A Guide to the Different Ways of Writing Numbers in Chinese
I've noticed a lot of posts on the various issues surrounding zero, and rather than comment on them all, I figured I'd post a general answer. And rather than just talk about zero, I figured I'd talk about all the numbers, because gosh darn it, this is interesting. So here's an overview for the numbering systems you are likely to encounter in daily life in the PRC. I'll cover six systems, the three major ones - arabic, hybrid, and chinese numerals, and three minor ones you'll still see - financial numerals, heavenly stems, and colloquial numbers.
The most common is actually "Arabic" numerals as used in English. These are used pretty universally, for everything from ATM machines to road signs and mailing addresses. Most people use these when they write, though it is worth noting that the way these numbers are written in Chinese handwriting can sometimes be confusing for a second or two. Thus, most dates would be written "2018" in most formats.
Another every common form is to use a hybrid of Arabic and Chinese numerals. This is especially common when dealing with large numbers. For instance, the population of China is 1.4 billion. In Chinese this would most frequently be written 14亿 with 亿 meaning 100 million. This hybrid system is also sort of what you see when you see the year written 二〇〇八 - the character is clearly taken from western 0, though in simplified Chinese, 〇 is actually supposed to be the correct character. This is most often used in numbers that would be read individually in a sequence rather than as a single number （二〇〇八 “er ling ling ba" instead of 两千零百零八). 〇 is also used on signage, or in places where the character 零 might be hard to read.
The third method is to use all Chinese numbers. This is the most cultural and traditional way to do things. Thus Chinese numbers will be used on things that people want to give an air of formality to, or to be more culturally evocative. Thus 零 is what you would use in calligraphy. Similarly, while street signs tend to use Arabic numerals, actual street names usually use Chinese numbers. Thus Beijing's Third Ring Road is called 三环 while a sign saying that the third ring road will be in three kilometers might just say "三环 3km." You can use Chinese numbers for basically everything. I did when I was a student, mostly to make sure I learned them, and because I thought they were cool. My friends thought I was a pain in the ass, though, and the post office once rolled their eyes at me and told me to at least write the Chinese "zip code" in arabic numbers. Another way to think of this is that when you are using numbers to count, or enter an integer value, or anything that you might be tempted to use in a calculator, use arabic values, whereas if you are transcribing speech, or other linguistic uses of numbers, you would usually write them out.
Another system that you will see is the financial system. This is a very old system of writing numbers in very complex ways to complicate forgery. It was also used on formal imperial documents. Today, you'll see it on coins and paper money. In this case the numbers are pronounced the same, but written 壹 贰 叁 肆 伍 陆 柒 捌 玖， while for the orders of magnitude ten is 拾, hundred is 佰， thousand is 仟, ten thousand is 萬, and hundred million is 億. Technically, in the PRC, 零 is part of the "Financial" numbering system, whereas 〇 is part of the "normal" system, however people still use 零 more often than the other financial numbers. The government, though scrupulously observes this difference, so you'll see 〇 on most non-financial / non-ceremonial documents: your visa, for example.
Another minor system you'll see are the "Heavenly Stems." This has its origins in the complex math of the Chinese traditional calendar's sexagenary cycle. But it's most commonly used today like we would use "A. B. C." for outlining. In this system, the numbers go 甲 (jia) 乙 (yi) 丙 (bing) 丁 (ding) 戊 (wu) 己 (ji) 庚 (geng) 辛 (xin) 壬 (ren) 癸 (gui). These are used in numbering lots of things, including train cars, seating sections in movie theaters, building entrances, classes of students, etc. They are also used in astrology and feng shui.
The last minor system is really a collection of various colloquial terms that can be used to replace numbers for greater verbal clarity, or simply as part of a local accent. Some of these are universal, and are also used by the Chinese military in radio operations, others are dialect. These are pronounced differently, and seldom seen written, but have their own characters. So for 1 they say "yao" 幺, for two, before a classifier, or in larger numbers, you say "liang" 两 (this one often is written, because it would be incorrect to say 二条路, for example, which would sound like "No. 2 Tiao Road," rather than "two roads"). In the Dongbei accent they tend to drop the final nasalizations (for example 什么 "shenme" in Dongbei is said 啥 "sha" as in "啥意思？" for 什么意思 meaning "what's your point?") so they pronunce "san" as "sa" which is written 仨. In military radio communications zero is 洞. In Classical Chinese there were also words for 20 (廿 nian) 30 (卅 sa) which you will see in place names, and maybe in Cantonese.
You could add some of the following uses to your list:
1.俩 and 仨 (two/three people):
我俩去北京 - the two of us are going to beijing.
咱哥仨喝酒去-(let us) three brothers go to drink some baijiu.
2.对and双 (pair) :
两个是一双(对) - two makes a pair
3.打(transliterated from ‘dozen'):
请给我一打啤酒 - please give me a dozen beer.
4.phonetic numerals - used in military and/reading out long string of numbers (over the phone/radio etc):
1 幺 yao
2 两 liang
3 三 san
4 四 si
5 五 wu
6 六/陆 liu
7 拐 guai
8 八 ba
9 狗 gou
0 洞 dong