Pi Day, 2021 by rhoing

Pi Day, 2021

A gift from friends several years ago, it used to hang in my campus office. Interestingly, though there would be many, many possibilities, π does not figure in any of these expressions for the numbers 1–12! (E.g., cos(π/2) = 1 so 2 = π/arccos(1).) Nevertheless, it's appropriate for Pi Day because the clock is a circle. :-)

  1 using the exact value of Legendre’s constant
  2 as a geometric series (with ½)
  3 encoded for html (HyperText Markup Language)
  4 is the modular inverse of 2 (mod 7) because 2 × 4 = 8 = 1 (mod 7).
  5 expressed in terms of the so-called “Golden ratio”
  6 = 3 factorial = 3! = 3 × 2 × 1 = 6
  7 A nice exercise in theorem-proving!*
  8 as a binary number (right-to-left: 0 units, 0 first-powers, 0 squares, 1 cube)
  9 as base 4 (“quaternary”) representation: 21 = 2 × 4¹ + 1 × 4⁰
10 is the number of “combinations” picking 2 things unordered from 5 things
11 in hexidecimal representation (0x indicates hexidecimal and 0B is 11)
12³ = 1728, so the cube root of 1728 is 12

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* Proof that 6.9999… = 7:
(a)   6.99999… = 7
Multiply both sides by 10:
(b) 69.9999… = 70
Subtract (a) from (b):
69.9999… – 6.9999… = 70 – 7 ==> 63 = 63. QED

Looking back
  1 year ago: “Inside a fitness machine”
 2 years ago: No post
 3 years ago: “Something circular”
 4 years ago: “Pi Day, 2017”
 5 years ago: “Pi Day, 2016”
 6 years ago: “Something round for Pi Day”
 7 years ago: “Pi Day, 2014”
 8 years ago: “Pi Day”
 9 years ago: “Pi in the Rocks?”
10 years ago: “Am I a bit ‘left-of-center’?”

[ PXL_20210315_021828773e280cwS7x7Atm :: cell phone ]
Wow! What a cool clock.
May 10th, 2021  
@dutchothotmailcom Thinking of your son…
May 10th, 2021  
Cool clock!
May 10th, 2021  
Neat clock!
May 11th, 2021  
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