Pi: A Source Book

Pi Day Is Upon Us Again and We Still Do Not Know if Pi Is Normal

“…With regards to item #3 above, there are many such "pi-mnemonics" or "piems" (i.e., phrases or verse whose letter count, ignoring punctuation, gives the digits of π) in the popular press [14,12]. Another is "Sir, I bear a rhyme excelling / In mystic force and magic spelling / Celestial sprites elucidate / All my own striving can't relate."…”

“…In 1761, using improper continued fractions, Lambert [12,Entry 20] proved that π is irrational, thus establishing that the digits of π never repeat. Then in 1882, Lindemann [12,Entry 22] proved that e α is transcendental for every nonzero algebraic number α, which immediately implied that π is transcendental (since e iπ = −1).…”

“…In 1761, using improper continued fractions, Lambert [12,Entry 20] proved that π is irrational, thus establishing that the digits of π never repeat. Then in 1882, Lindemann [12,Entry 22] proved that e α is transcendental for every nonzero algebraic number α, which immediately implied that π is transcendental (since e iπ = −1). This result settled in decisive terms the 2000-year-old question of whether a square could be constructed with the same area as a circle, using compass and straightedge (it cannot, because if it could then π would be a geometrically constructible number and hence algebraic).…”

Note on some infinite products for π

“…Multiplying by 2 (and grouping carefully to maintain a distinct formation law) leads to another version, now generally called "Wallis' product" (cf. [3,4]) or, more properly, "Wallis' second product", …”

EMBEDDING FINITELY GENERATED ABELIAN LATTICE-ORDERED GROUPS: HIGMAN'S THEOREM AND A REALISATION OF $\pi$

“…A similar technique can be applied to π = 2 + 4 ∞ n=1 (−1) n /(1 − 4n 2 ) (see [1, (13), p. 190] with z = 1/2), e π , etc., though far better recursive techniques are known for π (see [4], especially articles 70, 56, 62 and 64).…”