Comments by "MC116" (@angelmendez-rivera351) on "What are the Types of Numbers? Real vs. Imaginary, Rational vs. Irrational" video.

  1. This is a good question. There is no authoritative source that uses the symbol N or the name "natural numbers" and excludes 0, and the ISO does not even recognize the "whole numbers" nomenclature, since in most of the rest of the world, "whole numbers" refers to the set of integers Z. Mathematically, the only sets you make axiomatically are N and Z, and N includes 0, with the definition that 0 := {}, and if n is a natural number, then Union(n, {n}) is a natural number. There is no "whole numbers" nonsense, which is largely an invention of North American schools of the 20th century.
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  2. ♾ is NOT a number.
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  3. 0.(9) = 1. Note: 0.(9) is not the same as 0.999. The string 0.999 has four digits. The string 0.(9) has infinitely many digits, infinitely many repetitions of the digit 9.
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  4. Yes
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  5. You cannot divide by 0, because you cannot find any number such that, when you multiply by 0, you get 1.
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  6. I have a few pet peeves with how some of the information was delivered in the video. That numbers such as sqrt(2) or π are irrational is not because we have checked their decimal expansions and saw no repetition. They are irrational because we can definitely prove they are irrational: we do not need to check their decimal expansions, because we can definitively prove their decimal expansions are not periodic and do not terminate without having to check them: all we need to now is only the definition of the numbers to prove it. Also, the way "integers," "whole numbers," and "natural numbers" were presented was sort of confusing. For one, the set of integers is alos called "the set of whole numbers" in most places outside North America, even in places that use the English language, while no analogous distinction between "natural numbers without 0" and "whole numbers (natural numbers with 0)" exists in countries that do not use English as a language. On the other hand, most mathematicians today, and even authoratitative sources on mathematical notation, such as the ISO 80000-2, use the symbol N to refer to the natural numbers with 0, which in mathematics are just called "natural numbers." The set of natural number without 0 is not really considered any more fundamental than, say, the sequence of Fermat numbers, which is why mathematicians do this. So presenting a distinction between the two sets, especially with this very particular naming convention that is not even used by mathematicians, is definitely going to confuse people, and it could even be considered misleading, although I understand this obviously was not intentional. I also think it would have been informative to include an additional layer within the set of real numbers, talking about the algebraic numbers and nonalgebraic numbers, since their definition, while a little more complicated, is still intuitive enough to be understood from a simple presentation, and there is merit in doing so due to the historical importance of algebraic numbers and the types of implications it has in, for example, architecture. Even though irrational algebraic numbers cannot be written using finite decimal expansions, they still have decidedly nicer properties than transcendental numbers, making them much more convenient to work with. This also gives context as to why e and π are such important constants, but sqrt(3) is not, for example. Regardless, I understand why the decisions made in the video were made, and I think this was still informative and relatively good.
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  7. We can get its exact value fairly trivially, we just can never represent it by using a finite digital string.
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  8.  @abstractguy9  No, it is not. While the equation x^2 = y has two solutions, only one of those two solutions is actually called the square root.
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  9. The definition of the square root a real number y is the non-negative real number x such that x^2 = y, and the relation x |—> "square root of x" is a function, so the output is unique. It does not have two outputs.
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  10. By definition, root(z) for any number z is a single number, not two numbers. This is just how the symbol is defined. Look at any textbook and you will see that they agree with what I said.
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  11. "Fraction" is the colloquial English expression we use to refer to expressions of the form "a/b," but in mathematical language, it means nothing. In mathematical language, if you want to talk about expressions involving division, you need to specify what the dividend and divisor are, and from which side you are dividing.
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  12.  @KhanKhan-tp4ch  There is no meaningful way to plot numbers on a real line. The idea of "plotting numbers" is more of a visual aide for learning than an actual mathematically valid concept.
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  13. It is a ratio, but not a ratio of two integers, so it is not a rational number. If the radius is a rational number, then the circumference is not, and vice versa.
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  14.  @ProfessorDaveExplains  Ah, I know. I didn't mean to imply the video said that. This comment is addressed to the people who want to jump into the comments and complain about the fact that π being 22/7 contradicts it being irrational. This is also why I wrote this comment separately from my comment about the feedback of the video.
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  15. Imaginary
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  16. Yes. For example, the quaternions.
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  17.  @rickybobby5584  Yes, the square root of any prime number is irrational. This follows from the fact that the integer powers of any rational non-integer is always also a rational non-integer.
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  18. Uncomputable real number. Also, Chaitin's constants are not specific numbers, but a type of numbers.
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  19.  @abstractguy9  No, it does not.
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  20. Dear people, 1. π is NOT equal to 22/7. 22/7 is a rational approximation of π, because π is irrational. If π were rational, then we would not use an approximation. For the record, you can verify for yourselves that π is not equal to 22/7. Simply type both numbers into a calculator. 2. No, a number does not have two square roots, and root(4) is not equal to both 2 and –2. This is simply incorrect usage of the square root notation. BY DEFINITION, the square root of a real number y is equal to nonnegative real number x such that x^2 = y. This does not mean every solution x to the equation is the square root of y. Only the nonnegative solution is the square root of y. The other solutions is simply denoted with –root(y). This is all there is to it.
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  21.  @srisatya6563  π is not equal to 22/7.
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  22.  @abstractguy9  False. Even ancient socities used positional strings of digits to write their numbers. For example, the Babylonians did so in base 60, and this is how they developed their trigonometric tables.
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