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

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.

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

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

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.

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

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.

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.

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).

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

[...] 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.