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We know that it is impossible to divide a number by zero. And in the past, it is impossible to square root a negative number, but afterward, we defined that the square root of negative 1 is I. So we can almost do the same thing here.
Let's define that
1/0 = ζ
so the arithmetic calculations would look something like:
A/0 + B/0 = (A+B)/0 | Aζ + Bζ = (A+B)ζ |
A/0 - B/0 = (A-B)/0 | Aζ - Bζ = (A-B)ζ |
(A/0) * (B/0) = (A*B)/(0*0) | Aζ * Bζ = ABζζ = ABζ^2 |
(A/0) / (B/0) = A/B | Aζ / Bζ = A/B |
A/0 ≠ B/0 | Aζ ≠ Bζ |
here we have to treat 1/0 as a unit instead of a value
and also, since division is the opposite of multiplication, we also need to define 1*0,
so let's define that
1*0 = ξ
and so we would have:
A*0 + B*0 = (A+B)*0 | Αξ + Βξ = (Α+Β)ξ |
A*0 - B*0 = (A-B)*0 | Αξ - Βξ = (Α-Β)ξ |
A*0 * B*0 = AB00 = AB0^2 | Αξ * Βξ = ΑΒξξ = ΑΒξ^2 |
(A*0) / (B*0) = A/B | Αξ / Βξ = Α/Β |
A*0 ≠ B*0 | Αξ ≠ Βξ |
here we need to notice that
1*0 is 0, but not any 0, is the 0 made by using 1*0
2*0 is 0, but not any 0, is the 0 made by using 2*0, it is not the same as 1*0 or 3*0 etc.
n*0 is 0, but not any 0, it is the 0 made by using n*0, it is not the same as m*0 , where m ≠ n
this will be mentioned later.
but what will happen if we multiply ζ and ξ together?
we would have the following :
ζ * ξ = (1/0)ξ = (1/0)*0 = 1
ζξ = 1
so we know that ζ and ξ will cancel each other
with the result above, we can easily derive the following
n*(1/ξ) = nζ
n*(1/ζ) = nξ
then we can easily solve the following :
ζ / ξ = ζ * (1/ξ) = ζζ = ζ^2
ξ / ζ = ξ * (1/ζ) = ξξ = ξ^2
note that 1ζ is not the same as 2ζ, instead, 2ζ is 2 times bigger than 1ζ,
we can also say that Kζ is K times bigger than 1ζ
with the same idea, 1ξ is not the same as 2ξ, instead, 2ξ is 2 times bigger than 1ξ,
we can also say that Kξ is K times bigger than 1ξ
another way that we can understand the concept is Aζ and Bζ are not the same, they maintain some ratio between themselves. The same goes for Aξ and Bξ, they are not exactly the same, they maintain some ratio between themselves as well.
we can treat ζ and ξ as a special unit or vector.
Since we have the real number line and the imaginary number line,
it would be fun if we add a "Zerota Ksive line"
(derived from the words : Zero + Zeta + Ksi + Vector + Unit)
with this we can easily explain one of the famous proof that is used to say "you are no allowed to divide by zero", at some point you may have seen the following:
0 = 0
1*0 = 2*0
1*0/0 = 2*0/0
1 = 2
and they will say this does not make sense so you cannot divide but 0, but according to what we have defined earlier, the second line has already broken the rules, we have defined that "A*0 ≠ B*0", and so the second line "1*0 = 2*0" is wrong, it should have been "1*0 ≠ 2*0"
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