Every Number Explained in 6 minutes.

00:05:35
https://www.youtube.com/watch?v=sxPGDJ-uXV0

Sintesi

TLDRThe content outlines various types of numbers in mathematics, starting with counting numbers and whole numbers, followed by integers, rational and irrational numbers. It explains the properties of imaginary numbers, distinguishes between algebraic and transcendental numbers, and introduces higher-dimensional numbers like quaternions and octonions. Finally, it touches on constructible, computable, and definable numbers, highlighting their mathematical significance and applications in fields such as engineering, physics, and cryptography.

Punti di forza

  • 🔢 Counting numbers may or may not include zero.
  • 🟢 Whole numbers are non-negative integers.
  • 💡 Rational numbers can be expressed as fractions.
  • 🔍 Irrational numbers have non-terminating decimals.
  • 🔠 Imaginary numbers result in negative squares.
  • 📘 Algebraic numbers solve polynomial equations.
  • 🌌 Transcendental numbers are not algebraic solutions.
  • 🎮 Quaternions are used in 3D graphic representations.
  • 📐 Constructible numbers can be made with compass and straightedge.
  • 🧮 Not all real numbers are computable.

Linea temporale

  • 00:00:00 - 00:05:35

    Computable numbers are defined as those for which there is an algorithm to compute their digits with desired accuracy; this class includes both all algebraic numbers and some transcendental numbers. Definable numbers, on the other hand, can be described within a formal mathematical system, leading to a countably infinite set encompassing all rational and all algebraic numbers.

Mappa mentale

Video Domande e Risposte

  • What are counting numbers?

    Counting numbers are the oldest numbers used in counting and are often represented as 1, 2, 3, and may or may not include zero.

  • What are whole numbers?

    Whole numbers are non-negative integers, including zero, such as 0, 1, 2, 3.

  • What are rational numbers?

    Rational numbers can be expressed as fractions where both A and B in A/B are integers, with B not equal to zero.

  • What are irrational numbers?

    Irrational numbers cannot be expressed as fractions; their decimal expansions are non-terminating and non-repeating.

  • What are imaginary numbers?

    Imaginary numbers are numbers that yield a negative result when squared, denoted by the imaginary unit 'i'.

  • What is the difference between algebraic and transcendental numbers?

    Algebraic numbers are solutions to polynomial equations with integer coefficients, while transcendental numbers are not solutions to any such equations.

  • What are quaternions?

    Quaternions extend complex numbers to four dimensions and are used in 3D computer graphics and robotics.

  • What are constructible numbers?

    Constructible numbers are numbers that can be formed using only a compass and straightedge in a finite number of steps.

  • What are computable numbers?

    A number is computable if there is a finite algorithm to compute its digits with any desired accuracy.

  • What are definable numbers?

    Definable numbers have a finite description in a formal mathematical system and include all rational and algebraic numbers.

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Sottotitoli
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Scorrimento automatico:
  • 00:00:02
    these are the numbers used in counting
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    and ordering for example 1 2 3 may or
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    may not include zero depending on the
  • 00:00:11
    definition used they are closed under
  • 00:00:14
    addition and
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    multiplication they are the oldest of
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    the numbers they are also known as
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    counting
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    numbers the whole numbers are the non-
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    negative numbers without fraction hence
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    the name whole for example 0 1 2
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    3 the integers are the natural numbers
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    they're negatives and zero for example -
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    2 -1 0 1 2 they are closed under
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    addition subtraction and
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    multiplication used in temperature
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    scales and debt calculations
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    these are numbers that can be expressed
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    as a fraction ratio A over B where A and
  • 00:01:07
    B are not integers and B is not equal to
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    zero for example 1/3 45 54s - 3
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    0.26 they are closed under addition
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    subtraction multiplication and division
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    except by zero decimals either terminate
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    or repeat
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    irrationals are numbers that cannot be
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    expressed as
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    fractions their decimal expansions are
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    non-terminating and non-re repeating
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    example Square < tk2 pi and e used in
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    geometry and advanced
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    mathematics all the rational and
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    irrational numbers put together they
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    represent all points on the number line
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    for example 13 3 45 54 - 3
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    0.26 < tk2 piun n e they are closed
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    under addition subtraction
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    multiplication and division except by
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    zero imaginary numbers are numbers that
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    when squared give a negative result the
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    imaginary unit is denoted by I where I
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    isare < TK of
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    -1 for example I 2 I 3
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    I numbers of the form a plus b i where A
  • 00:02:38
    and B are real numbers and I is the
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    imaginary unit includes all the real and
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    imaginary
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    numbers for example 2 + 3 I 5 5 i -
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    1/4 used in engineering physics and
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    advanced math can represent 2D vectors
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    and rotations
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    real life example electrical engineering
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    AC circuits signal
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    processing an algebraic number is any
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    number that is a solution to a
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    polinomial equation with integer or
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    rational
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    coefficients for example I 5 < tk2 -
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    1/4 all rational numbers are algebraic
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    while some irrational numbers are
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    algebraic the set of algebraic numbers
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    is
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    countable a transcendental number is any
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    real or complex number that is not a
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    solution to any polinomial equation with
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    integer or rational
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    coefficients example natural log of 2 pi
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    and E they are uncountably infinite just
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    like real numbers they are always
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    irrational but not all irrationals are
  • 00:03:56
    transcendental
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    querian extend complex numbers to four
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    dimensions the symbol H stands for
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    Hamilton the mathematician who
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    discovered
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    querian used in 3D computer Graphics
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    Robotics and physics example
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    representing rotations in 3D
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    space octonian extend querian to eight
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    Dimensions used in theoretical physics
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    for example string theory and advanced
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    algebra
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    ptic numbers are an alternative number
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    system based on the concept of ptic
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    valuation used in advanced number Theory
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    cryptography and theoretical
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    physics constructible numbers are the
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    numbers that can be created using only a
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    compass and straight edge in a finite
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    number of steps all rational numbers are
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    constructible all constructible numbers
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    are algebraic but not all algebraic
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    numbers are constructible
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    a number is computable if there exists a
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    finite algorithm that can compute its
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    digits to any desired
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    accuracy this includes all algebraic
  • 00:05:09
    numbers and some transcendental
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    numbers most real numbers are not
  • 00:05:14
    computable a number is definable if
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    there exists a finite description of it
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    in a formal mathematical system the set
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    of definable numbers is countably
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    infinite unique finite in specification
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    and expressible in a formal language
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    all rational numbers and all algebraic
  • 00:05:33
    numbers are definable
Tag
  • numbers
  • mathematics
  • rational numbers
  • irrational numbers
  • imaginary numbers
  • algebraic numbers
  • transcendental numbers
  • quaternions
  • octonions
  • computable numbers
  • constructible numbers