to 9,999,999,999. And in hex, you can count up to FFFFFFFFFF (1,099,511,627,775 in decimal). In fact, binary is the LEAST efficient.

There is a thread on PF somewhere that discusses the highest you can count on your fingers and toes if you include the permutations of knuckles (3 per digit). This is not an unreasonable stretch to pretend you can manipulate all three digits independently, since even counting binarily using your fingers and toes is practically impossible in the first place.

No? I'll bet you a dollar you can't display the number 699050(dec) on your fingers and toes.

10. Mar 10, 2005

BobG

Unfortunately, my little toe isn't quite coordinated enough to do that in binary (geez, you picked that number on purpose, didn't you :rofl: ). But, I could do it on my fingers.

Manipulating each finger to represent hexadecimal would be quite a bit harder than you say. The method was a modification of the Chinese base 10 system of counting on your fingers. Each link of your finger had an inside, middle, and outside. It gave you 9 possible values on your finger. Not using that finger would give you your 10th digit. That means you need to manipulate each finger 10 different ways in order to remember what you're doing. The Chinese overcame that limit by using the fingers on one hand to just point to different locations on the fingers of the opposite hand, allowing them to count up to 99,999.

Adding the lines in your fingers (at the knuckles) adds 6 more possible values on each finger, raising to you hexadecimal. You still have to use the opposite hand to point at each section, meaning that even using hexadecimal, you wind up counting to 1,048,575.

You could use the fingers of both hands to point to different locations on your toes, which would get you to 1,099,511,627,775 in decimal, just as you said. Hopefully, a person doing that would be a fast counter. A slow counter (one who could only count up one number per second) would have a sore back by time they were done, since it would take them over 34,841 years to count that high.

The Chinese method is probably the most effective practical method. Counting to anything beyond 99,999 is more fun to think about than actually do . (Dang, when you think about it, counting to 99,999 isn't all that fun, even if it's at least a realistic possibility).

11. Mar 10, 2005

mathwonk

this question seems to have been about positional notation, not choice of base.

12. Mar 10, 2005

BobG

b would be written 10. This is true for whatever base b happens to be.
$$b^k$$ is calculated by its position. Since the series starts from 0, the first digit is:

$$a * b^0$$

Second is:

$$a * b^1$$

and then:

$$a * b^2$$
$$a * b^3$$
$$a * b^4$$

and so on.

Base 10, b^0=1, b^1=10, b^2=100, b^3=1000, and so on.

Base 16, the value of b^0=1, b^1=16, b^2=256, b^3=4096, and so on.
Note, when writing the result of the above in base 16, you'd write:
b^0=1, b^1=10, b^2=100, and so on.

The only time you get awkard numbers is if you're expressing the value of a number from one base in a different base.

13. Mar 30, 2005

C0nfused

Thanks for all your answers and sorry for not replying for such a long time! As mathwonk mentioned, this is indeed a question about positional notation and not choice of base. What i really want to ask is this:
if we forget all we know about numerical systems and we want to "construct"-define one ( for example the decimal) then we define some digits and the number of these digits is the base of our system and we also define how it will be notated? So when we talk about the decimal system the base is 10 (this notation is also defined by us) and ,considering that zero ("0") is also a digit, the symbol used for the base happens to represent the successor(the next natural) of the last(biggest) digit? So using all these and the properties of reals we come to give the definition:
$$((a_na_{n-1}...a_1a_0.c_1c_2c_3...)_b=\sum_{k=0}^{n} \ a_kb^k + \sum_{k=1}^{\infty} \ c_kb^{-k}$$
This is actually my understanding of the subject. What do u think?
Thanks again

14. Mar 30, 2005

HallsofIvy

Staff Emeritus
Yes, that's basically the definition of "positional notation". Also note that your basic "symbols" (the "a"s) must go from 0 up to b-1.