Tips for developing more powerful Mathematics
At present, we use only one type of complex numbers in mathematics which allows us to deal with only one type of complexity even though it has a potential of solving problems which may have almost any number of complexities.
I am going to tell — how.
If we look at the “a ± ib” format of complex numbers — it is not at all difficult to notice that such numbers may handle, at the most, only one type of complexity.
But is it not true that negative numbers do not have only “square roots”?
They have even “cube roots”, “fourth roots”, “fifth roots” — practically, unlimited types of roots.
Since, in physics, “i” is used as a symbol for current, we use “j” as a prefix for the square roots of the negative numbers instead of “i” is used as a symbol for “current” — suppose, we decide to use the prefix “j” for the square roots of negative numbers, “k” for the cube roots of negative numbers, “l” for the fourth roots of negative numbers, “m” for the fifth roots of negative numbers and so on.
Don’t you think — it should be possible for us to develop a next generation of mathematics, which may allow us to handle problems which require processing of several types of complexities using such complex numbers as “a ± jb ± kc ± ld ± me ± …..” where “jb” stands for the square roots of the negative numbers, “kc” for the cube roots of negative numbers, “ld” for the 4th root of the negative numbers and so on?
So, instead of using a “two-axial complex plane” which has “an axis of real numbers” and “an axis of j-numbers” — we may use an axis for even “k-numbers”.
If you think that negative numbers may have only square roots; I am sorry — it is not so.
Just have a look at the following evidences of the existence of the “cube roots” and even the “sixth roots” of negative numbers.
Evidences of the existence of the cube roots and the sixth roots of negative numbers
Though Gerolamo Cardano had discovered that negative numbers may have “real square roots”, through the equation “x² — 10x + 40 = 0” the roots “5 + √(–15)” and “5 — √(–15)” of which, embowel square roots of the negative number “–15”; Rafael Bombelli (1526–1572) had discovered that “∛{10 + √(108)} — ∛{–10 √(108)}” is the root of the equation “x3 + 6x –2 = 0”. [2]
Since “∛{10 + √(108)} — ∛{–10 √(108)}” embowels the cube root of “{–10 √(108)}” which happens to be a “negative number” — it proves that negative numbers may have even real cube roots.
He had also discovered that “√(–1) = ± [∛{(2 ±√(–121)} — 2}]” which embowels the square root of “–121” as well as the 6th root of “–121”, which proves that square roots of negative numbers may have not only “real square roots” but even “real cube roots” and “real sixth roots”.
Though Leonhard Euler (1707–1783) had standardized the use of “i” as a symbol for the square root of “–1” in the form “(a ± ib)” in the year 1777, he is also credited for discovering one more method of proving that it is wrong to think that only real numbers may have real square roots by discovering that “eix = cos x + isin x” in which, “e” is the base of the natural logarithm and “i” is the square root of “−1” when x = π . [3]
But what does not seem to have occurred to anybody as yet, is — since we know which equations prove that negative numbers may have real cube roots and real 6th roots of negative numbers; it should be also possible to discover such polynomial equations, the roots of which may embowel 5th roots, 7th roots and 11th roots onward of some negative numbers.
Nobody seems to have tried to discover such equations only because it may have appeared to them only a wasteful exercise.
Just think — if “–121” may have “cube roots”, is there any doubt, some negative numbers may have not only “5th roots”, “7th roots” and “11th roots, even all “upward prime roots”, as well because “–15” is not the only negative number which has real square roots?
I have found that the following equations embowel square roots of even some other negative numbers such as “–55” and “–75”.
(i) “5 ± √ (–55)” are the roots of the equation “x² — 10x + 80 = 0”, which embowel the square roots of “–55”.
(ii) “5 ± √ (–75)” are the roots of the equation “x² — 10x + 100 = 0”, which embowel the square roots of “–75”.
So, why should we doubt that it may not be impossible to discover not only such equations which may have 5th roots, 7th roots, 11th roots, 13th roots and 17th roots but even such equations which may have all prime roots of some negative numbers, the same way as I have found that not only “– 15” — even “–55” and “–75” have real square roots?
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[1 https://rossroessler.tripod.com/