heropupheropup 143k1515 gold badges113113 silver badges200200 bronze badges $endgroup$ two $begingroup$ Do you have any details about the initial one who proved this? $endgroup$
Think about the prolonged division algorithm we uncovered in quality university, where you are building the conditions on the very best separately as you might be dividing the dividend with the expression $one-r$, multiplying the freshly created expression with the divisor, subtracting, and iterating:
1 $begingroup$ @MSIS: Take into account an infinite field such as $mathbb Q$. Each and every industry is actually a Euclidean area. When there is a more elaborate set up to your trouble, inquiring a different Issue (with that context) will be constructive. $endgroup$
I believe you must elaborate when infinitesimal , and considerable finite means. It might be distinct from context to some but not to Other people. $endgroup$
For the reason that word "infinite" suggests "not finite", we start with the definition of "finite". Since we can establish
It is actually akin to inquiring, if John runs twice as speedy as Jack and both equally run off faraway from me, can I divide John's ultimate placement by Jack's last position, which might be the two more faraway from me than I am able to at any time go, and get $2=one$? (Naturally Infinite Craft neither John nor Jack on their own can access their "ultimate place", but the method by which they 'method' it explains the situation fairly well.)
Note that someone may define "transfinite" the same as "Dedekind infinite" which is an abuse of words in my opinion.
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two $begingroup$ Two points that I feel a freshman calc student demands to absorb: (1) Matters we'd produce as $infty/infty$ are named indeterminate forms, and calculus delivers specific approaches for studying them. (2) Is infinity is often a number? See this issue: math.
in concrete vogue and distinguish several situations according to the mother nature of numerator and denominator: infinitesimal, infinite, or appreciable finite, ahead of talking about the specialized Idea of limit which tends to be bewildering to rookies.
Asaf Karagila♦Asaf Karagila 402k4747 gold badges635635 silver badges1.1k1.1k bronze badges $endgroup$ one $begingroup$ To me "transfinite" strongly connotes ideas connected to ordinals, so I discover it a lousy option in contexts like nonstandard Investigation where it pitfalls contributing to misconceptions folks have that hyperreals have some particular link to infinite ordinals. $endgroup$