BENCHMARK REPORT MATHEMATICS GRADE 8
... Use square root symbols to represent solutions to equations of the form x2 = p, where p is a positive rational number Use cube root symbols to represent solutions to equations of the form x3 = p, where p is a positive rational number Evaluate square roots of small perfect squares Evaluate cube roots ...
... Use square root symbols to represent solutions to equations of the form x2 = p, where p is a positive rational number Use cube root symbols to represent solutions to equations of the form x3 = p, where p is a positive rational number Evaluate square roots of small perfect squares Evaluate cube roots ...
today3, 6, 10 marks
... What are the d.c.s of the vector equally inclined to the axes? Find direction cosines of the line joining (2, 3, 1) and (3, 1, 2). Find the angle between the following lines. ...
... What are the d.c.s of the vector equally inclined to the axes? Find direction cosines of the line joining (2, 3, 1) and (3, 1, 2). Find the angle between the following lines. ...
Complex 2.3
... Complex Concepts-Making “i” contact Products, Quotients, DeMoivre’s Theorem What happens when we multiply two complex numbers? We have observed a relationship between the radii and angles of the two complex numbers and the resulting product. Let’s prove this relationship in the general case: (r1 cis ...
... Complex Concepts-Making “i” contact Products, Quotients, DeMoivre’s Theorem What happens when we multiply two complex numbers? We have observed a relationship between the radii and angles of the two complex numbers and the resulting product. Let’s prove this relationship in the general case: (r1 cis ...
Linear Equations and Inequalities
... The answer to the previous example is 114, which can be written as a mixed number 2 34. In algebra, improper fractions are generally preferred. Unless the original problem has mixed numbers in it, or it is an answer to a real-world application, solutions will be expressed as reduced improper fractio ...
... The answer to the previous example is 114, which can be written as a mixed number 2 34. In algebra, improper fractions are generally preferred. Unless the original problem has mixed numbers in it, or it is an answer to a real-world application, solutions will be expressed as reduced improper fractio ...
Summary of lectures.
... Definition. A reduced residue system mod n is a set of φ(n) numbers r1 , r2 , . . . , rφ(n , all relatively prime to n such that no two are congruent mod n. That is: If ri ≡ rj mod n then i = j. The standard reduce residue system mod n are the remainders mod n which are relatively prime to n. For ex ...
... Definition. A reduced residue system mod n is a set of φ(n) numbers r1 , r2 , . . . , rφ(n , all relatively prime to n such that no two are congruent mod n. That is: If ri ≡ rj mod n then i = j. The standard reduce residue system mod n are the remainders mod n which are relatively prime to n. For ex ...
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.