Does the first two-digit number have to be 10?

[GladScientist]

Would it be more difficult if our number system went like this?

1-2-3-4-5-6-7-10...17-20...

If so, what makes ten the magic number for humans doing mathematics?

 

[DrewD]

I would probably be difficult to change, but any base is possible. Binary is probably the easiest to se where 10 is the numeral for two.

People have not always used base ten, but it seems logical to me that having ten fingers might have something to do with it. I don't know enough about math history though.

 

[micromass]

Nope, what you propose would not change our math at all. There is nothing special about 10 other than that we have 10 fingers on our hand.

The computers you use every day, for example, don't use a decimal numeral system but rather a binary. That is: in your computers the numbers are

$$0,~1, ~ 10, ~ 11, ~ 100, ~ 101, ...$$

You might want to take a look at:
http://en.wikipedia.org/wiki/Numeral_system
http://en.wikipedia.org/wiki/Binary_numeral_system

 

[the_emi_guy]

There is nothing mathematically special about using a radix of 10 (decimal numbering). Grouping in bunches of 10s evolved because we happen to have 10 fingers on our hands.
Decimal involves:
10 symbols
each place value represents 10^n (1, 10, 100, 1000 ...)

Your proposed scheme is called octal and was used on several early mainframe computers. It involves:
8 symbols (you needed to start your count at 0, 0 counts as a symbol).
each place value respresents 8^n (eights place, 64s place etc).

 

[Antiphon]

The first two digit number is always 10 if 0 and 1 are the first and second digits of the numeral set.

 

[Mentallic]

Except the use of base 10 is much better than base 2 for our purposes. It's easier to memorize 10 symbols and write down numbers with those than to remember 2 symbols but have to constantly deal with writing and reading many digits.

For example, 8740 is a relatively large number for practical purposes, and it only requires 4 digits which is easy to write, read, and manipulate.
The corresponding number in base 2 is 10 0010 0010 0100 which has 14 digits in total and is cumbersome to handle.

 

[HallsofIvy]

The first two digit number is always 10 if 0 and 1 are the first and second digits of the numeral set.

But the first two digit number is not necessarily "ten".

 

[AlephZero]

Your proposed scheme is called octal and was used on several early mainframe computers.

I don't think there was any computer hardware that used octal. There were electronic devices that used decimal arithmetic http://en.wikipedia.org/wiki/Dekatron

The octal numbers were just used as more convenient way for humans to "read" binary, and in fact they are still sometimes used for that purpose, for example the unix "chmod" command to set read/write permissions on files. http://en.wikipedia.org/wiki/Chmod

 

[GladScientist]

But the first two digit number is not necessarily "ten".

Yes my bad, I probably should have put "ten" in the title rather than "10." It can get a bit confusing.

 

[Mark44]

I don't think there was any computer hardware that used octal.

Not true. Quite a few computers in the early days used octal, among them Digital Equipment Corporation's PDP-8. This computer used 12-bit words composed of two 6-bit bytes. Each byte was made up of two 3-bit "nybbles."

In the wiki article for this computer (http://en.wikipedia.org/wiki/PDP-8), here is some PDP assembly code for outputting the string "Hello, world!" Notice the two characters just before the final 0 of the string: 015 and 012. These are the octal codes for carriage return (ASCII 13) and linefeed (ASCII 10) characters.

STRNG, "H               / A well-known message
          "e               /
          "l               / NOTE:
          "l               /
          "o               /   Strings in PAL-8 and PAL-III were "sixbit"
          ",               /   To use ASCII, we'll have to spell that out, character by character
          "                /
          "w               /
          "o               /
          "r               /
          "l               /
          "d               /
          "!               /
          [color="red"]015[/color]              /
          [color="red"]012[/color]              /
          0                /

The octal numbers were just used as more convenient way for humans to "read" binary, and in fact they are still sometimes used for that purpose, for example the unix "chmod" command to set read/write permissions on files. http://en.wikipedia.org/wiki/Chmod

 

[jbriggs444]

Although octal is a convenient radix within which to express 12 bit values, the PDP-8 hardware did not actually implement any mathematical operations that depended on the octal notation.

From memory, the architecture had one 12 bit register and an instruction set that consisted of the following instructions:

AND (and the contents of the target address into the accumulator)
TAD (add the contents of the target address into the accumulator)
ISZ (increment the contents of the target address. Skip the next instruction if non-zero)
DCA (store the accumulator into the target address and then clear the accumulator)
JMP (branch to the target address)
JSB (branch to the target address, but save the next instruction address in first)
The operate microinstructions
Some other set of microinstructions whose name escapes me.

Instructions were each coded into a 12 bit word, one instruction per word. 3 bits encoded the op code. The remaining bits encoded the target memory address. Within the address field, one bit encoded whether the memory reference was within the current page or within the zero page, one bit encoded whether the memory reference was direct or indirect and the remaining seven bits encoded the memory address within the target page.

Arithmetic was all on 12 bit words, not on 3 bit nybbles.

 

[DaveC426913]

Decimal has 10 symbols, then they must repeat.
0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21...

Octal has 8 symbols, then they must repeat.
0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17, 20, 21...

Binary has 2 symbols, then they must repeat.
0,1,10,11,100,101...

Hexadecimal has 16 symbols, then they must repeat.
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10,11,12,13,14,15,16,17,18,19,1A,1B,1C,1D,1E,1F,20,21...

etc.

 

[the_emi_guy]

Brings back memories. The first computer I got the privilege of programming was the Univac 1106. I have no idea how it worked "under the hood" but all of the user accessible numbers were in octal such as register numbers, interrupt levels, register dumps etc.

 

[GladScientist]

Decimal has 10 symbols, then they must repeat.
0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21...

Octal has 8 symbols, then they must repeat.
0,1,2,3,4,5,6,7,10,11,12,13,14,15,16,17, 20, 21...

Binary has 2 symbols, then they must repeat.
0,1,10,11,100,101...

Hexadecimal has 16 symbols, then they must repeat.
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,10,11,12,13,14,15,16,17,18,19,1A,1B,1C,1D,1E,1F,20,21...

etc.

Then I guess a better way of asking my question would to be ask: does the decimal system require 10 repeating symbols, or can it be done with 6 or something?

Although it has apparently been answered fairly thoroughly.

 

[Mentallic]

Then I guess a better way of asking my question would to be ask: does the decimal system require 10 repeating symbols, or can it be done with 6 or something?

It can be done with any integer number of digits, but what you want to aim for is to not have too many digits (binary, base 2) and not too many symbols. You want a nice value in between, and 10 is nice to go with since we have 10 fingers and toes.

Oh and in some other cases there are other reasons as well, such as with computers using base 2^n for some n.

 

[Diffy]

Why not a number system were every single number has it's own unique symbol?

 

[arildno]

Why not a number system were every single number has it's own unique symbol?

I wouldn't remember even half of them. Would you?

 

[Diffy]

I wouldn't remember even half of them. Would you?

Unless there was a method to creating the symbol. Could there be such a method for rational and irrational numbers?

It is just interesting to think about.

 

[Benn]

Unless there was a method to creating the symbol. Could there be such a method for rational and irrational numbers?

It is just interesting to think about.

There is such a system in place now. The character "2343678" uniquely determines that number, and, by a clever method, we know exactly where that number is in relation to all the other natural numbers.

The rationals have a similar system, but the representation isn't unique (i.e. 1/1 = 2/2 = 3/3...).

Most irrationals can't be defined (see http://en.wikipedia.org/wiki/Definable_real_number). So we certainly can't put a method in place to name them all.

 

[HallsofIvy]

I, II, III, IIII, IIIII... is precisely such a system. In effect, the Roman numeral system is such a system. They are not, of course, place number system, which is what you need to get "10" as the "first two digit number". And it is very awkward, though, of course, not impossible, to write very large numbers or fractional numbers.

 

[jbriggs444]

Why not a number system were every single number has it's own unique symbol?

You would have to print large numbers in a large font or risk running out of symbols. In a 4x6 font you would only have 24 pixels and accordingly could only count up to 16,777,216.

In addition, some of those numerals would be rather difficult to distinguish from one another.

There is a notion of radix "efficiency" where the cost of each digit is taken as proportional to the number of possible digits at that position. The cost of expressing a number x in radix r is then approximated by r * ln(x)/ln(r). For purposes of comparison a relevant figure of merit is e * ln(r)/r

By this measure of efficiency:

Radix 2 = 94.2%
Radix "e" = 100%
Radix 3 = 99.5%
Radix 4 = 94.2%
Radix 10 = 62.6%
Radix 16,777,216 = 0.0002.70%

 

[phinds]

Not true. Quite a few computers in the early days used octal,]

Mark, you have a serious misunderstanding of computer architecture. As jbriggs444 pointed out, octal is just a shorthand for 3 bits, so that people can manipulate things more easily. All normal computers use binary. Some use octal representations and some use hexidecimal representations, but all that is just shorthand for PEOPLE, not for the comptuer.

There actually HAVE been computers that didn't use binary, specifically one that used decimal (using what's called BCD --- binary coded decimal) and actually performed decimal, not binary arithmetic,

 

[Diffy]

There is such a system in place now. The character "2343678" uniquely determines that number, and, by a clever method, we know exactly where that number is in relation to all the other natural numbers.

The rationals have a similar system, but the representation isn't unique (i.e. 1/1 = 2/2 = 3/3...).

Most irrationals can't be defined (see http://en.wikipedia.org/wiki/Definable_real_number). So we certainly can't put a method in place to name them all.

That wiki article is fascinating. I've never really seen anything like that before. Thanks for posting it.

And yes, very clever saying that we have such a system in place now. I guess I should have been more specific in what I actually mean. Which is hard for me because I am not quite sure myself!

To the OPs point, we start repeating symbols in our numeral system. In essence, base 10 starts to repeat symbols for integers once we get to 10. The symbol one has been repeated for both 1 and 10. I consider here 10 a combination of two symbols 1 and 0.

Furthermore, Decimals like .2 reuse the symbol for 2. In the system I was trying to imagine, .2 would have reused the symbol for the number 2.

I am probably talking gibberish now. So I'll stop.

Thanks again for the article.

 

[Mentallic]

Why not a number system were every single number has it's own unique symbol?

Sure, if you can find a way of creating an infinite number of symbols that readers can remember (or at the very least deduce) and distinguish between.
I'm sure some primitive civilizations once used all different symbols for their number system, but people since then have learned of the power of combinations of symbols.

In fact, doesn't Mandarin (Chinese) use all different characters in its language to represent syllables? It must be a weird feeling to be able to read road signs and such but then you pick up a newspaper and get stuck on the first few characters.

 

[eumyang]

In fact, doesn't Mandarin (Chinese) use all different characters in its language to represent syllables? It must be a weird feeling to be able to read road signs and such but then you pick up a newspaper and get stuck on the first few characters.

Not quite. Chinese characters are used to represent the digits and some of the powers of 10 -- you have characters for 0 (sort of), 1-9, 10, 100, 1000, 10000... Chinese numbers are grouped by ten-thousands (not thousands).

Then you have a separate set of Chinese characters used by financial institutions, because the standard characters are geometrically simple and are easy to forge. See this Wikipedia article.

 

[eumyang]

All normal computers use binary. Some use octal representations and some use hexidecimal representations, but all that is just shorthand for PEOPLE, not for the comptuer.

Sometimes I wonder what would be like if we used the hexidecimal numbers instead of decimal from the start. I believe that the ancient Chinese used a system of weights based on hexadecimal instead of decimal.

There is website that advocates the use of hexadecimal, even changing the way we tell time (hexclock). Advocacy to use hexadecimal numbers is not new. See this book, published in 1862.

Of course, it's not practical for the world to change the way we count and do math today, but I find it interesting to think about.

 

[Mentallic]

Not quite. Chinese characters are used to represent the digits and some of the powers of 10 -- you have characters for 0 (sort of), 1-9, 10, 100, 1000, 10000... Chinese numbers are grouped by ten-thousands (not thousands).

Actually I was referring to their alphabet, not their numerical system, but I didn't know that so thanks for sharing

 

[dodo]

Why not a number system were every single number has it's own unique symbol?

This reminds me of a short-story by Jorge Luis Borges, "Funes the memorious", which can be found on the web. Some excerpts:

[...] He told me that toward 1886 he had devised a new system of enumeration and that in a very few days he had gone before twenty-four thousand. He had not written it down, for what he once meditated would not be erased. The first stimulus to his work, I believe, had been his discontent with the fact that "thirty-three Uruguayans" required two symbols and three words, rather than a single word and a single symbol. Later he applied his extravagant principle to the other numbers. In place of seven thousand thirteen, he would say (for example) Máximo Perez; in place of seven thousand fourteen, The Train; other numbers were Luis Melián Lafinur, Olimar, Brimstone, Clubs, The Whale, Gas, The Cauldron, Napoleon, Agustín de Vedia. In lieu of five hundred, he would say nine. [...]

Locke, in the seventeenth century, postulated (and rejected) an impossible idiom in which each individual object, each stone, each bird and branch had an individual name; Funes had once projected an analogous idiom, but he had renounced it as being too general, too ambiguous. In effect, Funes not only remembered every leaf on every tree of every wood, but even every one of the times he had perceived or imagined it. [...]

[...] It was not only difficult for him to understand that the generic term dog embraced so many unlike specimens of differing sizes and different forms; he was disturbed by the fact that a dog at three-fourteen (seen in profile) should have the same name as the dog at three-fifteen (seen from the front). [...]

Without effort, he had learned English, French, Portuguese, Latin. I suspect, nevertheless, that he was not very capable of thought. To think is to forget a difference, to generalize, to abstract. In the overly replete world of Funes there were nothing but details, almost contiguous details.