$$f_0(n) = n + 1$$ At the bottom of the ladder, the function simply adds one to the input. It has linear, slow growth.
): Each new level is defined by repeating the previous level times. For example, essentially becomes (multiplication). (exponentiation). approximates (power towers). Limit Step ( fλf sub lambda ): For "infinite" indices like fast growing hierarchy calculator
fλ(n)=fλ[n](n)f sub lambda of n equals f sub lambda open bracket n close bracket end-sub of n λ[n]lambda open bracket n close bracket refers to the $$f_0(n) = n + 1$$ At the bottom
The is a family of functions indexed by ordinal numbers, used to classify the growth rates of extremely large numbers. Calculating these values involves three fundamental recursive rules based on the type of ordinal used. Rules of the Hierarchy To compute , identify which rule applies to the ordinal Successor Case (Zero) : For the base level f0(n)=n+1f sub 0 of n equals n plus 1 For example, essentially becomes (multiplication)
f_ω²(3) ≈ 3→3→3→3 (Conway's 4-arrow notation) Magnitude: Incomprehensibly larger than Graham's Number.
Crucial: There is no single FGH. The calculator should let you toggle between:
When using a calculator, you might encounter the ( H_α(n) ). It is closely related: H_ω^α(n) = f_α(n) . Many calculators include a toggle between FGH and Hardy to show how multiplication shifts one level down.