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It probably wouldn't help, and TAOCP from the beginning is too many pages. It would help to read the division algorithm chapter.

I introduce my own notation from medium->small division onwards. If it's any help u'', v'' are the limbs that the division instruction sees (the ones pertinent to qhat) and u',v' are all the lower limbs.

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I appreciate the info but it’s unfortunately no help.

I have a passing interest in mathematics and especially comp sci but I routinely find myself stymied just by trying to understand the notation.

Get a copy of Vol 2 and read section 4.1 - Positional number systems - before tackling 4.3 - Multiple-precision arithmetic. It will give you the necessary grounding and notation. It's also an absolutely fascinating topic and will change the way you think about numbers.

If you have vol 2 you also might as well read at least the intro to chapter 3 which is beautiful.

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