What is this 10^40 thing?

Question

While checking out "正" (as in 正坐, specifically) I found that apparently one of the associated meaning would be 10^40 (i.e. 10 elevated at the 40th power).

Example -taken from Tangorin.com:

sei 【正】 noun / noun with genitive case particle の:

1. (logical) true;  regular
2. 10^40
3. original —Abbreviation. → せいほん【正本】
4. positive;  greater than zero; —Mathematics term. → ふ【負】

10^40 is a pretty large number (estimated number of water molecules on Earth: approx. 4.5x10^48) and I wonder if this is to be intended as "infinite", or if not, when is this specific "meaning" supposed to be used.

Answer 1

It is legit. From 大辞林

⑦　数の単位。澗（かん）の1万倍，すなわち10の40乗。 〔塵劫記〕

Also, from the Wikipedia JP page on "正 (数)".

正（せい）は漢字文化圏における数の単位の一つ。正がいくつを示すかは時代や地域により異なるが、現在では10⁴⁰を示す。

You can see a list of these numerical terms for large numbers on the Wikipedia JP page on "命数法". Oh, here's an English version.

That said, while 大辞林 has the corresponding entry for each in the list, 大辞泉 and 広辞苑 do not list the definition for everything from 𥝱 (10²⁴) to 極 (10⁴⁸) inclusive.

Additionally, the Wikipedia JP page on "京 (数)" mentions that

ただし、いずれの国でも京まで使われることは稀であり、台湾・韓国では京以上の命数はあまり知られていない。

i.e. the terms for　10¹⁶ (京) and upwards are rarely used nowadays.

Answer 2

I think it's just a number. In theory, you could use it any time you wanted to say 10⁴⁰, like if you were describing the number of ways you could choose 45 objects out of a bag of 160 objects, but in practice I doubt it would be written or spoken that way.

Japanese has lots of words for big numbers. Actually, English has some too! You could equally ask your question about vigintillion (10⁶³). Why does English need this word? Well, it probably doesn't, really, but we've got it anyway.

In practice, you can probably ignore this use of 正 the same way you likely ignore English vigintillion. If you ever come across one of them in print, you can look it up in a dictionary. These days, if you need to write that number, you can just write 10⁴⁰.