half, middle, odd number, semi-, part-
半 centers on the idea of dividing something into two equal parts: half, semi-, or part-. The sense of 'middle' comes from the point where something is split, and 'odd number' is a specific application where a number cannot be evenly halved.
half; semi-; partial
half
second half; latter half
first half
half a year; six months
majority; more than half; most (of)
remnant; fragment; incomplete set; incompleteness
middle; halfway; midway
lower half of the body; nether parts
half the amount (of money); half price; half fare
halfway; half measures; unfinished; incomplete; by halves; half-baked; half-cocked; half-hearted; half-arsed; half-assed
half the number; half (of an amount)
upper half of the body; upper body
peninsula
semiconductor
radius
half day
reduction by half; halving
majority
Korean peninsula
short sleeves
half-moon
half the body; half length
first half of the year; first half of the fiscal year
半 originally depicted an ox being split in half, with the top strokes representing the divided parts and the central vertical line indicating the split. The modern form retains the idea of division into two equal parts.
The two top strokes look like two halves of something, and the vertical line down the middle splits them apart. Picture cutting a cake exactly in half: the top strokes are the two equal pieces, and the vertical line is the knife.
For ハン, imagine a half-eaten ham sandwich: ham -> ハン, and only half of it is left.