第 dì = ⺮ (zhú, bamboo) over 弟 (dì, younger brother). The bamboo radical sits on top as the meaning element, and 弟 below carries the sound. But 弟 is doing double duty here. Beyond marking the pronunciation, 弟 already pictured a cord wound in even turns around a stake, an image of regular, stepped order, which is exactly why the younger brother (the next in line of siblings) took that glyph. Add bamboo, whose nodes climb the stalk in even joints one above the next, and you get 第: things arranged in fixed sequence, step after step.

From that core image of ordered steps come the senses that anchor the character:

1. The ordinal prefix. Placed before a number, 第 turns a count into a position in a series: 第一 dì-yī (the first), 第二 dì-èr (the second), 第三 dì-sān (the third). This is the everyday job of 第 and the reason it appears on nearly every page of Chinese a learner reads.
2. Grade, rank, order. A standalone noun for one place in a graded series: 等第 děngdì (grade, rank), and the examination sense below.
3. Examination success. In the imperial system, to place in the civil-service exam: 及第 jídì (to pass and be ranked), 落第 luòdì (to fail).
4. A grand residence. By extension, the ranked mansion granted to or built by an official: 府第 fǔdì (a mansion), 宅第 zháidì (a stately home).

The thread through every sense is sequence: the bamboo nodes counting upward, the numbered place in a list, the rank earned in an exam, the graded standing of a great house. Wherever there is an order with fixed steps, 第 marks the step.

The single most useful thing 第 does is convert a cardinal number into an ordinal. Chinese has no separate words for "first, second, third" the way English does; instead it sets 第 in front of the plain count. 一 yī is "one," and 第一 dì-yī is "the first." 二 èr is "two," and 第二 dì-èr is "the second." The rule never changes: 第 + number = the Nth in the series. 第十 is tenth, 第一百 is hundredth.

To name a specific item, a measure word usually follows: 第一个 dì-yī ge (the first one), 第一本书 dì-yī běn shū (the first book), 第三课 dì-sān kè (lesson three). Two very common fixed forms drop straight into daily speech: 第一次 dì-yī cì (the first time) and 第一名 dì-yī míng (first place, the winner).

To ask about position rather than state it, swap the number for 几 jǐ (how many / which): 第几 dì jǐ asks "which in order?" 你住第几层 "which floor do you live on?" 这是第几次 "what number time is this?" The answer comes back as another 第 form: 第八层 (the eighth floor), 第三次 (the third time).

Because the first place in any ranking is the top one, 第一 dì-yī slides easily from the neutral "first" into the evaluative "foremost, the very best, above all else." 世界第一 shìjiè dì-yī is "first in the world," that is, the world's best. 天下第一 tiānxià dì-yī, "first under heaven," is the classic boast of unmatched supremacy, the title a martial-arts master or a great dish or a famous scenic spot might claim.

The same 第一 also serves as a strong adverbial "above all, first and foremost," ranking a priority rather than a thing: 安全第一 ānquán dì-yī, "safety first," puts safety at the head of the list. Here 第一 has left counting behind entirely and become a way of saying "this matters most."

Before 第 settled into its modern job as a colorless prefix, it was a full noun meaning a graded place in a ranking, and it carried two weighty senses out of imperial life. The first is the examination. For more than a thousand years the civil-service exam (科举 kējǔ) ranked candidates in strict order, and to place in that order was 及第 jídì, literally "to reach a rank." Its opposite, to fail, was 落第 luòdì, "to fall from the rank." A candidate who took the very top place was 及第 of the highest grade, a life-changing result that could lift an entire family.

The second sense is the mansion. Officials of rank were granted or built graded residences, and the house itself came to be called a 第, a "ranked dwelling": 府第 fǔdì (an official's mansion), 宅第 zháidì or 宅邸 (a stately home), 门第 méndì (the standing of a family, literally "gate and house," that is, lineage and rank). These uses are now literary and historical rather than everyday, but they reveal the older weight behind a character a learner first meets as a humble numbering tool.

Picture a stalk of bamboo, ⺮, growing upward in even joints, one node stacked above the next. Each joint is a fixed step in a sequence, and that is the whole idea of 第: order with numbered places. Below the bamboo sits 弟, the younger brother, the next sibling in line, lending both the sound and the same sense of one-after-another.

So whenever you see 第 in front of a number, read it as "the node, the step, the place": 第一 is the first joint, 第二 the second, 第三 the third. Ask 第几 and you are asking which joint up the stalk. And in the old imperial world, to be 及第 was to reach a joint on the ranked ladder of the exam, to live in a 府第 was to hold a graded place among the great houses. Bamboo counting upward, step by ordered step: that is 第.