No., residence
第 centers on ordering and sequence: it marks a number in a series (No.) and by extension a graded rank or a place in a sequence. The 'residence' sense is an older, less common application of a place in a series, like a designated dwelling.
第 originally combined 竹 (bamboo) with 弟, which likely contributed the sound and the idea of order or sequence, as bamboo strips were tied in order to form early writing tablets. The modern form keeps this structure.
Picture bamboo strips 竹 tied in order with a string that loops around 丿 and through the inner 弔 shape. The strips are numbered in sequence: 第 marks the order.
For ダイ, imagine a line of bamboo strips forming a giant die (as in dice) that shows the number in the sequence: die -> ダイ, and the die tells you which number it is.
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prefix for forming ordinal numbers
depending on
third party; third person; outsider; disinterested person
gradually; by degrees; little by little; bit by bit
first step
leading person; foremost figure
the front (of a battlefield); forefront
failure (in an examination); failing to advance (to the next year)
first words said (e.g. when meeting someone); first thing out of one's mouth
passing (an examination); making the grade
a third country
the sixth sense; intuition; hunch
primary significance; primary importance; first principle
first; foremost; number one
thus; such being the case; in these circumstances
as soon as ... is completed; as soon as ... is done; as soon as one is able to ...; as soon as one has (free time)
as soon as it runs out; upon running out
first of all; in the first place; to begin with; for starters
America First (US policy)
if things permit; under certain circumstances