Euclidean Division of Integers: Theorem and Proof

3

  • 27
  • 0
  • 2.882
  • Reply

  • Download Download Torrent Open in the desktop app ADD TO PLAYLIST

    mes

    Published on Dec 31, 2020
    About :

    In this video I go over a pretty extensive “formal” proof of what otherwise seems to be a straight forward theorem known as the Euclidean Division. When an integer, known as the dividend, is divided by another integer, known as the divisor, we get an answer broken up into two parts. The first part is an integer known as the quotient that represents how many times the divisor can divide evenly into the dividend. The second part is the remaining fraction of the divisor that doesn’t divide cleanly, and in which the numerator is known as the Remainder. When working this out by hand such as the division 9/2 = 4+1/2 we can clearly see the breakdown of this division.

    The formulization of this process is known as Euclidean Division of Integers, and the theorem is as follows: The division a/b of two integers, where b is not equal to zero, involves the existence and uniqueness of two integers q and r, such that a = bq + r for 0 ≤ r less than |b|. Now the proof of this theorem is also in the algorithm in obtaining these integers q and r. First I show that whether a or b are positive or negative, the theorem gets reduced to just the positive case. Working in an incremental step by step method, I show the Division Algorithm needed to obtain these numbers, thus proving their existence.
    The uniqueness proof involves some out of the box thinking to first assume that there are other values that q and r can take to fit the theorem, but then showing that this is impossible. This is a very interesting part of the overall derivation so I highly recommend you watch and understand the very unique reasoning applied in it!

    This is a very interesting and extensive proof video of a seemingly basic division procedure but its applications are far-reaching, so make sure to watch this video!

    Download the notes in my video: https://1drv.ms/b/s!As32ynv0LoaIh4Bp4CwnSx7ecaSmQA

    View video notes on the Hive blockchain: https://peakd.com/mathematics/@mes/euclidean-division-of-integers-theorem-and-proof

    Related Videos:

    Types of Numbers: Natural, Integers, Rational, Irrational, and Real Numbers: http://youtu.be/U22Z1q_Ibqg
    Long Division by Hand - An in depth look: http://youtu.be/giBZg5Vqryo
    Polynomial Long Division - In depth Look on why it works!: http://youtu.be/E1H584xJS_Y .


    SUBSCRIBE via EMAIL: https://mes.fm/subscribe

    DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate

    Like, Subscribe, Favorite, and Comment Below!

    Follow us on:

    MES Truth: https://mes.fm/truth
    Official Website: https://MES.fm
    Hive: https://peakd.com/@mes
    Gab: https://gab.ai/matheasysolutions
    Minds: https://minds.com/matheasysolutions
    Twitter: https://twitter.com/MathEasySolns
    Facebook: https://fb.com/MathEasySolutions
    LinkedIn: https://mes.fm/linkedin
    Pinterest: https://pinterest.com/MathEasySolns
    Instagram: https://instagram.com/MathEasySolutions
    Email me: contact@mes.fm

    Free Calculators: https://mes.fm/calculators

    BMI Calculator: https://bmicalculator.mes.fm
    Grade Calculator: https://gradecalculator.mes.fm
    Mortgage Calculator: https://mortgagecalculator.mes.fm
    Percentage Calculator: https://percentagecalculator.mes.fm

    Free Online Tools: https://mes.fm/tools

    iPhone and Android Apps: https://mes.fm/mobile-apps

    Tags :

    mathematics calculus science messcience math

    Woo! This creator can upvote comments using 3speak's stake today because they are a top performing creator! Leave a quality comment relating to their content and you could receive an upvote worth at least a dollar.

    Their limit for today is $0!
    Comments:
    Time until mes can give away $0 to their commenters.
    0 Days 0 Hours 0 Minutes 0 Seconds
    Reply:

    To comment on this video please connect a HIVE account to your profile: Connect HIVE Account

    More Videos

    04:50
    12 views a year ago $
    21:58
    mes
    1 views a year ago $
    49:52
    123 views 2 years ago $