For anyone turned off by this document and its proofs, I recommend Numerical Methods for Scientists and Engineers (Hamming). Still a math text, but more approachable.

The five key ideas from that book, enumerated by the author:

(1) the purpose of computing is insight, not numbers

(2) study families and relationships of methods, not individual algorithms

(3) roundoff error

(4) truncation error

(5) instability