Rooting and to a lesser extent logarithms are the shittiest operations imaginable, and though their importance and utility cannot be denied, they indicate a massive logical and philosophical flaw in mathematics that remains unaddressed.
This is truthfully a problem with all inverse operators in a field. Exponents can be reduced to the traversal of a linear series, as it may be reduced to recursive multiplication, which itself can be reduced to recursive addition. Given a number line and the axiom of addition, then definitions for multiplication and exponentiation, any operation of these can be easily calculated through increments. Subtraction is easily axiomatic in and of itself, like addition, as it is simple decrementation along the number line.
The initial problem arises in the multiplication level, though it is not entirely unworkable. Just as multiplication can be imagined as recursive addition, division can be imagined as recursive subtraction with a stipulation. The stipulation being that the answer is the number of operations performed to transform the numerator into the denominator (and this rule even maps easily with remainders.) However despite this ease of calculation, it still breaks the pattern. The solution is derived from an external axis, so while the components of the operation are critical, they are detached from the number given as the quotient.
In the next level, perhaps "root"ing is the most significantly atrocious operator of this pattern. Unlike exponentiation it is unsolvable without precalculation, and guesswork is highly encouraged. It cannot be reduced to simple recursive division, it cannot be simplified to any bare arithmetic. The root operator is an enigma, and is entirely incalculable in and of itself. An operation that extends the base of a procedural system, that exists only in relation to an operation that DOES NOT extend the system, is an indicator of flawed rationale. Math needs to be revisited.
This is truthfully a problem with all inverse operators in a field. Exponents can be reduced to the traversal of a linear series, as it may be reduced to recursive multiplication, which itself can be reduced to recursive addition. Given a number line and the axiom of addition, then definitions for multiplication and exponentiation, any operation of these can be easily calculated through increments. Subtraction is easily axiomatic in and of itself, like addition, as it is simple decrementation along the number line.
The initial problem arises in the multiplication level, though it is not entirely unworkable. Just as multiplication can be imagined as recursive addition, division can be imagined as recursive subtraction with a stipulation. The stipulation being that the answer is the number of operations performed to transform the numerator into the denominator (and this rule even maps easily with remainders.) However despite this ease of calculation, it still breaks the pattern. The solution is derived from an external axis, so while the components of the operation are critical, they are detached from the number given as the quotient.
In the next level, perhaps "root"ing is the most significantly atrocious operator of this pattern. Unlike exponentiation it is unsolvable without precalculation, and guesswork is highly encouraged. It cannot be reduced to simple recursive division, it cannot be simplified to any bare arithmetic. The root operator is an enigma, and is entirely incalculable in and of itself. An operation that extends the base of a procedural system, that exists only in relation to an operation that DOES NOT extend the system, is an indicator of flawed rationale. Math needs to be revisited.
