Natural-number powers and principal square roots

Mathematical statement

For a real number a and a natural number n, use the empty-product convention a^0=1 (including 0^0=1 in this algebraic setting) and define powers recursively by a^(n+1)=a^n·a. For r≥0, the principal square root √r is the unique real number y satisfying y≥0 and y²=r.

Representative formula (exact TeX)

a^{n}

Assumptions

• n∈N={0,1,2,…}; negative exponents, noninteger exponents, and complex branches are outside this entry.
• The real principal square root is defined here only for r≥0. The convention 0^0=1 is algebraic and does not prescribe a rule for limits.

Symbol scope

• a — real base
• n — natural-number exponent

Intuition, definition, and misconceptions

a^n does not mean a·n. At n=0, the empty product is 1; for n>0, it is a product of n copies of a. A negative base has a negative positive-odd power and a positive positive-even power. A zero base raised to a positive power remains zero.

√0=0, and the principal square root of positive r selects the nonnegative branch. The real solution set of y²=r has two roots ±√r when r>0, only zero when r=0, and is empty when r<0. Consequently √(a²)=|a|, and it cannot be replaced by a without a sign condition.

The exercises use exact instances with integer bases, nonnegative integer exponents, and perfect squares, and separately check the domain condition for negative radicands. A future bridge to Mathlib Real.sqrt must impose r≥0 because its zero value for negative inputs is not the textbook root value defined here.

Common errors include giving ± values for a principal square root, replacing √(a²) with a without checking its sign, or applying the local 0^0 convention to limits.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Arithmetic operations — Natural-number powers are defined recursively using multiplication.
• Real number — The principal square-root branch is selected using the nonnegative real domain and its order.

Verification boundary

The stated instances can be recomputed exactly; formal Evidence for general square-root existence and uniqueness and power laws has not yet been attached.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Bind the empty-product recursion for natural powers, the nonnegative principal-root premise, sqrt_sq_eq_abs, and the zero-value boundary of Real.sqrt.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.