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Thread: Natural numbers vs. whole numbers vs. integers

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    Natural numbers vs. whole numbers vs. integers

    What is a natural number? A whole number? An integer?

    I've discovered:
    • Natural number: positive integer, nonnegative integer
    • Whole number: positive integer, nonnegative integer, integer


    For counting, we always use positive integers: 1, 2, 3, 4, 5, ...

    But for set theory, we also need zero, giving the nonnegative integers: 0, 1, 2, 3, 4, 5, ...

    One also needs zero to get addition and multiplication in a straightforward way from Peano's axioms of arithmetic. Those axioms define natural numbers from some initial number and a successor for each number.


    As to the word "integer", it comes from Latin "integer" and I find
    integer - Wiktionary - Latin adjective
    • complete, whole, intact
    • uninjured, sound, healthy

    Charlton T. Lewis, Charles Short, A Latin Dictionary, intĕger - numerous meanings, including "Undiminished, whole, entire, complete, perfect" - also has lots of usage examples

    So "integer" is from a word that means "whole".

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    What are positive numbers and negative numbers?

    "Positive" in general mean "assured of truth or reality".

    "Negative" in general means "denied or rejected".

    That means that negative numbers do not seem as real as positive numbers. For counting, yes, but there are plenty of other applications for numbers. Consider the numberline, a common illustration of numbers. One goes along it 0, 1, 2, 3, ... If one goes the opposite way, one gets 0, -1, -2, -3, ...

    Brahmagupta (598 - 670) - Biography - MacTutor History of Mathematics - he possibly lived in Ujjain, west-central India

    He had a good understanding of positive and negative numbers, calling positive ones "fortunes" or "assets" and negative ones "debts".

    Some Indian and Chinese mathematicians before him also had a good understanding of negative numbers.

    But Western mathematicians did not consider negative numbers completely legitimate until the 19th cy.

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    Zero? That number's name is from Arabic "sift" meaning "empty". It got borrowed as Medieval Latin "zephirum" and then into several more languages.

    Looking at words for that number in different languages, some of them are versions of "null". That is from Latin "nullus", "not any". Another such borrowing is "nil", from Latin "nihil", "nothing".

    Still others are words for "nothing" in those languages.

    So zero is the nothing number.


    Zero started out as a way of indicating nothing or none of something, and it is a necessity for place systems of numbers. One place system was developed in India, and it spread from there to the medieval Arabic world and then to the Western world, and it is our usual way of writing numbers. History of the Hindu–Arabic numeral system

    Brahmagupta recognized zero as a full-scale number, and not just a way of marking absent digits in place systems.

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    Fractions? Their name comes from a Latin word for breaking, because fractions are amounts of broken-apart entities instead of complete ones. Like 1/2 for half of something.

    Rational numbers? They get their names from being ratios of numbers.


    Irrational numbers are numbers that are not rational numbers.


    Imaginary numbers got their name because they did not seem like legitimate numbers, and they were also called impossible numbers for that reason. Like negative numbers, they were not completely accepted as legitimate numbers by Western mathematicians until the 19th cy.

    That is why more ordinary sorts of numbers are called real numbers.

    Mathematician Carl Friedrich Gauss has a famous quote on them:
    If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, √−1 positive, negative, or imaginary (or even impossible) units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness. — Gauss (1831)
    Complex number has a history, though I can't tell from it when "imaginary number" was restricted to real-number multiples of sqrt(-1). "Complex" refers to numbers that are (real number) + (imaginary number).

    I like Gauss's terminology: direct for positive numbers, inverse for negative numbers, and lateral for imaginary numbers.

    That last one fits in well with a 2D extension of the numberline for complex numbers - one axis for real numbers and one axis for imaginary numbers.

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    Adding confusion is the redundant-looking term "rational integer." It is used by algebraicists to denote what others call "integers"; so that they can use the latter term for (usually complex-number) generalizations of integer like Gaussian integers or Eisenstein integers.

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    I have no depth in math foundations, but the question I would ask is there any algebraic functional difference across definitions?


    All things algebraic, add mult div sub, reduces to a number line , or am I missing something? It is common in technology to coopt a name from something else and assign a specialized narrow meaning.

    The word imaginary causes confusion when people take the word literally.

    The top number class being complex in the (R,i) coordinate domain, reals are complex with 0 imaginary, integers are a class of reals.

    From grammar school I seem to remember N,C, and R.

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    Quote Originally Posted by steve_bank View Post
    I have no depth in math foundations, but the question I would ask is there any algebraic functional difference across definitions?


    All things algebraic, add mult div sub, reduces to a number line , or am I missing something?
    You're not missing anything. Addition and subtraction are easy to picture on a numberline. Multiplication by an integer can be understood as repeated addition or subtraction, and division by a positive integer as cutting into equal-sized parts.
    The word imaginary causes confusion when people take the word literally.
    Yes, it's an awful name. CF Gauss had a much better name: lateral numbers. I also like his direct and inverse numbers for positive and negative ones, and also Brahmagupta's asset and debt numbers.

    From grammar school I seem to remember N,C, and R.
    What's what:
    • N - natural numbers: positive or nonnegative integers.
    • Z - integers
    • Q - rational numbers
    • R - real numbers
    • C - complex numbers

    For algebraic numbers A, one must distinguish real and complex ones.

    There's also a notation for some set extended with some number outside of it.

    Z(1/2) is all integers and half-odd rational numbers.
    Z(i) is Gaussian integers or complex integers, or C(Z)
    Q(sqrt(2)) is rational numbers extended with the square root of 2
    C = R(i)
    i = sqrt(-1) here

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    When I see Z I see z = r + i a complex number.

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