My summary (intended as an incentive to read the Feynman, not a replacement for
reading it):
We start with addition of discrete objects (“I have two apples; you have
three apples. How many apples do we have between us?”). No fractions, no
negative numbers, no problem.
We get other operations by repetition—multiplication is repeated
addition, exponentiation is repeated multiplication.
We get yet more operations by reversal—subtraction is reversed
addition, division is reversed multiplication, roots and logarithms are
reversed exponentiation. These operations also let us define new kinds of
numbers (fractions, negative numbers, reals, complex numbers) that are not
necessarily useful for counting apples or sheep or pebbles but are useful
in other contexts.
Rules for how to work with these new kinds of numbers are motivated by
keeping things as consistent as possible with already-existing rules.
I recommend chapter 22 (“Algebra”) of volume 1 of The Feynman Lectures on Physics. Here’s a PDF.
My summary (intended as an incentive to read the Feynman, not a replacement for reading it):
We start with addition of discrete objects (“I have two apples; you have three apples. How many apples do we have between us?”). No fractions, no negative numbers, no problem.
We get other operations by repetition—multiplication is repeated addition, exponentiation is repeated multiplication.
We get yet more operations by reversal—subtraction is reversed addition, division is reversed multiplication, roots and logarithms are reversed exponentiation. These operations also let us define new kinds of numbers (fractions, negative numbers, reals, complex numbers) that are not necessarily useful for counting apples or sheep or pebbles but are useful in other contexts.
Rules for how to work with these new kinds of numbers are motivated by keeping things as consistent as possible with already-existing rules.