number theory and methods of proof
... i.e. if p q and q p then p q and we say that p is true if and only if q is true. If p q then we say that p is a sufficient condition for q. If q p then we say that p is a necessary condition for q. If p q and q p then we say that p is a necessary and sufficient condition for q. ...
... i.e. if p q and q p then p q and we say that p is true if and only if q is true. If p q then we say that p is a sufficient condition for q. If q p then we say that p is a necessary condition for q. If p q and q p then we say that p is a necessary and sufficient condition for q. ...
Least Common Multiple
... Least Common Multiple Find the LCM of 180 and 378. When the numbers are too large to list the multiples of the two numbers, find the prime factorization of each number. The LCM is the product of the common prime factors and the factors that are not common. ...
... Least Common Multiple Find the LCM of 180 and 378. When the numbers are too large to list the multiples of the two numbers, find the prime factorization of each number. The LCM is the product of the common prime factors and the factors that are not common. ...
Sequences and Series
... It is easy to determine the sum of the following arithmetic sequence: 3 + 5 + 7 = 15 This is an arithmetic series with common difference 2 Now let’s use the formula ...
... It is easy to determine the sum of the following arithmetic sequence: 3 + 5 + 7 = 15 This is an arithmetic series with common difference 2 Now let’s use the formula ...
Day5-Arrays
... (will have name, address, phone, …) * MovieClip Record (will have length, width, rotation, alpha, …) ...
... (will have name, address, phone, …) * MovieClip Record (will have length, width, rotation, alpha, …) ...
![[2014 question paper]](http://s1.studyres.com/store/data/008844916_1-6bcf832b30821138ae4b8d8f7d9aa434-300x300.png)
[2014 question paper]
... (a) If the equation (1) admits a solution for all b ∈ Rm then n must be greater than or equal to m. (b) If the equation (1) admits a unique solution for some b ∈ Rm , then n must be greater than or equal to m. (c) If the equation (1) admits two distinct solutions for some b ∈ Rm , then m must be gre ...
... (a) If the equation (1) admits a solution for all b ∈ Rm then n must be greater than or equal to m. (b) If the equation (1) admits a unique solution for some b ∈ Rm , then n must be greater than or equal to m. (c) If the equation (1) admits two distinct solutions for some b ∈ Rm , then m must be gre ...
From Rainbow to the Lonely Runner
... Suppose k runners running on a circular field of circumference r. Suppose each runner keeps a constant speed and all runners have different speeds. A runner is called “lonely” at some moment if he or she has (circular) distance at least r/k apart from all other runners. Conjecture: For each runner, ...
... Suppose k runners running on a circular field of circumference r. Suppose each runner keeps a constant speed and all runners have different speeds. A runner is called “lonely” at some moment if he or she has (circular) distance at least r/k apart from all other runners. Conjecture: For each runner, ...
Exam Review 08
... For i = 0 To 7 'collect data from input box into sample sample(i) = Val(InputBox("Enter Number")) Next i 'counted loop from 0 to 6 (ubound-1) Why 6??? For Outter = 0 To 6 ' counted loop from 1 past Outter to 7 (ubound) For Inner = (Outter + 1) To 7 'compare which number is greater to put highest on ...
... For i = 0 To 7 'collect data from input box into sample sample(i) = Val(InputBox("Enter Number")) Next i 'counted loop from 0 to 6 (ubound-1) Why 6??? For Outter = 0 To 6 ' counted loop from 1 past Outter to 7 (ubound) For Inner = (Outter + 1) To 7 'compare which number is greater to put highest on ...
Revised Version 080113
... In the array below, the first row pairs 1 red dot with 9 black dots, the second row pairs 2 red dots with 8 black dots, the third row pairs 3 red dots with 7 black dots, and the fourth row pairs 4 red dots with 6 black dots. Because 5 is not paired with another number, there are only 5 red dots in t ...
... In the array below, the first row pairs 1 red dot with 9 black dots, the second row pairs 2 red dots with 8 black dots, the third row pairs 3 red dots with 7 black dots, and the fourth row pairs 4 red dots with 6 black dots. Because 5 is not paired with another number, there are only 5 red dots in t ...
Elementary mathematics
Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. The most basic topics in elementary mathematics are arithmetic and geometry. Beginning in the last decades of the 20th century, there has been an increased emphasis on problem solving. Elementary mathematics is used in everyday life in such activities as making change, cooking, buying and selling stock, and gambling. It is also an essential first step on the path to understanding science.In secondary school, the main topics in elementary mathematics are algebra and trigonometry. Calculus, even though it is often taught to advanced secondary school students, is usually considered college level mathematics.