Beginning of the Year Math Review
... MULTIPLY (x) • When you want to find the quotient of two or more numbers, you: DIVIDE (/) ...
... MULTIPLY (x) • When you want to find the quotient of two or more numbers, you: DIVIDE (/) ...
The Impossibility of Trisecting an Angle with Straightedge and
... Today, the proof that straightedge–and–compass trisection is impossible is usually done using algebraic methods. The following fact, which can be found in any good abstract algebra textbook, is needed. Theorem A: If a real number α is constructible using only straightedge and compass, then α satisfi ...
... Today, the proof that straightedge–and–compass trisection is impossible is usually done using algebraic methods. The following fact, which can be found in any good abstract algebra textbook, is needed. Theorem A: If a real number α is constructible using only straightedge and compass, then α satisfi ...
File
... states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between –2 and –3. ...
... states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between –2 and –3. ...
Full text
... Lemma 5: Let m b e a natural number prime to 10. For each integer i from 0 to (m -1) / 2, let us write rt for the residue of 10/ when reduced mod m, and let n be the number of T-'S that are greater than mil. With this notation, n is an even number iff m is like 1. Proof: Numerical checks show that t ...
... Lemma 5: Let m b e a natural number prime to 10. For each integer i from 0 to (m -1) / 2, let us write rt for the residue of 10/ when reduced mod m, and let n be the number of T-'S that are greater than mil. With this notation, n is an even number iff m is like 1. Proof: Numerical checks show that t ...
Clint`s group handout
... We can split the odd primes into two distinct groups: those of the form 4k + 1 (the first few being 5, 13, 17, 29, 37, . . .), and those of the form 4k + 3 (the first few being 3, 7, 11, 19, 23, . . .). Since we know there are infinitely many primes (and only one even prime!), at least one of these ...
... We can split the odd primes into two distinct groups: those of the form 4k + 1 (the first few being 5, 13, 17, 29, 37, . . .), and those of the form 4k + 3 (the first few being 3, 7, 11, 19, 23, . . .). Since we know there are infinitely many primes (and only one even prime!), at least one of these ...
Full text
... We seek conditions under which terms ja and U+joz] are not consecutive in s: spppose ja < w + va < [1 + ja]. ^Equivalently, (j - v)a < u < 1 + \Ja] - va. Such an integer u exists if and only if [(/ - v)aJ < [1 + ija] - vaj or, equivalently, LO* ~ ^)^J < L / a J - L v a X o r Yet again, (((j - v)a )) ...
... We seek conditions under which terms ja and U+joz] are not consecutive in s: spppose ja < w + va < [1 + ja]. ^Equivalently, (j - v)a < u < 1 + \Ja] - va. Such an integer u exists if and only if [(/ - v)aJ < [1 + ija] - vaj or, equivalently, LO* ~ ^)^J < L / a J - L v a X o r Yet again, (((j - v)a )) ...
1-2
... **Every rational number that is not an integer can be written as a fraction. 1. A common fraction is written with a _____________ and ____________. 2. Even decimals can be converted into fractions. 3. _____________ decimals are decimals with a finite number of digits.(They _______.) 4. ___________ o ...
... **Every rational number that is not an integer can be written as a fraction. 1. A common fraction is written with a _____________ and ____________. 2. Even decimals can be converted into fractions. 3. _____________ decimals are decimals with a finite number of digits.(They _______.) 4. ___________ o ...
![arXiv:math/0511682v1 [math.NT] 28 Nov 2005](http://s1.studyres.com/store/data/014696627_1-8def914a5ac3ed74bde3727e1309931c-300x300.png)
arXiv:math/0511682v1 [math.NT] 28 Nov 2005
... this question was first considered by Khintchine in [21] (see also [6, 38, 40] for surveys including a discussion on this subject). A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The first result of this type goes back to the ...
... this question was first considered by Khintchine in [21] (see also [6, 38, 40] for surveys including a discussion on this subject). A preliminary step towards its resolution consists in providing explicit examples of transcendental continued fractions. The first result of this type goes back to the ...
Counting
... 4. (Putnam 2005, A2) Let S = {(a, b)|a = 1, 2, . . . , n, b = 1, 2, 3}. A rook tour of S is a polygonal path made up of line segments connecting points p1 , p2 , . . . , p3n in sequence such that (a) pi ∈ S, (b) pi and pi+1 are unit distance apart for 1 ≤ i < 3n, (c) for each p ∈ S there is unique i ...
... 4. (Putnam 2005, A2) Let S = {(a, b)|a = 1, 2, . . . , n, b = 1, 2, 3}. A rook tour of S is a polygonal path made up of line segments connecting points p1 , p2 , . . . , p3n in sequence such that (a) pi ∈ S, (b) pi and pi+1 are unit distance apart for 1 ≤ i < 3n, (c) for each p ∈ S there is unique i ...
Patterns and Sequences
... • Patterns refer to usual types of procedures or rules that can be followed. • Patterns are useful to predict what came before or what might come after a set a numbers that are arranged in a particular order. • This arrangement of numbers is called a sequence. For example: 3,6,9,12 and 15 are number ...
... • Patterns refer to usual types of procedures or rules that can be followed. • Patterns are useful to predict what came before or what might come after a set a numbers that are arranged in a particular order. • This arrangement of numbers is called a sequence. For example: 3,6,9,12 and 15 are number ...
Full text
... numbers. Finally, in Subsection E, we use the summation formula to determine the so-called internal path length of the trees {TL } , which determination was one of the motivations for studying the profile numbers. Our investigation will then have gone full circle. In what follows, we shall refer oft ...
... numbers. Finally, in Subsection E, we use the summation formula to determine the so-called internal path length of the trees {TL } , which determination was one of the motivations for studying the profile numbers. Our investigation will then have gone full circle. In what follows, we shall refer oft ...