8.1 Symbols and Sets of Numbers
... Order Property for Real Numbers: For any two real numbers a and b, 1. a is less than b if a is to the left of b on the number line. 2. a is greater than b if a is to the right of b on the number line. Example 1. Insert <, >, or = in the space between the paired numbers to make each statement true. a ...
... Order Property for Real Numbers: For any two real numbers a and b, 1. a is less than b if a is to the left of b on the number line. 2. a is greater than b if a is to the right of b on the number line. Example 1. Insert <, >, or = in the space between the paired numbers to make each statement true. a ...
Grade 7/8 Math Circles Series Sequence Recap
... 5. *These two questions are geometric series that start at a term other then 1, because N P j−1 −r N of this they use a slightly different formula arn−1 = a r 1−r n=j ...
... 5. *These two questions are geometric series that start at a term other then 1, because N P j−1 −r N of this they use a slightly different formula arn−1 = a r 1−r n=j ...
Worksheet #5 on Quadratics
... A quadratic equation is an equation that can be written in the form ax 2 bx c 0 where a, b and c are numbers. Some quadratic equations can be solved by factoring and using the fact that if the product of two numbers is 0, then one of the numbers must be 0: EXAMPLE 1: ...
... A quadratic equation is an equation that can be written in the form ax 2 bx c 0 where a, b and c are numbers. Some quadratic equations can be solved by factoring and using the fact that if the product of two numbers is 0, then one of the numbers must be 0: EXAMPLE 1: ...
Solutions 2
... Case 2: Suppose b − a ≤ 1. Since b − a > 0 we know that (Axiom of Archimedes) ∃k ∈ N s.t. b − a > 1/k, which means that bk − ak > 1. So we can apply case 1 to ak and bk instead of a and b and again, find an integer, say m s.t. ak < m < bk, which can be rewritten a < m/k < b. Since m/k is rational, w ...
... Case 2: Suppose b − a ≤ 1. Since b − a > 0 we know that (Axiom of Archimedes) ∃k ∈ N s.t. b − a > 1/k, which means that bk − ak > 1. So we can apply case 1 to ak and bk instead of a and b and again, find an integer, say m s.t. ak < m < bk, which can be rewritten a < m/k < b. Since m/k is rational, w ...
PDF
... Dedekind cuts are particularly appealing for two reasons. First, they make it very easy to prove the completeness, or continuity of the real line. Also, they make it quite plain to distinguish the rationals from the irrationals on the real line, and put the latter on a firm logical foundation. In t ...
... Dedekind cuts are particularly appealing for two reasons. First, they make it very easy to prove the completeness, or continuity of the real line. Also, they make it quite plain to distinguish the rationals from the irrationals on the real line, and put the latter on a firm logical foundation. In t ...