irrationality and transcendence 4. continued fractions.
... language as, for example, “the continued fraction α = [a0 , a1 , a2 , . . .] ” instead of saying more precisely, “the continued fraction [a0 , a1 , a2 , . . .] which represents the number α”. Continued fractions and their convergents have many fascinating properties. Lemma. Let α be the finite simpl ...
... language as, for example, “the continued fraction α = [a0 , a1 , a2 , . . .] ” instead of saying more precisely, “the continued fraction [a0 , a1 , a2 , . . .] which represents the number α”. Continued fractions and their convergents have many fascinating properties. Lemma. Let α be the finite simpl ...
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... If multiplication results in poly. of degree > n-1 Reduce it by modulo some irreducible poly. m(x) f(x) = q(x) m(x) + r(x) => r(x) = f(x) mod m(x) ...
... If multiplication results in poly. of degree > n-1 Reduce it by modulo some irreducible poly. m(x) f(x) = q(x) m(x) + r(x) => r(x) = f(x) mod m(x) ...
Real Numbers and the Number Line - peacock
... • A number line is a line with marks on it that are placed at equal distances apart. • One mark on the number line is usually labeled zero and then each successive mark to the left or to the right of the zero represents a particular unit such as 1 or ½. • On the number line above, each small mark re ...
... • A number line is a line with marks on it that are placed at equal distances apart. • One mark on the number line is usually labeled zero and then each successive mark to the left or to the right of the zero represents a particular unit such as 1 or ½. • On the number line above, each small mark re ...
§2 Group Actions Definition. Let G be a group, and Ω a set. A (left
... divides |P|, it is a power of p greater than 1, and so it is divisible by p. Therefore Ω is the union of P-orbits whose sizes are divisible by p, together with one part of size 1, and it follows that |Ω| ≡ 1 mod p. Let Q be a p-subgroup of G, and consider the action of Q on Ω. Since the size of any ...
... divides |P|, it is a power of p greater than 1, and so it is divisible by p. Therefore Ω is the union of P-orbits whose sizes are divisible by p, together with one part of size 1, and it follows that |Ω| ≡ 1 mod p. Let Q be a p-subgroup of G, and consider the action of Q on Ω. Since the size of any ...
Somewhat More than Governors Need to Know about Trigonometry1
... of n are genuine exclusions. For cosine, we exclude n = 1, 2, corresponding to multiples of 180 degrees. The cosines of such angles are 0 or ±1, hence of degree 1 (and not 1/2). For sine, we exclude n = 4, corresponding to 90 and 270 degrees, for which sine is 0, hence of degree 1 (and not 1/2). Not ...
... of n are genuine exclusions. For cosine, we exclude n = 1, 2, corresponding to multiples of 180 degrees. The cosines of such angles are 0 or ±1, hence of degree 1 (and not 1/2). For sine, we exclude n = 4, corresponding to 90 and 270 degrees, for which sine is 0, hence of degree 1 (and not 1/2). Not ...
