The Chinese counting system is fairly easy and straightforward. There are not as many number words as in English (where the numbers zero up to twenty require twenty-one different words, plus more words for the tens, e.g. 'forty'). The Chinese characters for numbers are used much like the Arabic numerals (0 1 2 3 4 5 6 7 8 9). There are a few more characters to say ten, hundred, thousand, and so forth.
Note that most of the number characters have another more complicated form representing exactly the same meaning. This form is used when we do not want the numbers to be altered after being written, like on bank checks. This will be explained in later parts of this lesson. However, the simpler form (小寫) and complicated form (大寫) do not correspond to Simplified Chinese and Traditional Chinese.
Basic Numbers 
The 小寫 are common daily use, as they are easier to write. In fact, it will be rare to see the 大寫 numbers in normal lives. So, for the time being, you can ignore the 大寫 numbers unless you need it for the uses described later. The only exceptional case is 0 (zero). The character 零 is considered for normal use along with the other numbers, so do remember this - it will be weird to use 〇 in normal circumstances unless speed is needed, though everybody will still understand it.
Looking at the 小寫 numbers, 0 through 3 is fairly straightforward. In fact 0 does not change at all in Chinese characters, except that it seems more circular if you take it very seriously. The characters for numbers 1, 2, and 3 have one, two and three strokes. The other numbers are not as simple, but each has only a few strokes, so although the only way to learn them is by memorizing, you should not encounter much difficulty. As for the 大寫 numbers, it is not necessary to learn them actually when you only want to achieve basic communication with other Chinese people. However, read the following uses of the 大寫 numbers. If you find yourself in any these case(s), do learn them for your own use.
Tens, Hundreds, etc 
The following table lists out the corresponding Chinese characters for tens, hundreds, etc. The way to express them will be explained afterwards.
As a note, 万 and 亿 are not acceptable for Traditional Chinese. That is, both 小寫 and 大寫 are 萬 and 億.
Expressing Numbers 
When we express a number to represent a value, or for counting (that is, cardinal numbers) we just use the number directly, for example, "three cups of tea" written in Chinese is "三(three)杯(cups)茶(tea)". Note that the word "of" is understood.
More digits 
Consider this number: 42359
In Chinese: 四萬二千三百五十九
Analysing the Chinese number, we can get 4(四) "10,000s"(萬) + 2(二) "1,000s"(千) + 3(三) "100s"(百) + 5(五) "10s"(十) + 9(九).
So, in this example we can see that the Chinese numbers are expressed from the largest digit to the smallest, in a quite straight-forward translation. We write the number of a digit (42359 四), and then add the corresponding digit indicators (42359 四萬) and do the rest in this form. So we can express the numbers gradually as we analyse each digit in the Arabic number: 4-2-3-5-9 四萬-二千-三百-五十-九.
Larger numbers 
Up to "10,000s" (萬), there are corresponding Chinese characters for each digit. However, as the number gets larger, the digits are not that much available. Look at the following case to solve this problem.
In English you have only tens, hundreds, and thousands - what about the "ten thousands" and "hundred thousands"? You express it just as if you're counting the "thousands", that is, you count that there are "four hundred and thirty" thousands. So that's how you write it.
In Chinese we do the same: we have the most "10,000s" (萬) available in this case. So, we count the 10,000s, which gives us 43 "10,000s". So we write 四十三萬 to express these larger numbers.
Place holders 
Consider this number: 4,560,082
To express this in Chinese we first write the "10,000s" we see, that is, 四百五十六萬. Now we see nil digits. To express this we simply replace all the zeros with "零", and continue expressing the number: 四百五十六萬零八十二.
For multiple place holders required, we use one "零" for each place holder, while for consecutive place holders only one "零" is required.
Consider this number: 50,237,005
Applying the above rules we can express it as 五千零二十三萬七千零五.
In addition to the above rules, there is one left: when expressing 10~19, we write 十,十一...十九 - we don't say 一十,一十一,一十九 alone. However in other numbers like 110~119, we have to add a 一 in front of the 十, that is, 一百一十,一百一十一...一百一十九. This only affects 10s to simplify the form in daily uses.
The use of 大寫 numbers 
The convenience of using the simplier 小寫 numbers can be clearly seen, but the drawback of these numbers also comes from its simplicity. Imagine you're writing a cheque, and the payable amount is supposed to be 三十萬 (300,000). However, just by adding three strokes, others can easily change it to 五千萬 (50,000,000)! This might be an extreme case, but you can see its drawbacks. The simplicity such as 一(1), 二(2), 三(3),etc, could cause much problems.
For this reason, the 大寫 numbers should be used whenever you don't want the numbers to be modified. In the above example, if you have written 參拾萬 instead, nobody could have changed your writings simply because of the complexity of the characters.