The problem is that these two counters for the Japanese numbers refer to different notions of numbering (what being the number one means), and both these notions are translated the same way in English.
The word ichiban 一番 refers to the "number one" in ranking. So the ichiban is the "best," the one at the top, the "number one." In this way, niban 二番 would be the "number two" in ranking, sanban 三番, the third, and so on.
When ichiban is combined with the me 目 suffix it changes slightly. The ichibanme 一番目 usually refers to the "number one" in turn. That is, the number one in a given "ordering," junban 順番. The nibanme 二番目 is the one that comes after the ichibanme in order, and so on.
ichigou 一号 is a slightly different concept. It refers to the "number one" in a number series. This is usually used for machines and versions of machines, just like in English they are labelled like Mk. I, Mk. II, Mk. III, and so on. So something ichigou 一号 would be "something number one," "something #1" or "something Mk. I." Then nigou 二号 would be "something number two."
To have a better idea, in the anime Dragon Ball Z, the character "number 18" (or "android 18," though she's actually a cyborg) is called jyuu hachi gou 18号 in Japanese. Her creator made the androids and their names reflect the order they were made, so number 18 is the 18th in a series of androids (and cyborgs). The first one is called ichigou 1号, the second one nigou 2号, the third sangou 3号, and so on.
As you can see, the counters ban and gou are very similar in meaning thought different. The most interesting thing is what happen when you join their kanji: bangou 番号. That is, a "number," like the number 18, or somebody's telephone number. (not a "number (of things)", as in quantity, like 18 androids, for that the word is kazu 数)
Another thing: ban 番 can sometimes refer to a "role" or "duty" someone can take. For example:
- deban 出番
Someone's turn to enter (a show, a battle, etc.) - rusuban 留守番
Taking care of the house
(while others are "away from home," rusu 留守) - miseban 店番
Taking care of the shop.
(while others are away from the "shop," mise 店)
Lastly. daiichi 第一 refers to the "first" of a topic. As in "First off, ...blahblahblah." Or "First rule." And so on. The word ichime 一目 refers to the "first" in an ordering, but not necessarily a turn order (which is what ichibanme means).
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