4 ms·

'Zero' was most likely invented for the same reason Complex Numbers were invented: to make some rare cases of contradictory mathematical formulas work. Zero is

'Zero' was most likely invented for the same reason Complex Numbers were invented: to make some rare cases of contradictory mathematical formulas work. Zero is more a mathematical device than a number.

Sharlin 4y ago
On the contrary, complex numbers are one of the most "number" things that exist. It could be convincingly argued that either ℕ or ℂ are the most "real" of all sets of numbers; ℝ is a rather artificial restriction of the true richness of structure of its less popular superset.
- sumitviii 4y ago
  >It could be convincingly argued Go on. Argue that please.
  - Sharlin 4y ago
    Your comment has a needlessly confrontational tone. Anyway, the field of complex numbers is algebraically closed. Reals are not.
- dahart 4y ago
  FWIW I don’t think this is contrary to the comment you replied to, but I started reading your comment hoping you’d have the convincing argument. Are there some examples you’re thinking of? There’s no denying the mathematical utility of complex numbers, they are a great mathematical device. There’s also no question that natural and real numbers existed in math and language for thousands of years before complex numbers were invented. But why would complex be more “number” or more “real” than scalars, what does that mean? How are reals an artificial restriction of complex numbers? (Does it make real number sense to have 2+3i dollars or apples?) Surely the choice to include the real number line in the complex plane isn’t the only reason? Otherwise, I’d argue that vectors are the more natural choice for what you’re trying to say.
  - Sharlin 4y ago
    > There’s also no question that natural and real numbers existed in math and language for thousands of years before complex numbers were invented. Real numbers very definitely were not known to humans until the 1700s, except in the vaguest sense that rational numbers seemed to have gaps in them that are the roots of certain polynomials. It was only Newton and Leibniz who really had to start thinking about what continuity and limits mean, and what they did manage was very hand-wavy, just enough to sort of convince themselves that analysis works. Fully rigorous construction of the reals was not achieved until 1871, by Cantor, although Cauchy's work earlier in the 19th century, particularly the introduction of 𝛅–𝛆 reasoning, was certainly vital. It is important to note that by that time, complex numbers had already been introduced, and indeed were of great help in figuring out the reals, exactly because reals themselves were incomplete. But really, I digress. > Does it make real number sense to have 2+3i dollars or apples? Does it make sense to have 𝛑 dollars or apples? Tangible quantities are integral or rational. Every extension of number systems was inspired by the fact that some reasonable calculations did not seem to have an answer within the set of what was at the time understood as "numbers". Zero was introduced to reason about the question of what remains if you have n sheep and give all of them away. Negative numbers were introduced to reason about what happens if you want to buy a sheep that costs n coins and you only have n-1 coins, ie. debt. Rational numbers were introduced to reason about dividing a deceased father's sheep and coins among several brothers, or a year or a day into sub-periods. Real numbers were introduced to… Well, at this point it becomes more abstract, because the only justification for the reals was the philosophical desire to talk about the length of the hypothenuse of an ideal right isosceles triangle or the length of the circumference of an ideal circle. Complex numbers were introduced because not all polynomials of degree n > 1 seemed to have real roots at all, and others only had some number < n. It was found that this, again, was only because reals were in some way incomplete, and the roots were all there if only we could have more numbers in some sense. Today we say that the field of complex numbers is algebraically closed [1] and reals are not. This is a tremendously important property and the very reason that ℂ is seen as mathematicians as the real deal and ℝ only interesting as a subfield of ℂ. [1] https://en.wikipedia.org/wiki/Algebraically_closed_field https://en.wikipedia.org/wiki/Algebraically_closed_field
    - 4y ago

Sharlin 4y ago

On the contrary, complex numbers are one of the most "number" things that exist. It could be convincingly argued that either ℕ or ℂ are the most "real" of all sets of numbers; ℝ is a rather artificial restriction of the true richness of structure of its less popular superset.

sumitviii 4y ago
>It could be convincingly argued Go on. Argue that please.
- Sharlin 4y ago
  Your comment has a needlessly confrontational tone. Anyway, the field of complex numbers is algebraically closed. Reals are not.
dahart 4y ago
FWIW I don’t think this is contrary to the comment you replied to, but I started reading your comment hoping you’d have the convincing argument. Are there some examples you’re thinking of? There’s no denying the mathematical utility of complex numbers, they are a great mathematical device. There’s also no question that natural and real numbers existed in math and language for thousands of years before complex numbers were invented. But why would complex be more “number” or more “real” than scalars, what does that mean? How are reals an artificial restriction of complex numbers? (Does it make real number sense to have 2+3i dollars or apples?) Surely the choice to include the real number line in the complex plane isn’t the only reason? Otherwise, I’d argue that vectors are the more natural choice for what you’re trying to say.
- Sharlin 4y ago
  > There’s also no question that natural and real numbers existed in math and language for thousands of years before complex numbers were invented. Real numbers very definitely were not known to humans until the 1700s, except in the vaguest sense that rational numbers seemed to have gaps in them that are the roots of certain polynomials. It was only Newton and Leibniz who really had to start thinking about what continuity and limits mean, and what they did manage was very hand-wavy, just enough to sort of convince themselves that analysis works. Fully rigorous construction of the reals was not achieved until 1871, by Cantor, although Cauchy's work earlier in the 19th century, particularly the introduction of 𝛅–𝛆 reasoning, was certainly vital. It is important to note that by that time, complex numbers had already been introduced, and indeed were of great help in figuring out the reals, exactly because reals themselves were incomplete. But really, I digress. > Does it make real number sense to have 2+3i dollars or apples? Does it make sense to have 𝛑 dollars or apples? Tangible quantities are integral or rational. Every extension of number systems was inspired by the fact that some reasonable calculations did not seem to have an answer within the set of what was at the time understood as "numbers". Zero was introduced to reason about the question of what remains if you have n sheep and give all of them away. Negative numbers were introduced to reason about what happens if you want to buy a sheep that costs n coins and you only have n-1 coins, ie. debt. Rational numbers were introduced to reason about dividing a deceased father's sheep and coins among several brothers, or a year or a day into sub-periods. Real numbers were introduced to… Well, at this point it becomes more abstract, because the only justification for the reals was the philosophical desire to talk about the length of the hypothenuse of an ideal right isosceles triangle or the length of the circumference of an ideal circle. Complex numbers were introduced because not all polynomials of degree n > 1 seemed to have real roots at all, and others only had some number < n. It was found that this, again, was only because reals were in some way incomplete, and the roots were all there if only we could have more numbers in some sense. Today we say that the field of complex numbers is algebraically closed [1] and reals are not. This is a tremendously important property and the very reason that ℂ is seen as mathematicians as the real deal and ℝ only interesting as a subfield of ℂ. [1] https://en.wikipedia.org/wiki/Algebraically_closed_field https://en.wikipedia.org/wiki/Algebraically_closed_field
  - 4y ago