Japanese high numbers

… can your brain process them? I just came across 45 oku, 3000 man (on NHK easy) and I can not process this without writing it down and then reading it in millions, billions, whatever

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That’s 45*100002 + 3000*100001 + 0*100000.

Or in other words, 45.3*(104)2 = 4.53*109 = 4.53 billion

Honestly, I’ve got about an inherent understanding of how big 100002 is as I do about 10003 (which is to say, pretty much none at all), so it doesn’t really make a huge difference in the long run.

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Me thinking back to front:
3000 man - quite a lot
45 oku - more than I can comprehend of anything in real life, so…

basically what @belthazar said

and you might be interested to know this isn’t really straightforward for people attempting to communicate/translate big numbers between UK and US English and European languages…

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Oh I know, I’m from Germany. For me a billion is 10^12, that just adds even more to the confusion. We call 10^9 milliarde…

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lol, yes, it was through a very confusing conversation in German that I first realised this

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and to answer your original question, all jokes aside, no I still can’t manage to precisely get large Japanese numbers without thinking hard about it or writing it down. My one useful shortcut is to remember 百万 as a million and go up or down from there, but you probably already got that one down.

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I saw a pretty nice way to handle the conversion from George Trombley that he calls the “zero method” which he used when doing interpretation to be able to quickly convert in real time. It has worked well for me. Maybe give it a try?

He covers it well in the following video.

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I’ll do some shilling, Jotoba can do number parsing. If you input 45億3000万 for example, it will tell you what it is with arabic numerals:

Currently it sometimes splits up numbers, which makes it not work as well (already reported it as a bug), but in those cases you can fix the splitting manually and it will likely work.

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Just a quick reply for now (I want to check out that video posted by @MikeyDC65 before I delve deeper).

There are two distinct issues at play here.

1. Having an intuitive ‘feeling’ for large numbers
2. Being able to work with large numbers arithmetically, even if you don’t really have an intuition for them.

Issue 1 is something that I think is more-or-less independent of the language being spoken/written/read, and will be an issue regardless of even the number system involved. So, I’m afraid I don’t have much help to offer here except to ‘dive in’ and ‘get familiar’ with different scales of large quantities.

Issue 2 can perhaps be illustrated by considering how people express things in Scientific Notation and/or Engineering Notation.

Scientific notation is where you express things like ‘Avogadro’s number’ as 6.022 x 1023, or on a calculator as 6.022 E 23. This allows you to keep the ‘mantissa’ (the decimal part) within a familar range between 1 and less than 10, while pushing all the large-scale-ness into the ‘exponent’ (the power that 10 is raised to; or, the part after the E).

Engineering notation is really just a variation of scientific notation where you always express the exponent (the power of 10) in multiples of 3 – like 0, 3, 6, 9, etc., which corresponds to 1; 1000; 1,000,000; 1,000,000,000; etc. So, thousands, millions, billions (milliards, whatever you want to call them), etc. In engineering notation, we would represent ‘Avogadro’s number’ as 602.2 E 21, since 21 is a multiple of 3, namely 7 times 3.

In the West, we tend to align decimal numbers in groups of three digits, since we’re familiar with those (again, thousands, millions, billions/milliards, etc.)

So, when we do arithmetic with large numbers, we can more-easily ‘line up’ the numbers to add or subtract them. That may be physically on a piece of paper, or typing on a calculator, or if you practice you can do some of this in your head.

For example, if someone asks you to add “2 billion 120 million plus 10 billion 630 million”, it might take some concentration, but in your head you could probably get the right answer, “12 billion 750 million”.

That’s what I mean by ‘being able to work with’ large numbers. It doesn’t matter if we can’t intuitively understand how much ‘2 billion’ really is, let alone ‘12 billion 750 million’. The math will still work out.

So, for comprehending Japanese large numbers, I would suggest exploring the idea of a (making up a term here for convenience) ‘Japanese Engineering Notation’, or just ‘Japanese Notation’, where instead of aligning numbers by grouping their digits into groups of three, we group them into groups of four.

In ‘Japanese’ notation, Avogadro’s number would be 6022 E 20, because 20 is a multiple of 4, specifically 5 times 4.

If you start expressing numbers with this grouping-by-4 method, then you can again learn to ‘be able to work with’ large numbers, but you’ll just have to get used to grouping by 4 instead of 3.

And then you would ‘just’ have to learn the common names of common groupings by 4. So, instead of [thousand, million, billion, …], it’s [man, oku, … and I’ve already run out, …], but you get the idea.

My main point is that having intuition for large numbers is an issue regardless of the system. Nobody really has any intuition about the actual magnitude of Avogadro’s number, except that we can use scientific notation, 6.022 x 1023, to get a numerical ‘reading’ on the ‘power of 10’, namely 23 powers of 10. But what does that really ‘mean’ intuitively?

(In fact, ‘Avogadro’s number’ is ‘a thing’ precisely because it allows us to regain some form of intuition for crazy-vast quantities like the number of molecules or atoms consumed or produced in a chemical reaction. But that’s another story. I don’t want to make a mountain out of a mole hill. )

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I think it helps to parse numbers in scientific notation. I try to remember numbers as powers of ten instead of names like “ten million” or “one hundred million.” I know that

is equivalent to \underbrace{百}_{10^2} \times \underbrace{万}_{10^4} = 10^6, which is a million, and I don’t need to remember every possible combination.

I would think of this as \left( 4.5 \times \underbrace{\underbrace{十}_{10^1} \times \underbrace{億}_{10^8}}_{10^9 = \text{1 billion}} \right) + \left( 3 \times \underbrace{\underbrace{千}_{10^3} \times \underbrace{万}_{10^4}}_{10^7 = \text{10 million}} \right).

Now, I know that 10^9 is two orders of magnitude larger than 10^7, so this number is approximately 4.5 billion if I want an intuitive understanding.

Edit: I am an engineer and I work with very large and very small numbers in scientific notation every day, so this works well enough for me. I don’t have a problem writing “one hundred million” as 10^8 immediately, for instance. If you’re not used to it, though, I can see that this would take some time to get used to. As @wct said, we are typically used to numbers grouped by 10^3 in western languages. It could help if you remember (with numbers that are multiples of 10^3 in bold)

Number Equivalent in scientific notation
one 10^0
ten 10^1
hundred 10^2
thousand 10^3
million 10^6
billion 10^9
trillion 10^{12}
quintillion 10^{18}

Notice that we, as English speakers, really like multiples of 10^3=\text{1 thousand} for our large numbers. The exceptions to this rule are smaller numbers, and this makes sense as an exception because we encounter small numbers frequently in daily life and it is more intuitive to say \text{twenty} instead of \text{0.02 thousand}. (Technically, it is actually necessary to have a name for the n powers of ten below the number used to group powers n. This is why English has one, ten, and hundred—three “small” numbers not bold in the above table—and Japanese has four “small” numbers before the pattern starts: 一, 十, 百, and 千.) If you encounter a number like 10^8 and don’t immediately know what it is, try doing division of the power by 3. You get 8 / 3 = 2\text{, remainder }2, which is the product of two thousands—also known as a million—and the remainder, one hundred. Together, then, 10^8 is one hundred million. This just takes practice.

In Japanese, you will see numbers grouped as (with numbers that are multiples of 10^4 in bold)

Number Reading Equivalent in scientific notation
いち 10^0
じゅう 10^1
ひゃく 10^2
せん 10^3
まん 10^4
おく 10^8
ちょう 10^{12}
けい 10^{16}

This is actually remarkably similar to the English-language table, except it’s based on powers of 4 instead of 3. Once again, there are exceptions for smaller numbers. In Japanese, there is a grouping by ten thousand—a “myriad”—instead of one thousand. Larger numbers will also follow this pattern.

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which for engineers/maths kind of people is

oh that is so nice, thanks both for linking!

I had not noticed that and it’s actually so easy now. As a very brief visual, I’d simplify this as:

English parsing:
1,000,000,000,000
trillion, billion, million, thousand

Japanese parsing (I’m just putting in the commas as a visual!)
1,0000,0000,0000
chou (corrected), oku, man

and with that now I finally understand why you can put a 千 in front of a 万! Because in Japanese you can go up to 4 places above the last counter whereas in English you just go 3 places above the last counter.

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A 千 in front of a 万 works because the next-highest number with a name is 億, which is 10^8. 千万 is a thousand ten thousands, or \underbrace{10^3}_{\text{thousand, 千}} \times \underbrace{10^4}_{\text{ten thousands, 万}} = 10^7 (what we’d call “ten million,” \underbrace{10^1}_{\text{ten}} \times \underbrace{10^6}_{\text{million}} = 10^7, in English). It is smaller than 億. To illustrate this, let’s just count up by powers of ten:

Scientific notation Japanese Japanese note English English note
10^3 thousand
10^4 ten thousand No independent name; one power of ten higher than thousand
10^5 十万 No independent name; one power of 十 higher than 万 hundred thousand No independent name; two powers of ten higher than thousand
10^6 百万 No independent name; two powers of 十 (or 100, written as 百) higher than 万 million
10^7 千万 No independent name; three powers of 十 (or 1000, written as 千) higher than 万 ten million No independent name; one power of ten higher than million
10^8 hundred million No independent name; two powers of ten higher than million
10^9 十億 No independent name; one power of 十 higher than 億 billion
10^{10} 百億 No independent name; two powers of 十 (or 100, written as 百) higher than 億 ten billion No independent name; one power of ten higher than billion
10^{11} 千億 No independent name; three powers of 十 (or 1000, written as 千) higher than 億 hundred billion No independent name; two powers of ten higher than billion
10^{12} trillion Both Japanese and English have a name for this number because 4 and 3 are both factors of 12

This is the same kind of thing as when we say “ten million” in English. We don’t have an independent name for ten million; we know what a million is, so we multiply it by ten. Whenever you see two numbers together like this to describe the power of ten, just multiply them together.

The big difference here is that there are three unnamed orders of magnitude between names in Japanese and two unnamed orders of magnitude between names in English.

This is exactly right. In English, we are totally fine with saying “ten million” or “hundred million,” but “thousand million” doesn’t make sense because we name orders of magnitude in groups of three and we’d just call that a “billion.” In Japanese, orders of magnitude are grouped in fours, so you have “intermediate” numbers that are not just multiples of ten and one hundred, but also of one thousand.

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I just want to point out that George Trombley, the guy from the Japanese From Zero video, made an error when he used the ‘kei’ (けい, 京 = 16 zeroes) as the next unit up from the ‘oku’ (おく, 億 = 8 zeroes). The correct next step up is the ‘chou’ (ちょう, 兆 = 12 zeroes).

And I only realized this when I saw @northpilot’s table where he lists the Japanese units:

So, it would actually go:
1,0000,0000,0000
chou, oku, man

Just thought it was important to point out that little blooper from the video. Once you get above the billions or okus, I’m sure it’s common for folks to mix things up.

BTW: The video was actually great, IMO, and his ‘method of zeroes’ is great – equivalent to talking about ‘powers of ten’, but with more intuitive ‘number of zeroes (after the first digit)’ concept instead; remarkably similar to ‘pure’ scientific notation actually. Didn’t want to come across as just knocking on him for this one relatively minor error.

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Good catch I corrected mine to avoid more confusion!

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I think learning some tricks with scientific notation can really help you understand how to convert between English and Japanese representations of large numbers, so I’m going to continue with a couple more examples if you want to practice.

Let’s take a number like 25,394,000,000. I read this in English as “twenty-five billion, three hundred ninety-four million.” If I were to write this in scientific notation, I would write 2.5394 \times 10^{10} (3 orders of magnitude for each of 3 commas, plus 1 for the tens place in the billions, for a total of 10). In English, we would split this up into orders of magnitude as \underbrace{10^1}_{\text{ten}} \times \underbrace{10^9}_{\text{billion}} = 10^{10}. In Japanese, however, we’d look for multiples of four instead of three in the second number. I’d instead write 10^2 \times 10^8 = 10^{10} to immediately know that the number is on the order of \underbrace{百}_{10^2}\underbrace{億}_{10^8}.

We now know the number starts at 2百億 (the first number and the order of magnitude we found), but we now have to reason through the rest of the conversion, which is probably the hardest part of thinking about numbers across languages. We have 2百億 for the order of magnitude, but the rest of the numbers are unaccounted for. They fit into decreasing orders in the table I posted earlier, so we have 2百億+5十億+3億+9千万+4百万, which you’d probably write as 253億9400万.

You can do this in reverse to estimate the order of magnitude in English, which is probably what you’ll be doing if you encounter a number in writing or speech and you’re more comfortable with a western language. Let’s do it in reverse with the original number:

I’m only going to look at the first number, which is the larger part, to estimate the magnitude (which is way faster if you’re trying to do this while someone is talking, or while trying to read quickly). I’d mentally note that this is \underbrace{4.5 \times 10^1}_{\text{divide out as many factors of 10 as you can}} \times \underbrace{10^8}_{ 億}. I’d then look for the largest multiple of three that I can make with the sum of the powers of ten, which in this case is 1+8=9 without any remainder. I know 10^9 is a billion, so I can very quickly reason that this number is approximately 4.5 billion.

If you have more time to parse the number, you can continue with the next part, which is 3 \times 10^3 \times \underbrace{10^4}_{ 万}. To convert to English, we’d look for the largest multiple of 3 at or below 3+4=7, which is \underbrace{6}_{10^6=1\text{ million}} with a remainder of \underbrace{1}_{10^1=10}. This tells us the second part of the number is thirty million.

Let’s put the two pieces together: We now have four billion, five hundred thirty million for the whole number.

If I encountered a number like 5315億, I’d do the same thing: I would first mentally rewrite it as 5.315 \times 10^3 \times 10^8, note that 3+8=11, and quickly note 10^{11} is the product of 10^9 (a billion, with 9 being the largest multiple of 3 at or below 11) and 10^2 (a hundred), so the number is 5.315 hundred billion, or 531 billion, 500 million.

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Holy moly, you don’t have to explain the math, I know the math. The question was if you guys process the number without math-ing it out.

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No, I don’t; I think that comes with time. I do the math. I wrote it out in case others are curious about how to do it.

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Damn, y’all are going crazy.

Pretty easy, no?

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I guess it’s more like this: let’s say you’re reading a fantasy novel that says the population of the planet is 56億. Do you instantaneously and intuitively know if that is large or small for the population of a planet? Personally I still have to think about it, even if only for a few seconds.

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Thought this would be about stupidly large numbers like 那由多.