short note
... Claim 2. Suppose x is rational and y is irrational. Then (i) x + y is irrational. (ii) xy is irrational unless x = 0 (in which case xy = 0 is rational). (iii) x/y is irrational unless x = 0 (in which case x/y = 0 is rational). To see (i), suppose x + y were rational. Then by Claim 1, − x = (−1) x ∈ ...
... Claim 2. Suppose x is rational and y is irrational. Then (i) x + y is irrational. (ii) xy is irrational unless x = 0 (in which case xy = 0 is rational). (iii) x/y is irrational unless x = 0 (in which case x/y = 0 is rational). To see (i), suppose x + y were rational. Then by Claim 1, − x = (−1) x ∈ ...
Full text
... might well have been influenced by the Pythagorean philosophy, for $ occurs conspicuously in the theory of the pentagram, which was the badge of the Pythagoreans [7, p. 28]. Reference [4, p. 162], refers to the "perfect phi pyramid" whose square base is 2 by 2 units, the length of the apothem (the s ...
... might well have been influenced by the Pythagorean philosophy, for $ occurs conspicuously in the theory of the pentagram, which was the badge of the Pythagoreans [7, p. 28]. Reference [4, p. 162], refers to the "perfect phi pyramid" whose square base is 2 by 2 units, the length of the apothem (the s ...
n - UOW
... Question14 Show that in any group of 367 people at least two people must have the same birthday. Does it work for 366 people? Question15 Seven different integers are chosen from 1 to 12 inclusive. Must at least one be odd? Question16 Let T = {1, 2, 3, 4, 5, 6, 7, 8, 9} . Five integers are chosen fro ...
... Question14 Show that in any group of 367 people at least two people must have the same birthday. Does it work for 366 people? Question15 Seven different integers are chosen from 1 to 12 inclusive. Must at least one be odd? Question16 Let T = {1, 2, 3, 4, 5, 6, 7, 8, 9} . Five integers are chosen fro ...
syllabus - Cambridge International Examinations
... Cambridge IGCSE Mathematics syllabus is designed as a two-year course for examination at age 16-plus. The goals of this syllabus should enable students to: 1. develop their mathematical knowledge and oral, written, and practical skills in a way that encourages confidence and provides satisfaction an ...
... Cambridge IGCSE Mathematics syllabus is designed as a two-year course for examination at age 16-plus. The goals of this syllabus should enable students to: 1. develop their mathematical knowledge and oral, written, and practical skills in a way that encourages confidence and provides satisfaction an ...
Grade Seven Outlined
... SPI 0706.1.2 Generalize a variety of patterns to a symbolic rule from tables, graphs, or words. GLE 0706.1.5 Use mathematical ideas and processes in different settings to formulate patterns, analyze graphs, set up and solve problems and interpret solutions. SPI 0706.1.1 Use proportional reasoning to ...
... SPI 0706.1.2 Generalize a variety of patterns to a symbolic rule from tables, graphs, or words. GLE 0706.1.5 Use mathematical ideas and processes in different settings to formulate patterns, analyze graphs, set up and solve problems and interpret solutions. SPI 0706.1.1 Use proportional reasoning to ...
A Quick Change of Base Algorithm for Fractions
... only provides information about the behavior of 17 , 27 , and 47 . If we are flexible about the base b chosen to represent a number, then for every odd prime p we can always choose b so that 1p has a maximum period in that base. In other words, every prime generates a b-cyclic number for some base b ...
... only provides information about the behavior of 17 , 27 , and 47 . If we are flexible about the base b chosen to represent a number, then for every odd prime p we can always choose b so that 1p has a maximum period in that base. In other words, every prime generates a b-cyclic number for some base b ...
Math 232 - Discrete Math Notes 2.1 Direct Proofs and
... Prove the statement: For all sets X, Y, Z prove that if X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z then Y = Z. proof: Let X, Y, Z be sets. Assume X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z. We need to show that Y and Z are subsets of each other: (Y is a subset of Z) Let y ∈ Y . Then y ∈ X ∪ Y . So y ∈ X ∪ Z. That means ...
... Prove the statement: For all sets X, Y, Z prove that if X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z then Y = Z. proof: Let X, Y, Z be sets. Assume X ∩ Y = X ∩ Z and X ∪ Y = X ∪ Z. We need to show that Y and Z are subsets of each other: (Y is a subset of Z) Let y ∈ Y . Then y ∈ X ∪ Y . So y ∈ X ∪ Z. That means ...
