If d = 0, 1, 4, 5, 6, or 9, then d^3 ends in d.
If d = 2, 3, 7, or 8, then d^3 ends in 10-d.
(2/10)
How to mentally compute the cube root of the cube of a two digit number.
Thread (1/10)
If d = 0, 1, 4, 5, 6, or 9, then d^3 ends in d.
If d = 2, 3, 7, or 8, then d^3 ends in 10-d.
(2/10)
For example, if 50653 is the cube of an integer, it's the cube of a number ending in 7.
(3/10)
Then 10a <= n <= 10(a+1), and so n^3 is between (10a)^3 and (10(a+1))^3, i.e. between 1000 a^3 and 1000 (a+1)^3.
(4/10)
Chop off the last three digits of x.
Then a, the first digit of n, is the largest number whose cube is no greater than what's left of x.
(5/10)
(6/10)
Lopping off the last three digits gives 50.
3^3 < 50 < 4^3, so the first digit of the cube root of 50653 is 3.
(7/10)
(8/10)
More from Maths
OK, I may be guilty of a DoS attack attempt on mathematicians' brains here, so lest anyone waste too much precious brain time decoding this deliberately cryptic statement, let me do it for you. •1/15
First, as some asked, it is to be parenthesized as: “∀x.∀y.((∀z.((z∈x) ⇒ (((∀t.((t∈x) ⇒ ((t∈z) ⇒ (t∈y))))) ⇒ (z∈y)))) ⇒ (∀z.((z∈x) ⇒ (z∈y))))” (the convention is that ‘⇒’ is right-associative: “P⇒Q⇒R” means “P⇒(Q⇒R)”), but this doesn't clarify much. •2/15
Maybe we can make it a tad less abstruse by using guarded quantifiers (“∀u∈x.(…)” stands for “∀u.((u∈x)⇒(…))”): it is then “∀x.∀y.((∀z∈x.(((∀t∈x.((t∈z) ⇒ (t∈y)))) ⇒ (z∈y))) ⇒ (∀z∈x.(z∈y)))”. •3/15
Maybe a tad clearer again by writing “P(u)” for “u∈y” and leaving out the quantifier on y, viꝫ: “∀x.((∀z∈x.(((∀t∈x.((t∈z) ⇒ P(t)))) ⇒ P(z))) ⇒ (∀z∈x.P(z)))” [✯]. Now it appears as an induction principle: namely, … •4/15
… “in order to prove P(z) for all z∈x, we can assume, when proving P(z), that P(t) is already known for all t∈z∩x” (n.b.: “(∀z.(Q(z)⇒P(z)))⇒(∀z.P(z))” can be read “in order to prove P(z) for all z, we can assume Q(z) known when proving P(z)”). •5/15
\u2200x.\u2200y.((\u2200z.((z\u2208x) \u21d2 ((\u2200t.((t\u2208x) \u21d2 (t\u2208z) \u21d2 (t\u2208y)))) \u21d2 (z\u2208y))) \u21d2 (\u2200z.((z\u2208x) \u21d2 (z\u2208y))))
— Gro-Tsen (@gro_tsen) February 12, 2021
First, as some asked, it is to be parenthesized as: “∀x.∀y.((∀z.((z∈x) ⇒ (((∀t.((t∈x) ⇒ ((t∈z) ⇒ (t∈y))))) ⇒ (z∈y)))) ⇒ (∀z.((z∈x) ⇒ (z∈y))))” (the convention is that ‘⇒’ is right-associative: “P⇒Q⇒R” means “P⇒(Q⇒R)”), but this doesn't clarify much. •2/15
Maybe we can make it a tad less abstruse by using guarded quantifiers (“∀u∈x.(…)” stands for “∀u.((u∈x)⇒(…))”): it is then “∀x.∀y.((∀z∈x.(((∀t∈x.((t∈z) ⇒ (t∈y)))) ⇒ (z∈y))) ⇒ (∀z∈x.(z∈y)))”. •3/15
Maybe a tad clearer again by writing “P(u)” for “u∈y” and leaving out the quantifier on y, viꝫ: “∀x.((∀z∈x.(((∀t∈x.((t∈z) ⇒ P(t)))) ⇒ P(z))) ⇒ (∀z∈x.P(z)))” [✯]. Now it appears as an induction principle: namely, … •4/15
… “in order to prove P(z) for all z∈x, we can assume, when proving P(z), that P(t) is already known for all t∈z∩x” (n.b.: “(∀z.(Q(z)⇒P(z)))⇒(∀z.P(z))” can be read “in order to prove P(z) for all z, we can assume Q(z) known when proving P(z)”). •5/15
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I just finished Eric Adler's The Battle of the Classics, and wanted to say something about Joel Christiansen's review linked below. I am not sure what motivates the review (I speculate a bit below), but it gives a very misleading impression of the book. 1/x
The meat of the criticism is that the history Adler gives is insufficiently critical. Adler describes a few figures who had a great influence on how the modern US university was formed. It's certainly critical: it focuses on the social Darwinism of these figures. 2/x
Other insinuations and suggestions in the review seem wildly off the mark, distorted, or inappropriate-- for example, that the book is clickbaity (it is scholarly) or conservative (hardly) or connected to the events at the Capitol (give me a break). 3/x
The core question: in what sense is classics inherently racist? Classics is old. On Adler's account, it begins in ancient Rome and is revived in the Renaissance. Slavery (Christiansen's primary concern) is also very old. Let's say classics is an education for slaveowners. 4/x
It's worth remembering that literacy itself is elite throughout most of this history. Literacy is, then, also the education of slaveowners. We can honor oral and musical traditions without denying that literacy is, generally, good. 5/x
As someone\u2019s who\u2019s read the book, this review strikes me as tremendously unfair. It mostly faults Adler for not writing the book the reviewer wishes he had! https://t.co/pqpt5Ziivj
— Teresa M. Bejan (@tmbejan) January 12, 2021
The meat of the criticism is that the history Adler gives is insufficiently critical. Adler describes a few figures who had a great influence on how the modern US university was formed. It's certainly critical: it focuses on the social Darwinism of these figures. 2/x
Other insinuations and suggestions in the review seem wildly off the mark, distorted, or inappropriate-- for example, that the book is clickbaity (it is scholarly) or conservative (hardly) or connected to the events at the Capitol (give me a break). 3/x
The core question: in what sense is classics inherently racist? Classics is old. On Adler's account, it begins in ancient Rome and is revived in the Renaissance. Slavery (Christiansen's primary concern) is also very old. Let's say classics is an education for slaveowners. 4/x
It's worth remembering that literacy itself is elite throughout most of this history. Literacy is, then, also the education of slaveowners. We can honor oral and musical traditions without denying that literacy is, generally, good. 5/x
