From Counting to Rational Exponents

Unit 1 tells one story: each new kind of number and notation is added so the old rules stay true, from counting numbers to rational exponents.

Start where Video 002 started, with 1, 2, 3. Subtraction demands answers the counting numbers lack, and the integers arrive; division demands more, and the rational numbers arrive; then Video 005's π and √2 show the line still has occupied points no fraction names, and the real numbers close the gaps. Nothing old is discarded — every natural number keeps every property it had, and the property list of Video 014 through Video 018 reads like a constitution the new arrivals must sign: commute, associate, distribute, keep your identities and inverses. Exponents replay the same pattern in miniature. The notation begins as a tally of repeated factors — 2⁵ counts five twos and equals 32 — and the product, quotient, and power rules follow from counting alone. Then the quotient rule generates cases the tally cannot picture: 5³/5³ points at 5⁰, and 5²/5⁵ points at 5⁻³. The zero and negative rules answer the only way that keeps the quotient rule true — a⁰ = 1, a⁻ⁿ = 1/aⁿ — extension by loyalty to old law, not decree. Radicals ask the reverse question — which number, squared, lands here? — and the principal-root convention of Video 041 keeps the answer single: √25 is 5, √(x²) is |x|, bars intact. Rational exponents close the loop: 8^(1/3) is the cube root of 8, and 16^(3/4) runs a root and a power in either order because Video 025's rule licenses the swap. One system, no exceptions granted, each extension paying its dues to the rules before it.