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

1. Jul 14, 2012

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?

Last edited: Jul 14, 2012
2. Jul 14, 2012

### 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.

3. Jul 14, 2012

### micromass

Staff Emeritus
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

4. Jul 14, 2012

### 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).

5. Jul 14, 2012

### Antiphon

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

6. Jul 14, 2012

### 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.

7. Jul 15, 2012

### HallsofIvy

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

8. Jul 15, 2012

### AlephZero

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

9. Jul 15, 2012

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

10. Jul 15, 2012

### Staff: Mentor

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.
Code (Text):

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                /

11. Jul 16, 2012

### 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)
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.

Last edited: Jul 16, 2012
12. Jul 16, 2012

### 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.

13. Jul 16, 2012

### 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.

14. Jul 20, 2012

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.

15. Jul 20, 2012

### Mentallic

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.

16. Jul 20, 2012

### Diffy

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

17. Jul 20, 2012

### arildno

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

18. Jul 20, 2012

### Diffy

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.

19. Jul 20, 2012

### Benn

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.

20. Jul 20, 2012

### HallsofIvy

Staff Emeritus
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.