Shunyaya Symbolic Mathematics — Elementary algebraic functions — powers & roots (2.31)

Abstract
We lift powers and roots to symbolic numerals (m, a) so that classical magnitude rules hold while alignment is carried safely. Integer and fractional exponents use domain guards (sign/branch), with special handling for m < 0, k < 0, and 0^0. Under collapse phi(m,a) = m, all formulas reduce to the classical ones.

Definitions

Notation. A symbolic numeral is x = (m, a) with magnitude m in R and alignment a in (-1, 1) (reported; internally clamp near edges when using atanh). Unary function lifts use alignment carry on monotone domains:

f*(m, a) = ( f(m) , a )

1) Integer powers (k in Z)

Domain.

  • If k > 0: allowed for all m (no guard).
  • If k = 0: defined for m != 0 as m^0 = 1; the pair is (1, a).
    By default 0^0 is undefined (see Notes).
  • If k < 0: require m != 0.

Lift.

pow_k*(m, a) = ( m^k , a )   on the chosen monotone domain

Monotonicity partitions.

  • k odd, k > 0: increasing on (-inf, +inf).
  • k even, k > 0: decreasing on (-inf, 0], increasing on [0, +inf); treat sides as separate branches.
  • k < 0: treat (-inf, 0) and (0, +inf) as separate branches;
    on (0, +inf) the map is decreasing; on (-inf, 0) it is increasing for odd k (e.g., 1/m) and even-valued-in-magnitude for even k.

2) Fractional powers and nth roots (k = p/q in lowest terms, q > 0)

Principal real branch.

pow_{p/q}*(m, a) = ( m^(p/q) , a )

with guards:

  • If q even: require m >= 0 (principal real result).
  • If q odd: allow all real m.
  • If p < 0: also require m != 0.

Named roots.

sqrt*(m, a)     = ( sqrt(m) , a )            with m >= 0
cuberoot*(m, a) = ( sign(m) * |m|^(1/3) , a )   for all m

Collapse and order

Collapse check.

phi( pow_* (m, a) ) = m^k
phi( root_* (m, a) ) = root(m)

Order and ties.

  • On any interval where f(m) = m^k is monotone increasing: (m1, a1) <=_m (m2, a2) ⇒ f*(m1, a1) <=_m f*(m2, a2)
  • On intervals where f is decreasing, magnitude order reverses.
  • When magnitudes tie (e.g., (-2)^2 = (+2)^2), break ties by your declared symbolic preorder (e.g., strength via S_beta then documented rule). Do not infer cross-branch order from magnitude alone for even powers.

Continuity and edge cases

  • Near 0.
    sqrt*(m,a) is right-continuous as m -> 0+. Odd roots are continuous through 0; derivatives may blow up classically, but alignment carry keeps a bounded.
  • Negative bases.
    Even-denominator fractional powers are not real for m < 0 on the principal branch; odd-denominator roots allow m < 0.
  • Zero and reciprocals.
    For k < 0, require m != 0.
    Canonical zero remains (0, +1) when results land at zero.
  • 0^0.
    Default: undefined. If you need a totalized convention (e.g., 0^0 = 1 in combinatorics), declare it explicitly in the manifest for that page only.

Worked examples

A. Odd power (global monotone).
x = (-3, +0.4)
pow_3*(x) = ( (-3)^3 , +0.4 ) = ( -27 , +0.4 )
Collapse: -27.

B. Even power with branch tie.
y1 = (-2, -0.6) on (-inf, 0], y2 = (+2, +0.1) on [0, +inf)
pow_2*(y1) = ( 4 , -0.6 ), pow_2*(y2) = ( 4 , +0.1 )
Magnitudes tie; the symbolic preorder handles the tie using alignment and your declared rule.

C. Roots with guards.
sqrt*( (9, +0.8) ) = ( 3 , +0.8 ) (allowed; m >= 0)
cuberoot*( (-8, -0.3) ) = ( -2 , -0.3 ) (odd root permits m < 0)

D. Negative exponent (reciprocal).
pow_{-1}*( (2, 0.5) ) = ( 1/2 , 0.5 ) with guard m != 0.

Implementation notes (manifest)

  • Function and domain. Record k and the exact branch domain ((-inf,0), {0}, (0,+inf), or m >= 0 for even roots).
  • Alignment policy. Unary lifts carry a unchanged; composite expressions (products, quotients) use your default multiplicative rule (M2).
  • Edge clamps. When any downstream step needs atanh(a), use a_clamped = clamp(a, -1+eps, +1-eps) with eps = 1e-6; report a = tanh(u) only at presentation.
  • Zero policy. Use canonical zero (0, +1).

Takeaway

Powers and roots lift cleanly by alignment carry on monotone domains. Domain guards (for m = 0, k < 0, and even-denominator roots) make the lift collapse-safe, and symbolic ties encode stability when magnitudes coincide. Under collapse, everything is exactly classical.

Navigation
Previous → General Extensions (2.30)
Next → Rational functions (2.32)

Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2, clamp_eps = 1e-6, |a| < 1. All formulas are presented in plain text. Collapse uses phi(m,a) = m.