> Let us think of x as a quantity that can grow by a small amount so as to become x+dx, where dx is the small increment added by growth. The square of this is x2+2x⋅dx+(dx)^2. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of x^2.
It seems to me that the third term is actually a bit of a bit of x, rather than of x^2.
>Now if, for such a purpose, we regard 1/1,000,000 (or one millionth) as a small quantity, then 1/1,000,000 of 1/1,000,000, that is 1/1,000,000,000,000 (or one billionth) ..
1/1,000,000,000,000 is actually one trillionth
Check https://en.wikipedia.org/wiki/Long_and_short_scales for more
1,000,000,000, i.e. one thousand million, or 109 (ten to the ninth power), as defined on the short scale. This is now the meaning in both British and American English.
Historically, in British English, 1,000,000,000,000, i.e. one million million, or 1012 (ten to the twelfth power), as defined on the long scale. This is one thousand times larger than the short scale billion, and equivalent to the short scale trillion.