Maths Shortcuts2
... total ways of getting 729 are, 1*729, 3*243, 9*81, 27*27. So 4 ways Funda is that, all four ways of expressing can be used to findout distinct a,b values, for example take 9*81 now since 9*81 = (a-b)(a+b) by solving the system a-b = 9 and a+b = 81 we can have 45,36 as soln. 7) How many times the dig ...
... total ways of getting 729 are, 1*729, 3*243, 9*81, 27*27. So 4 ways Funda is that, all four ways of expressing can be used to findout distinct a,b values, for example take 9*81 now since 9*81 = (a-b)(a+b) by solving the system a-b = 9 and a+b = 81 we can have 45,36 as soln. 7) How many times the dig ...
Chapter 1: The Foundations: Logic and Proofs Section 1.7
... a) a proof by contraposition. b) a proof by contradiction. Solution: 1. Proof by contraposition: We need to prove that if n is odd, then 3n + 2 is odd. Assume that n is odd. Then we can write n = 2k + 1 for some integer k. Then 3n + 2 = 3(2k + 1) + 2 = 6k + 5 = 2(3k + 2) + 1. – odd number Thus 3n + ...
... a) a proof by contraposition. b) a proof by contradiction. Solution: 1. Proof by contraposition: We need to prove that if n is odd, then 3n + 2 is odd. Assume that n is odd. Then we can write n = 2k + 1 for some integer k. Then 3n + 2 = 3(2k + 1) + 2 = 6k + 5 = 2(3k + 2) + 1. – odd number Thus 3n + ...
CIS 260 Recitations 3 Feb 6 Problem 1 (Complete proof of example
... definition a 2k 1 for some integer k . When we square both sides, we obtain that a 2 (2k 1) 2 4k 2 4k 1 2(2k 2 2k ) 1 2s 1 By definition this means that a 2 is even, which leads to contradiction with the assumption that it is odd. This shows that if a 2 is even then a is also ...
... definition a 2k 1 for some integer k . When we square both sides, we obtain that a 2 (2k 1) 2 4k 2 4k 1 2(2k 2 2k ) 1 2s 1 By definition this means that a 2 is even, which leads to contradiction with the assumption that it is odd. This shows that if a 2 is even then a is also ...
Arithmetic with Decimals
... Order of Operations To avoid having different results for the same problem, mathematicians have agreed on an order of operations when simplifying expressions that contain multiple operations. 1. Perform any operation(s) inside grouping symbols. (Parentheses, brackets above or below a fraction bar) ...
... Order of Operations To avoid having different results for the same problem, mathematicians have agreed on an order of operations when simplifying expressions that contain multiple operations. 1. Perform any operation(s) inside grouping symbols. (Parentheses, brackets above or below a fraction bar) ...
Math 236H Final exam
... b. An element of C4 × C4 × C4 has order 4 if it is not contained in C2 × C2 × C2 . The number of such elements is 43 − 23 = 56. 11. (10 points) Let n be an integer such that 6n − 1 and 6n + 1 are both primes. (Such primes are called twin primes.) Show that every group G of order 36n2 − 1 is cyclic. ...
... b. An element of C4 × C4 × C4 has order 4 if it is not contained in C2 × C2 × C2 . The number of such elements is 43 − 23 = 56. 11. (10 points) Let n be an integer such that 6n − 1 and 6n + 1 are both primes. (Such primes are called twin primes.) Show that every group G of order 36n2 − 1 is cyclic. ...
9 Review
... 13. The height of a soccer ball that is kicked from the ground can be approximated by the function , where y is the height of the soccer ball in feet x seconds after it is kicked. Graph this function. Find the time it takes the soccer ball to reach its maximum height, the soccer ball’s maximum heigh ...
... 13. The height of a soccer ball that is kicked from the ground can be approximated by the function , where y is the height of the soccer ball in feet x seconds after it is kicked. Graph this function. Find the time it takes the soccer ball to reach its maximum height, the soccer ball’s maximum heigh ...
Factorization
In mathematics, factorization (also factorisation in some forms of British English) or factoring is the decomposition of an object (for example, a number, a polynomial, or a matrix) into a product of other objects, or factors, which when multiplied together give the original. For example, the number 15 factors into primes as 3 × 5, and the polynomial x2 − 4 factors as (x − 2)(x + 2). In all cases, a product of simpler objects is obtained.The aim of factoring is usually to reduce something to “basic building blocks”, such as numbers to prime numbers, or polynomials to irreducible polynomials. Factoring integers is covered by the fundamental theorem of arithmetic and factoring polynomials by the fundamental theorem of algebra. Viète's formulas relate the coefficients of a polynomial to its roots.The opposite of polynomial factorization is expansion, the multiplying together of polynomial factors to an “expanded” polynomial, written as just a sum of terms.Integer factorization for large integers appears to be a difficult problem. There is no known method to carry it out quickly. Its complexity is the basis of the assumed security of some public key cryptography algorithms, such as RSA.A matrix can also be factorized into a product of matrices of special types, for an application in which that form is convenient. One major example of this uses an orthogonal or unitary matrix, and a triangular matrix. There are different types: QR decomposition, LQ, QL, RQ, RZ.Another example is the factorization of a function as the composition of other functions having certain properties; for example, every function can be viewed as the composition of a surjective function with an injective function. This situation is generalized by factorization systems.