MTH304 - National Open University of Nigeria
... abbreviate the rational number (n,1) by n, there is absolutely no danger of confusion: 2 + 3 = 5 stands for (2,1)+(3,1)=(5,1). The equation 3x = 8 that started this all may then be interpreted as shorthand for the equation (3,1)(u, v)=(8,1), and one easily verifies that x =(u, v)=(8,3) is a solution ...
... abbreviate the rational number (n,1) by n, there is absolutely no danger of confusion: 2 + 3 = 5 stands for (2,1)+(3,1)=(5,1). The equation 3x = 8 that started this all may then be interpreted as shorthand for the equation (3,1)(u, v)=(8,1), and one easily verifies that x =(u, v)=(8,3) is a solution ...
Theorems here
... graph can be plotted quickly. 3. Make a table of values using synthetic division. 4. Find the y-intercept and as many x-intercepts as possible (the latter are roots of the polynomial). In doing this, recall the theorems about roots, including Descartes’ rule of signs. 5. Plot the points and connect ...
... graph can be plotted quickly. 3. Make a table of values using synthetic division. 4. Find the y-intercept and as many x-intercepts as possible (the latter are roots of the polynomial). In doing this, recall the theorems about roots, including Descartes’ rule of signs. 5. Plot the points and connect ...
Fraction IX Least Common Multiple Least Common Denominator
... least common denominator or least common multiple. ...
... least common denominator or least common multiple. ...
Document
... Modeling with Recurrence Relations • Example 5: The Tower of Hanoi A popular puzzle of the late nineteenth century invented by the French mathematician Édouard Lucas, called the Tower of Hanoi, consists of three pegs mounted on a board together with disks of different sizes. Initially there disks a ...
... Modeling with Recurrence Relations • Example 5: The Tower of Hanoi A popular puzzle of the late nineteenth century invented by the French mathematician Édouard Lucas, called the Tower of Hanoi, consists of three pegs mounted on a board together with disks of different sizes. Initially there disks a ...
2017 Discrete Math Pacing Guide
... The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the ...
... The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the ...
Weighted Catalan Numbers and Their Divisibility Properties
... 3). In [Pos00], Postnikov shows that the Morse links are counted by an integer sequence Ln = 1, 10, 325, 22150, . . ., called the weighted Catalan numbers, which generalize the Catalan numbers [OEI01b]. The weighted Catalan numbers are an integer sequence which satisfies a recurrence similar to that ...
... 3). In [Pos00], Postnikov shows that the Morse links are counted by an integer sequence Ln = 1, 10, 325, 22150, . . ., called the weighted Catalan numbers, which generalize the Catalan numbers [OEI01b]. The weighted Catalan numbers are an integer sequence which satisfies a recurrence similar to that ...
Congruent number problems and their variants
... R EMARK 2.1. In contrast, the Diophantine equation X n C Y n Z n D 0 of degree n 3 has no nontrivial solutions in integers .a; b; c/. Here, by a nontrivial solution we mean a triple of integers .a; b; c/ with abc ¤ 0 satisfying the equation. This is the celebrated proof of Fermat’s Last Theorem [W ...
... R EMARK 2.1. In contrast, the Diophantine equation X n C Y n Z n D 0 of degree n 3 has no nontrivial solutions in integers .a; b; c/. Here, by a nontrivial solution we mean a triple of integers .a; b; c/ with abc ¤ 0 satisfying the equation. This is the celebrated proof of Fermat’s Last Theorem [W ...
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.