Methods of Proof
... Existence: that such a value does indeed exist Either via a constructive or non-constructive existence ...
... Existence: that such a value does indeed exist Either via a constructive or non-constructive existence ...
Full text
... This description gives a very fast method for computing the coefficients a(m) recursively. Once we have computed them for 0 ≤ m < Fn we can immediately compute them for Fn ≤ m < Fn+1 using Proposition 1. Also, since the coefficient of xm in A(x) is equal to −1, 0 or 1 for all non-negative integers m ...
... This description gives a very fast method for computing the coefficients a(m) recursively. Once we have computed them for 0 ≤ m < Fn we can immediately compute them for Fn ≤ m < Fn+1 using Proposition 1. Also, since the coefficient of xm in A(x) is equal to −1, 0 or 1 for all non-negative integers m ...
Full text
... Each row of Pascal's triangle is formed by addition of adjacent elements of the preceding row, producing binomial coefficients and counting combinations. Each row of Stern's diatomic array is formed by addition of adjacent elements of the preceding row, but interspersed with elements of the precedin ...
... Each row of Pascal's triangle is formed by addition of adjacent elements of the preceding row, producing binomial coefficients and counting combinations. Each row of Stern's diatomic array is formed by addition of adjacent elements of the preceding row, but interspersed with elements of the precedin ...
CS1101: Programming Methodology
... language named after him (though in no real sense related to him). More of him in this website (for those who are interested): http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html ...
... language named after him (though in no real sense related to him). More of him in this website (for those who are interested): http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Pascal.html ...
Complex Eigenvalues
... numbers which are complex conjugates of each other have the same absolute value. The result is that the versions of power iteration that we have discussed so far won't separate them if one only uses real shifts. The key to approximating the complex eigenvalues is to allow shifts that are complex. We ...
... numbers which are complex conjugates of each other have the same absolute value. The result is that the versions of power iteration that we have discussed so far won't separate them if one only uses real shifts. The key to approximating the complex eigenvalues is to allow shifts that are complex. We ...
Ready Set Math!
... A.1 Counting review - 0 to 10 A.2 Counting review - up to 20 A.3 Counting tens and ones - up to 20 A.4 Counting - up to 30 A.5 Counting - up to 100 A.6 Counting by tens - up to 100 A.7 Counting tens and ones - up to 99 A.8 Skip-counting - with pictures A.9 Counting by twos, fives, ...
... A.1 Counting review - 0 to 10 A.2 Counting review - up to 20 A.3 Counting tens and ones - up to 20 A.4 Counting - up to 30 A.5 Counting - up to 100 A.6 Counting by tens - up to 100 A.7 Counting tens and ones - up to 99 A.8 Skip-counting - with pictures A.9 Counting by twos, fives, ...
Addition
Addition (often signified by the plus symbol ""+"") is one of the four elementary, mathematical operations of arithmetic, with the others being subtraction, multiplication and division.The addition of two whole numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two apples together; making a total of 5 apples. This observation is equivalent to the mathematical expression ""3 + 2 = 5"" i.e., ""3 add 2 is equal to 5"".Besides counting fruits, addition can also represent combining other physical objects. Using systematic generalizations, addition can also be defined on more abstract quantities, such as integers, rational numbers, real numbers and complex numbers and other abstract objects such as vectors and matrices.In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others. In algebra, addition is studied more abstractly.Addition has several important properties. It is commutative, meaning that order does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition is performed does not matter (see Summation). Repeated addition of 1 is the same as counting; addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication.Performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months and even some non-human animals. In primary education, students are taught to add numbers in the decimal system, starting with single digits and progressively tackling more difficult problems. Mechanical aids range from the ancient abacus to the modern computer, where research on the most efficient implementations of addition continues to this day.