<div dir="auto">Rational numbers whose decimal representations have a finite length are well-ordered. In third grade they divide integers. This may lead to rational quotients but they just write 34R3, for example, if the remainder is 3.<br><br><div data-smartmail="gmail_signature">---<br>Frank C. Wimberly<br>140 Calle Ojo Feliz, <br>Santa Fe, NM 87505<br><br>505 670-9918<br>Santa Fe, NM</div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, Apr 30, 2021, 12:01 PM jon zingale <<a href="mailto:jonzingale@gmail.com">jonzingale@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Mmm, long division is an interesting one. Who am I to say how things must be proved, but the proofs of the division algorithm with which I am familiar involve the well-ordering principle. There, in this one idea, lies two problematic details:<p>
1. The non-algebraic nature of the <a href="https://en.wikipedia.org/wiki/Well-ordering_principle" rel="nofollow noreferrer" link="external" target="_blank">well-ordering principle</a>, and its correlative <a rel="nofollow noreferrer" link="external">controversies</a>. As outlined in the paper, "It has been shown that if you want to believe
the well-ordering theorem, then it must be taken as an axiom."<p>
2. The first significant moment where intension in the form of computational complexity enters an otherwise extensional number theory.
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