The document defines linear equations in two variables as equations that can be written in the form ax + by + c = 0, where a, b, and c are real numbers and a and b are not equal to 0. It provides an example linear equation as 2x + 3y = 18 and explains how to determine if a given ordered pair (3, 4) is a solution by substituting the values into the equation.
The document announces a mathematics project competition open to students in forms 3 and 4 at Maria Regina College Boys' Junior Lyceum. Teams of two students can participate by creating one of the following: a statistics project, charts, or a PowerPoint presentation on a given theme related to mathematics history or concepts. The top five entries will represent the school in the national competition and prizes will be awarded to the top teams nationally. Proposals are due by November 30th and completed projects by January 18th.
This document provides a high-level overview of the history of mathematics, levels of mathematics taught at different grades, famous mathematicians, unsolved math problems, and resources for math games and jokes. It discusses important developments like the first evidence of counting 50,000 BC, the definition of the 360 degree circle in 180 BC, the first trigonometry in 140 BC, and the proofs of Fermat's Last Theorem in 1994 and the Four Color Theorem in the 1970s using computers.
Rene Descartes lay awake one night and noticed a fly on his ceiling. He wanted to describe the fly's exact position but could not use vague terms like "to the left". He decided to draw perpendicular horizontal and vertical lines across the ceiling, assigning numbers to each. This allowed him to precisely say the fly's position as a pair of numbers, like (4,5), representing the distance across and up. This Cartesian plane, mapping coordinates to locations, became an important mathematical concept.
Cartesian Coordinate Plane - Mathematics 8Carlo Luna
This document explains the Cartesian coordinate plane. It describes how the plane is divided into four quadrants by the x and y axes which intersect at the origin. It provides examples of plotting points using ordered pairs with coordinates (x,y). The document also notes that Rene Descartes developed this system by combining algebra and geometry. It includes an activity for students to physically position themselves on the x and y axes to learn the coordinate system.
This document provides information about various landmarks and their locations: Rizal Park in Manila, New York City in the USA, the Eiffel Tower in Paris, the Merlion in Singapore, and the Basilica of Saint Peter in Rome. It then instructs the reader to locate each of these places on a map according to their latitude and longitude. The document explains how Cartesian coordinates use a grid system with x and y axes to precisely locate points on a plane or map. It provides examples of plotting points in the Cartesian plane's four quadrants and identifying points' coordinates. The document distinguishes between points that lie within the quadrants versus along the axes. It concludes with an activity asking the reader to identify quadrant locations for
René Descartes is credited with developing the Cartesian plane by joining algebra and geometry. The Cartesian plane is formed by intersecting two perpendicular number lines, called the x-axis and y-axis, dividing the plane into four quadrants. Each point on the Cartesian plane is associated with an ordered pair of coordinates (x,y) representing its distance from the origin, where the axes intersect.
The document announces a mathematics project competition open to students in forms 3 and 4 at Maria Regina College Boys' Junior Lyceum. Teams of two students can participate by creating one of the following: a statistics project, charts, or a PowerPoint presentation on a given theme related to mathematics history or concepts. The top five entries will represent the school in the national competition and prizes will be awarded to the top teams nationally. Proposals are due by November 30th and completed projects by January 18th.
This document provides a high-level overview of the history of mathematics, levels of mathematics taught at different grades, famous mathematicians, unsolved math problems, and resources for math games and jokes. It discusses important developments like the first evidence of counting 50,000 BC, the definition of the 360 degree circle in 180 BC, the first trigonometry in 140 BC, and the proofs of Fermat's Last Theorem in 1994 and the Four Color Theorem in the 1970s using computers.
Rene Descartes lay awake one night and noticed a fly on his ceiling. He wanted to describe the fly's exact position but could not use vague terms like "to the left". He decided to draw perpendicular horizontal and vertical lines across the ceiling, assigning numbers to each. This allowed him to precisely say the fly's position as a pair of numbers, like (4,5), representing the distance across and up. This Cartesian plane, mapping coordinates to locations, became an important mathematical concept.
Cartesian Coordinate Plane - Mathematics 8Carlo Luna
This document explains the Cartesian coordinate plane. It describes how the plane is divided into four quadrants by the x and y axes which intersect at the origin. It provides examples of plotting points using ordered pairs with coordinates (x,y). The document also notes that Rene Descartes developed this system by combining algebra and geometry. It includes an activity for students to physically position themselves on the x and y axes to learn the coordinate system.
This document provides information about various landmarks and their locations: Rizal Park in Manila, New York City in the USA, the Eiffel Tower in Paris, the Merlion in Singapore, and the Basilica of Saint Peter in Rome. It then instructs the reader to locate each of these places on a map according to their latitude and longitude. The document explains how Cartesian coordinates use a grid system with x and y axes to precisely locate points on a plane or map. It provides examples of plotting points in the Cartesian plane's four quadrants and identifying points' coordinates. The document distinguishes between points that lie within the quadrants versus along the axes. It concludes with an activity asking the reader to identify quadrant locations for
René Descartes is credited with developing the Cartesian plane by joining algebra and geometry. The Cartesian plane is formed by intersecting two perpendicular number lines, called the x-axis and y-axis, dividing the plane into four quadrants. Each point on the Cartesian plane is associated with an ordered pair of coordinates (x,y) representing its distance from the origin, where the axes intersect.
Cartesian coordinates use a grid system to precisely locate points in space. A point is identified by its x and y coordinates, which indicate the distance from the origin point along the x-axis and y-axis. For example, the point (3,2) is located 3 units to the right of the origin along the x-axis and 2 units above the origin along the y-axis. The axes divide the plane into four quadrants, with points falling into different quadrants based on whether their x and y values are positive or negative. Cartesian coordinates provide a way to pinpoint locations using simple numbers.
This document provides information about the Cartesian plane (or Cartesian coordinate system) including:
- It specifies each point uniquely using a pair of numerical coordinates that represent the distance from the point to two fixed perpendicular axes.
- Rene Descartes invented the Cartesian coordinate system to plot ordered pairs (x,y) on a plane with perpendicular x and y axes intersecting at the origin (0,0).
- The x-coordinate represents the horizontal axis and the y-coordinate represents the vertical axis. Ordered pairs written as (x, y) locate a point by moving left/right along the x-axis and up/down along the y-axis from the origin.
This document discusses Pascal's triangle and its relationship to the Fibonacci sequence. It explains that Pascal's triangle is an ancient mathematical pattern where each number is the sum of the two numbers above it. The diagonals of the triangle relate to important numerical sequences like the counting numbers, triangular numbers, and the Fibonacci sequence. The document also outlines various properties of Pascal's triangle, such as the horizontal rows summing to powers of two and representing powers of eleven. It provides examples of how Pascal's triangle can be used to solve probability and binomial expansion problems.
This document provides an overview of linear equations in two variables and the rectangular coordinate system. It defines key terms like the x-axis, y-axis, and quadrants. It describes how René Descartes developed the coordinate plane and revolutionized mathematics. The document explains how to locate and plot points on the plane using x and y coordinates. It includes examples of plotting points and determining the coordinates of points. Students are assigned questions about plotting points in different quadrants and how the signs of coordinates affect a point's location.
The rectangular coordinate system, also known as the Cartesian coordinate system, was developed by the French mathematician René Descartes. It uses two perpendicular number lines, the x-axis and y-axis, that intersect at the origin (0,0) to locate points in a plane. Each point is identified with an ordered pair of numbers known as Cartesian coordinates that represent the distance from the origin on the x-axis and y-axis. The system divides the plane into four quadrants and allows points to be easily plotted and located.
The document discusses Pascal's Triangle, including its history, patterns, and applications. The triangle was used in the 11th century by Chinese and Persian mathematicians, though the French mathematician Blaise Pascal studied its properties more extensively in the 1600s. The triangle exhibits several patterns, such as the Fibonacci sequence appearing in its diagonals and horizontal lines doubling in sum. It is used in algebra, probability, and combinatorics.
History,applications,algebra and mathematical form of plane in mathematics (p...guesta62dea
The document provides information about planes and equations of planes. It defines a plane as a flat surface that extends indefinitely in width and height but has no thickness. Various plane shapes and their area formulas are described. Different forms of equations for a straight line including slope-intercept, point-slope, two-point, and standard forms are derived from the general linear equation. Two and three-dimensional Cartesian coordinate systems are also explained.
The document discusses Pascal's triangle, a mathematical pattern named after French mathematician Blaise Pascal. It contains properties like the Fibonacci sequence and uses like binomial expansion. Pascal was not the first to document the triangle - others like the Persians and Chinese used it earlier. The document also explains how Pascal's triangle can be used to calculate combinations and probabilities in events like coin tosses.
The various visual and numeric patterns, seen in the Pascal's Triangle. Includes a brief introduction and help on constructing the Pascal's Triangle. Binomial Theorem is not discussed. Though, the n C r formula has been described. Hope you enjoy it !
This document defines and explains the rectangular coordinate system. It begins by introducing Rene Descartes as the "Father of Modern Mathematics" who developed the Cartesian plane. The Cartesian plane is composed of two perpendicular number lines that intersect at the origin (0,0) and divide the plane into four quadrants. Each point on the plane is identified by an ordered pair (x,y) denoting its distance from the x-axis and y-axis. The sign of the x and y values determines which quadrant a point falls into. Examples are provided to identify points and their corresponding quadrants.
The document explores Pascal's triangle, named after French mathematician Blaise Pascal. While Pascal did not discover it, he popularized it in Europe. The triangle arranges the binomial coefficients and was known in other ancient cultures as well. It exhibits several mathematical properties like generating triangular and tetrahedral numbers.
The document discusses the Cartesian coordinate plane and functions. It defines the Cartesian plane as being formed by two perpendicular number lines called the x-axis and y-axis that intersect at the origin (0,0). It describes how each point on the plane is associated with an ordered pair (x,y) denoting its coordinates and how the plane is divided into four quadrants. It then demonstrates how to plot various points on the plane by starting at the origin and moving right or left along the x-axis and up or down along the y-axis. Finally, it discusses relations and functions, defining a function as a relation where each x-value is mapped to only one y-value.
Pascal's triangle is a triangular array of the binomial coefficients where each number is the sum of the two numbers directly above it. It was studied by mathematicians as early as ancient Greece, India, and China, but Pascal organized the information and popularized its uses in probability. The triangle demonstrates many mathematical properties and patterns such as the Fibonacci sequence, triangular numbers, binomial coefficients, and it can be used to calculate combinations and probabilities of outcomes.
Pascal's triangle is a mathematical concept originally developed by the Chinese that was later popularized by French mathematician Blaise Pascal. It arranges numbers in triangular form according to binomial coefficients and can be used for applications in algebra, probability, and counting patterns like Catalan numbers. Pascal discovered the importance of the triangle in 1653 and it is constructed by adding numbers above following the rule of their sum being the numbers directly next to them.
Pascal's Triangle is a triangular array of the binomial coefficients formed by taking powers of 11. It was named after French mathematician Blaise Pascal but had been discovered earlier by Chinese and Indian mathematicians. The triangle exhibits several interesting numerical and geometric properties including symmetry, doubling of row sums, generation of the Fibonacci sequence and triangular, tetrahedral and counting numbers along diagonals. It can also be used to calculate probabilities of outcomes from events like coin tosses.
Pascal’s triangle and its applications and propertiesJordan Leong
Pascal's Triangle is named after Blaise Pascal, though it was discovered centuries earlier in China and India. It is a triangular array where each number is the sum of the two numbers directly above it. It has many interesting properties, including relationships to binomial expansions, probabilities, combinations, and sequences like Fibonacci numbers. Some of its applications include representing probabilities of coin toss outcomes and finding combinations of objects selected from a group.
The document discusses magic squares, which are grids of numbers arranged so that the sums of each row, column and diagonal are equal. It provides examples of 3x3 and 4x4 magic squares and notes that magic squares were historically used for protection and as amulets. It also highlights some unique magic squares, including one created by Benjamin Franklin that contains interesting additional features beyond just the row/column/diagonal sums.
The document discusses graphing ordered pairs on a coordinate plane. It defines key terms like coordinate, plane, axes, quadrants, and ordered pairs. It explains that the x-axis is horizontal, the y-axis is vertical, and they intersect at the origin (0,0). Ordered pairs use the format (x,y) to locate points by giving the x-coordinate first and y-coordinate second. Several examples are given to demonstrate graphing points in the quadrants of the coordinate plane.
1. The document defines triangles and their properties including three sides, three angles, and three vertices.
2. It explains five criteria for determining if two triangles are congruent: side-angle-side (SAS), angle-side-angle (ASA), angle-angle-side (AAS), side-side-side (SSS), and right-angle-hypotenuse-side (RHS).
3. Some properties of triangles discussed are: angles opposite equal sides are equal, sides opposite equal angles are equal, and the sum of any two sides is greater than the third side.
Triangles are three-sided polygons that have three angles and three sides. There are three main types of triangles based on side lengths: equilateral (all sides equal), isosceles (two sides equal), and scalene (no sides equal). The interior angles of any triangle always sum to 180 degrees. Important triangle properties include the exterior angle theorem, Pythagorean theorem, and congruency criteria like SSS, SAS, ASA. Common secondary parts are the median, altitude, angle bisector, and perpendicular bisector. The area of triangles can be found using Heron's formula or other formulas based on side lengths and types of triangles.
Cartesian coordinates use a grid system to precisely locate points in space. A point is identified by its x and y coordinates, which indicate the distance from the origin point along the x-axis and y-axis. For example, the point (3,2) is located 3 units to the right of the origin along the x-axis and 2 units above the origin along the y-axis. The axes divide the plane into four quadrants, with points falling into different quadrants based on whether their x and y values are positive or negative. Cartesian coordinates provide a way to pinpoint locations using simple numbers.
This document provides information about the Cartesian plane (or Cartesian coordinate system) including:
- It specifies each point uniquely using a pair of numerical coordinates that represent the distance from the point to two fixed perpendicular axes.
- Rene Descartes invented the Cartesian coordinate system to plot ordered pairs (x,y) on a plane with perpendicular x and y axes intersecting at the origin (0,0).
- The x-coordinate represents the horizontal axis and the y-coordinate represents the vertical axis. Ordered pairs written as (x, y) locate a point by moving left/right along the x-axis and up/down along the y-axis from the origin.
This document discusses Pascal's triangle and its relationship to the Fibonacci sequence. It explains that Pascal's triangle is an ancient mathematical pattern where each number is the sum of the two numbers above it. The diagonals of the triangle relate to important numerical sequences like the counting numbers, triangular numbers, and the Fibonacci sequence. The document also outlines various properties of Pascal's triangle, such as the horizontal rows summing to powers of two and representing powers of eleven. It provides examples of how Pascal's triangle can be used to solve probability and binomial expansion problems.
This document provides an overview of linear equations in two variables and the rectangular coordinate system. It defines key terms like the x-axis, y-axis, and quadrants. It describes how René Descartes developed the coordinate plane and revolutionized mathematics. The document explains how to locate and plot points on the plane using x and y coordinates. It includes examples of plotting points and determining the coordinates of points. Students are assigned questions about plotting points in different quadrants and how the signs of coordinates affect a point's location.
The rectangular coordinate system, also known as the Cartesian coordinate system, was developed by the French mathematician René Descartes. It uses two perpendicular number lines, the x-axis and y-axis, that intersect at the origin (0,0) to locate points in a plane. Each point is identified with an ordered pair of numbers known as Cartesian coordinates that represent the distance from the origin on the x-axis and y-axis. The system divides the plane into four quadrants and allows points to be easily plotted and located.
The document discusses Pascal's Triangle, including its history, patterns, and applications. The triangle was used in the 11th century by Chinese and Persian mathematicians, though the French mathematician Blaise Pascal studied its properties more extensively in the 1600s. The triangle exhibits several patterns, such as the Fibonacci sequence appearing in its diagonals and horizontal lines doubling in sum. It is used in algebra, probability, and combinatorics.
History,applications,algebra and mathematical form of plane in mathematics (p...guesta62dea
The document provides information about planes and equations of planes. It defines a plane as a flat surface that extends indefinitely in width and height but has no thickness. Various plane shapes and their area formulas are described. Different forms of equations for a straight line including slope-intercept, point-slope, two-point, and standard forms are derived from the general linear equation. Two and three-dimensional Cartesian coordinate systems are also explained.
The document discusses Pascal's triangle, a mathematical pattern named after French mathematician Blaise Pascal. It contains properties like the Fibonacci sequence and uses like binomial expansion. Pascal was not the first to document the triangle - others like the Persians and Chinese used it earlier. The document also explains how Pascal's triangle can be used to calculate combinations and probabilities in events like coin tosses.
The various visual and numeric patterns, seen in the Pascal's Triangle. Includes a brief introduction and help on constructing the Pascal's Triangle. Binomial Theorem is not discussed. Though, the n C r formula has been described. Hope you enjoy it !
This document defines and explains the rectangular coordinate system. It begins by introducing Rene Descartes as the "Father of Modern Mathematics" who developed the Cartesian plane. The Cartesian plane is composed of two perpendicular number lines that intersect at the origin (0,0) and divide the plane into four quadrants. Each point on the plane is identified by an ordered pair (x,y) denoting its distance from the x-axis and y-axis. The sign of the x and y values determines which quadrant a point falls into. Examples are provided to identify points and their corresponding quadrants.
The document explores Pascal's triangle, named after French mathematician Blaise Pascal. While Pascal did not discover it, he popularized it in Europe. The triangle arranges the binomial coefficients and was known in other ancient cultures as well. It exhibits several mathematical properties like generating triangular and tetrahedral numbers.
The document discusses the Cartesian coordinate plane and functions. It defines the Cartesian plane as being formed by two perpendicular number lines called the x-axis and y-axis that intersect at the origin (0,0). It describes how each point on the plane is associated with an ordered pair (x,y) denoting its coordinates and how the plane is divided into four quadrants. It then demonstrates how to plot various points on the plane by starting at the origin and moving right or left along the x-axis and up or down along the y-axis. Finally, it discusses relations and functions, defining a function as a relation where each x-value is mapped to only one y-value.
Pascal's triangle is a triangular array of the binomial coefficients where each number is the sum of the two numbers directly above it. It was studied by mathematicians as early as ancient Greece, India, and China, but Pascal organized the information and popularized its uses in probability. The triangle demonstrates many mathematical properties and patterns such as the Fibonacci sequence, triangular numbers, binomial coefficients, and it can be used to calculate combinations and probabilities of outcomes.
Pascal's triangle is a mathematical concept originally developed by the Chinese that was later popularized by French mathematician Blaise Pascal. It arranges numbers in triangular form according to binomial coefficients and can be used for applications in algebra, probability, and counting patterns like Catalan numbers. Pascal discovered the importance of the triangle in 1653 and it is constructed by adding numbers above following the rule of their sum being the numbers directly next to them.
Pascal's Triangle is a triangular array of the binomial coefficients formed by taking powers of 11. It was named after French mathematician Blaise Pascal but had been discovered earlier by Chinese and Indian mathematicians. The triangle exhibits several interesting numerical and geometric properties including symmetry, doubling of row sums, generation of the Fibonacci sequence and triangular, tetrahedral and counting numbers along diagonals. It can also be used to calculate probabilities of outcomes from events like coin tosses.
Pascal’s triangle and its applications and propertiesJordan Leong
Pascal's Triangle is named after Blaise Pascal, though it was discovered centuries earlier in China and India. It is a triangular array where each number is the sum of the two numbers directly above it. It has many interesting properties, including relationships to binomial expansions, probabilities, combinations, and sequences like Fibonacci numbers. Some of its applications include representing probabilities of coin toss outcomes and finding combinations of objects selected from a group.
The document discusses magic squares, which are grids of numbers arranged so that the sums of each row, column and diagonal are equal. It provides examples of 3x3 and 4x4 magic squares and notes that magic squares were historically used for protection and as amulets. It also highlights some unique magic squares, including one created by Benjamin Franklin that contains interesting additional features beyond just the row/column/diagonal sums.
The document discusses graphing ordered pairs on a coordinate plane. It defines key terms like coordinate, plane, axes, quadrants, and ordered pairs. It explains that the x-axis is horizontal, the y-axis is vertical, and they intersect at the origin (0,0). Ordered pairs use the format (x,y) to locate points by giving the x-coordinate first and y-coordinate second. Several examples are given to demonstrate graphing points in the quadrants of the coordinate plane.
1. The document defines triangles and their properties including three sides, three angles, and three vertices.
2. It explains five criteria for determining if two triangles are congruent: side-angle-side (SAS), angle-side-angle (ASA), angle-angle-side (AAS), side-side-side (SSS), and right-angle-hypotenuse-side (RHS).
3. Some properties of triangles discussed are: angles opposite equal sides are equal, sides opposite equal angles are equal, and the sum of any two sides is greater than the third side.
Triangles are three-sided polygons that have three angles and three sides. There are three main types of triangles based on side lengths: equilateral (all sides equal), isosceles (two sides equal), and scalene (no sides equal). The interior angles of any triangle always sum to 180 degrees. Important triangle properties include the exterior angle theorem, Pythagorean theorem, and congruency criteria like SSS, SAS, ASA. Common secondary parts are the median, altitude, angle bisector, and perpendicular bisector. The area of triangles can be found using Heron's formula or other formulas based on side lengths and types of triangles.
This document provides information about mensuration and geometry topics such as trapezoids, rhombuses, cubes, cuboids, cylinders, and their formulas for area, surface area, and volume. It includes definitions and examples of each shape. There are also example problems, tables summarizing the formulas, and a multiple choice and short answer question bank related to mensuration. The document was created by Arnav Gosain of VIII-C at Tagore International School for the purpose of learning about geometry topics involving area, surface area, and volume calculations.
A circle is defined as all points in a plane that are equidistant from a fixed center point. The center point is called the center of the circle, and the fixed distance from the center is called the radius. The longest chord that can be drawn through the center is the diameter. If two chords of a circle are equal in length, then their distances from the center are also equal, as proven using the Side-Side-Side congruence rule for triangles.
The document provides a lesson plan for teaching algebraic expressions and identities to 8th grade students. It outlines objectives to help students understand identities in algebraic expressions, the relationship between algebra, geometry and arithmetic, and how to apply identities to solve problems. Example activities are presented to show representing algebraic expressions geometrically and applying identities to evaluate expressions and arithmetic problems. Key identities introduced are (a + b)2, (a - b)2, and (a + b)(a - b). Students are given practice problems to solve using the identities.
This document discusses the prevalence and importance of mathematics in everyday life. It provides examples of how mathematics is used in areas like health, weather, transportation, society, and more. While some applications are directly observable, others involve more complex systems that are still being understood mathematically, like DNA. The document also discusses the historical foundations of mathematics over centuries, with concepts building upon each other like a pyramid, and provides a brief biography of the mathematician Aryabhata, who made important contributions in astronomy and mathematics.
This document provides an overview of triangles, including definitions, types, properties, secondary parts, congruency, and area calculations. It defines a triangle as a 3-sided polygon with three angles and vertices. Triangles are classified by side lengths as equilateral, isosceles, or scalene, and by angle measures as acute, obtuse, or right. Key properties discussed include the angle sum theorem, exterior angle theorem, and Pythagorean theorem. Secondary parts like medians, altitudes, perpendicular bisectors, and angle bisectors are also defined. Tests for triangle congruency such as SSS, SAS, ASA, and RHS are outlined. Formulas are provided for calculating the areas of
The document defines quadrilaterals as polygons with 4 sides and 4 angles. It states that the sum of the interior angles of any quadrilateral is always 360 degrees. It provides an example of using this fact to find the value of an unknown angle. Finally, it lists and defines the different types of quadrilaterals, including general quadrilaterals, trapezoids, parallelograms, rectangles, squares, rhombuses.
1) A quadrilateral is a polygon with four sides and four vertices. There are over 9 million types of quadrilaterals that can be classified as simple or complex.
2) Quadrilaterals include parallelograms, trapezoids, kites, and more. A parallelogram has two sets of parallel sides and opposite/adjacent angles are equal. A square is a special type of rectangle and parallelogram.
3) The interior angles of any quadrilateral sum to 360 degrees. The line segment between the midpoints of two sides of a triangle is parallel to the third side and half its length.
This document provides guidance on developing effective lesson plans. It discusses key components to consider, including knowing your students, the content, and available materials and equipment. Lesson plans should have clear objectives, outline the procedure and activities, and include assessments tied to the objectives. The document also presents several common lesson plan models, such as Gagne's nine events of instruction and the 5E model. Readers are encouraged to design lesson plans that incorporate useful instructional strategies and techniques.
This document discusses various theorems and properties related to triangles. It explains the Basic Proportionality Theorem, also known as Thales' Theorem, which states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally. It also covers similarity criteria like AAA, SSA, and SSS. The Area Theorem demonstrates that the ratio of areas of similar triangles equals the square of the ratio of corresponding sides. Additionally, it proves Pythagoras' Theorem, which relates the sides of a right triangle, and its converse. In summary, the document outlines key triangle theorems regarding proportional division, similarity, areas, and the Pythagorean relationship between sides.
This document provides guidance on writing effective questionnaires. It recommends that questionnaires include a title, introduction explaining the purpose and how responses will help, instructions in Japanese, questions divided into topics with headings, and a thank you message. Sample language is given for introductions, explanations, instructions, and thank you messages to help writers construct questionnaires that clearly convey the purpose and how to respond.
A PowerPoint presentation on circles defines key terms like diameter, radius, circumference, chord, tangent, and sectors. It presents a theorem stating that for any external point, the lengths of the two tangents drawn to a circle are equal, and the angles between each tangent and the line segment joining the point to the circle's center are also equal. A proof of the theorem is provided using properties of congruent triangles.
The sand dunes show different stages of development as you move inland from the shoreline. Embryo dunes near the beach are less than 1 meter tall and only stabilized by debris. Foredunes a little further inland reach up to 5 meters tall with grasses like marram grass stabilizing the sand. Yellow dunes have greater plant diversity as conditions improve, with dunes 5-10 meters tall and 80% plant coverage.
1. The document provides an overview of common pediatric surgical conditions, including congenital diaphragmatic hernia, esophageal atresia with tracheoesophageal fistula, congenital hypertrophic pyloric stenosis, intussusception, Meckel's diverticulum, Hirschsprung disease, and anorectal malformations.
2. It describes the definition, etiology, pathophysiology, clinical presentation, diagnostic workup, and management of each condition. Diagrams and images are provided to illustrate key aspects.
3. The review is intended for 6th year medical students to familiarize them with important pediatric surgical topics.
This document discusses ectopic pregnancy, which occurs when a fertilized egg implants outside the uterus, usually in a fallopian tube. It presents risks to a woman's health. Ectopic pregnancies are often caused by factors that delay the transit of the fertilized egg through the fallopian tubes. Diagnosis can be made through a combination of clinical history, examination, and investigations like ultrasound and blood tests. Early diagnosis through increased awareness and diagnostic tests has lowered the risks associated with ectopic pregnancies.
The document provides information about the cerebellum including its anatomical subdivisions, development, functional organization, and connections. It discusses the phylogenetic organization of the spinocerebellum, pontocerebellum, and vestibulocerebellum. It also summarizes the functions of the archicerebellum, paleocerebellum, and neocerebellum as well as cerebellar abnormalities caused by lesions in different areas.
The document outlines the syllabus for Class IX mathematics for the academic session 2011-2012. It is divided into two semesters. The first semester covers chapters on number systems, polynomials, coordinate geometry, introduction to Euclid's geometry, lines and angles, triangles, and Heron's formula. The second semester covers chapters on linear equations in two variables, quadrilaterals, areas of parallelograms and triangles, circles, constructions, statistics, probability, and surface areas and volumes. Mental maths practice is scheduled every Monday based on the concerned topic. Related activities are provided at the end of each chapter.
The document summarizes key aspects of contact lens fitting and evaluation. It discusses the anatomy relevant to contact lenses including the tear film and cornea. It then covers common contact lens materials and parameters like oxygen permeability. The document outlines a typical contact lens examination including case history, fitting evaluation, and patient education on proper lens care.
Infinity refers to concepts that are boundless or larger than any natural number. It plays a role in philosophy, theology, mathematics and cosmology. In mathematics, early Greek thinkers like Anaximander explored the concept of infinity. Later, Zeno of Elea contributed paradoxes that helped develop rigorous concepts of infinity. Modern set theory, developed by Cantor, established different types and sizes of infinite sets. Infinity is used in fields like real analysis to denote unbounded limits and in cosmology to explore whether aspects of the universe are infinite.
A square meets all the properties of a rectangle - it has four sides, four right angles, opposite sides that are parallel and equal in length. Additionally, all four sides of a square are equal in length. In mathematics, categories are defined inclusively so that a square is considered a special case of a rectangle. This makes theorems and proofs simpler by avoiding separate cases for different shapes.
This document provides an overview of different number systems and concepts in mathematics related to numbers. It defines real numbers, rational numbers, integers, whole numbers, and natural numbers. It discusses that rational numbers can be divided into integers, whole numbers, and natural numbers. Irrational numbers are also introduced. Important mathematicians who contributed to the study and understanding of numbers are referenced, including Pythagoras, Archimedes, Aryabhatta, Dedekind, Cantor, Babylonians, and Euclid.
This document discusses the history and key concepts of real numbers. It provides background on how real numbers developed from ancient civilizations working with simple fractions to the formal acceptance of irrational numbers. Key figures discussed include Euclid, Hippasus, and developments in ancient Egypt, India, Greece, the Middle Ages, and Alexandria. Fundamental ideas covered include Euclid's lemma, the fundamental theorem of arithmetic, prime factorisation, and the distinction between rational and irrational numbers.
1) The document contains a final round of answers to questions about mathematics. It includes identifying the Poincare Conjecture and Grigori Perelman, decoding a message about math being fun, and solving puzzles linked to the death of Jim Moriarty.
2) The next set of questions involves figuring out connections between Isaac Newton, measurements of the Louvre Pyramid, Armstrong numbers, and the password "LUNE".
3) The final question asks to identify figures formed when a circle is rotated around a line in 3D space, and to name the Abel Prize, transcendental numbers, and Al-Khwarizmi's contributions to algebra.
Middle School Mathematics quiz final.pptxDheerajKL1
This document appears to be a summary of rounds from an online mathematics quiz competition. It outlines the rules and structure for 4 rounds:
Round 1 has miscellaneous multiple choice questions on topics like Fermat's Last Theorem, Rubik's Sudoku, algebra and algorithms, Mobius strips, calculus, and the number 13.
Round 2 is team-based and asks for the names of mathematicians like Brahmagupta, Fibonacci, and Manjul Bhargava.
Round 3 is a buzzer round with questions about mathematicians commemorated in Google Doodles like George Boole, Pi Day, and Ramanujan.
Round 4 allows teams to identify false statements, solve grid
What impact did Pythagoras have on EuclidSolutionPythagorasP.pdfformaxekochi
What impact did Pythagoras have on Euclid?
Solution
Pythagoras
Probably the most famous name during the development of Greek geometry is Pythagoras, even
if only for the famous law concerning right angled triangles. This mathematician lived in a secret
society which took on a semi-religious mission. From this, the Pythagoreans developed a number
of ideas and began to develop trigonometry. The Pythagoreans added a few new axioms to the
store of geometrical knowledge.
1)The sum of the internal angles of a triangle equals two right angles 180*.
2)The sum of the external angles of a triangle equals four right angles 360*
3)The sum of the interior angles of any polygon equals 2n-4 right angles, where n is the number
of sides.
4)The sum of the exterior angles of a polygon equals four right angles, however many sides.
5)The three polygons, the triangle, hexagon, and square completely fill the space around a point
on a plane - six triangles, four squares and three hexagons. In other words, you can tile an area
with these three shapes, without leaving gaps or having overlaps.
6)For a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of
the other two sides.
Most of these rules are instantly familiar to most students, as basic principles of geometry and
trigonometry. One of his pupils, Hippocrates, took the development of geometry further. He was
the first to start using geometrical techniques in other areas of maths, such as solving quadratic
equations, and he even began to study the process of integration. He solved the problem of
Squaring a Lune and showed that the ratio of the areas of two circles equalled the ratio between
the squares of the radii of the circles.
Euclid
Alongside Pythagoras, Euclid is a very famous name in the history of Greek geometry. He
gathered the work of all of the earlier mathematicians and created his landmark work, \'The
Elements,\' surely one of the most published books of all time. In this work, Euclid set out the
approach for geometry and pure mathematics generally, proposing that all mathematical
statements should be proved through reasoning and that no empirical measurements were
needed. This idea of proof still dominates pure mathematics in the modern world.
The reason that Euclid was so influential is that his work is more than just an explanation of
geometry or even of mathematics. The way in which he used logic and demanded proof for every
theorem shaped the ideas of western philosophers right up until the present day. Great
philosopher mathematicians such as Descartes and Newton presented their philosophical works
using Euclid\'s structure and format, moving from simple first principles to complicated
concepts. Abraham Lincoln was a fan, and the US Declaration of Independence used Euclid\'s
axiomatic system.
Apart from the Elements, Euclid also wrote works about astronomy, mirrors, optics, perspective
and music theory, although many of his works are lost to posterity. Certainl.
For a long time, mathematicians tried unsuccessfully to find a number whose square is negative one. In the 1500s, some work with square roots of negative numbers began again. The first major work occurred in 1545, though the mathematician greatly disliked imaginary numbers. Later, Descartes standardized complex numbers as a + bi, but also doubted their usefulness. Today, complex numbers are used extensively in engineering, physics, and computing.
This document provides a brief history of mathematics from ancient civilizations like Egypt and Babylon through modern times. It outlines key developments and contributors to mathematics over time, including the Greeks who established foundations of geometry and number theory, Islamic mathematicians who advanced algebra and algorithms, and modern mathematicians who developed calculus, probability, logarithms, and other critical concepts. The document suggests mathematics will continue having applications in fields like biology, cybernetics, and help solve open problems like the P vs. NP and Riemann hypothesis.
Middle School MathempaSWVWrwWuiz[1].pptxDheerajKL1
The document provides information on several math-related topics:
1) It describes a Rubik's Sudoku toy that allows playing sudoku puzzles with interlocking pieces in different colors to make it easier.
2) It discusses important mathematicians like Al-Khwarizmi, whose contributions led to the words "algebra" and "algorithm", and Brahmagupta who made advances in arithmetic.
3) It presents shapes like the Mobius strip that has only one side and inside/outside.
This document contains 20 math-related questions with answers. It includes questions about Pi Day, Gottfried Leibniz, Thales of Miletus, zero, Richard Feynman, RD Sharma, the number 100, Brook Taylor, Bhaskara-I, Sophie Germain, the Rule of Three, the Plimpton-322 tablet, the Ulam spiral, quaternions, infix notation, the Koch snowflake, the dangerous bend symbol, Evariste Galois, infinity, and Isaac Newton. The questions cover a wide range of mathematicians, concepts, theorems, and historical discoveries.
Order in the distribution of the prime numbers over the natural numbersGert Kramer
Long time ago a famous 'prince' mathematician called Leonhard Euler said following: "Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the human mind will never penetrate."
Could it be that mathematicians have blinded themselves by a one way approach of numbers using only the decimal system rather than as well the ancient nine based system.
In this presentation I show and give prove that there exists a perfect order in the distribution of the prime numbers over natural numbers.
Of course as chaos is a 'state of mind' depending on the consciousness of the observer towards his/her perceived reality in which the way the observer observes it determined by his/her own assumptions that are often the result of generations of taught behavior.
As Albert Einstein said problems can't be solved by the same level of thinking that was used to create them. So we need to put on another pair of spectacles and have to learn to see the world anew to find solutions that get rid of our own created problems and boundaries/limitations.
I hope you enjoy reading and when you have any clue of the current day applications in society that use prime numbers, applicatiosn that are all based on the perceived chaos, you will understand the majority of the impact of finding order.
For instance the Riemann hypothesis that stands at the basis of many models becomes completely obsolete. As well logistics systems and computer processors can be accelerated and encryption technology industry will have its 'black Monday'.
It is the difference of walking in a forest with or without a map. Guess who makes it home earlier and who runs the risk of getting lost?
If you are interested in receiving the algorithm that creates prime numbers based on the found order than please go to www.thesharingsociety.com or contact me at gertkramer[ @]nucleus9.com
Enjoy reading and spread the news. Order is found.
Geometry is a branch of mathematics concerned with shapes, sizes, positions, and properties of space. It arose independently in early cultures and emerged in ancient Greece where Euclid formalized it in his influential Elements text around 300 BC. Elements defined geometry axiomatically and influenced mathematics for centuries. It included proofs of theorems like two triangles being congruent if they share two equal angles and one equal side. Euclid's work defined much of the rules and language of geometry still used today.
The document discusses medieval mathematics from the 12th-14th centuries. It provides biographies of several important medieval mathematicians including Fibonacci, who introduced the Fibonacci sequence to Western Europe and studied rabbit populations. It also discusses Nicole Oresme who proved the divergence of the harmonic series and Giovanni di Casali who analyzed accelerated motion graphically. The document notes that during this time, Europeans learned mathematics from Arabic sources that had been translated to Latin.
This document provides a summary of a student's paper on the topic of the Golden Ratio and its use in Dan Brown's novel The Da Vinci Code. It discusses what the Golden Ratio is mathematically, its history and appearances in art, architecture and nature. It describes how Brown incorporated the Fibonacci sequence and Golden Ratio symbols in the plot of the novel. The conclusion discusses how mathematics and science have become part of literature.
The document discusses real numbers and their history. It provides background on real numbers, including that they represent quantities on a continuous line and include rational and irrational numbers. It then discusses the history of real numbers, including their use by ancient Egyptians and Indians and the recognition of irrational numbers by Greek mathematicians like Pythagoras. It also discusses important mathematicians who contributed to the development and understanding of real numbers, such as Euclid, Hippasus, and Arabic mathematicians.
The role mathematics has played in changing the world has been very much underplayed. This slide was made with intention to show the inventions of some of the greatest mathematicians who have graced the surface of this Earth
π (pi) is the ratio of a circle's circumference to its diameter. It is an irrational and transcendental number represented by the Greek letter π. William Jones is believed to have first used π in its modern sense in 1706 to represent the constant ratio, rather than a varying circumference. Important formulas using π include the circumference of a circle (2πr), area of a circle (πr^2), and volume/surface area of a sphere. While π cannot be expressed exactly as a fraction, it is commonly approximated as 22/7 or 3.14159.
Euclid's Geometry outlines Euclid's influential work on geometry from around 300 BCE. It defines Euclidean geometry as the study of plane and solid figures using axioms and theorems. It also distinguishes between axioms, which are general mathematical assumptions, and postulates, which are specific geometric assumptions. Finally, it briefly discusses several influential mathematicians throughout history and their contributions, including Euclid, Ramanujan, Descartes, Aryabhatta, and Thales.
A review of the growth of the Israel Genealogy Research Association Database Collection for the last 12 months. Our collection is now passed the 3 million mark and still growing. See which archives have contributed the most. See the different types of records we have, and which years have had records added. You can also see what we have for the future.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
The event will cover the following::
Walmart Business + (https://business.walmart.com/plus) is a new shopping experience for nonprofits, schools, and local business customers that connects an exclusive online shopping experience to stores. Benefits include free delivery and shipping, a 'Spend Analytics” feature, special discounts, deals and tax-exempt shopping.
Special TechSoup offer for a free 180 days membership, and up to $150 in discounts on eligible orders.
Spark Good (walmart.com/sparkgood) is a charitable platform that enables nonprofits to receive donations directly from customers and associates.
Answers about how you can do more with Walmart!"
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
3. Natural numbers : the numbers which starts from 1 these numbers
are callled natural numbers.
Whole Numbers : The numbers which starts from 0 are called
whole numbers.
Rational Numbers : A numbers r is called a rational number, if it
can be written in the form of p/q , where p & r integers & q is not
equal to 0.
Irrational numbers : A number s is called a irrational number, if it
can not be written in the form p/q, where p and q are integers
and q is not equal to 0.
The decimal expansion of a rational number is either terminating
or non terminating recurring moreover a number whose decimal
expansion is terminating or non-terminating recurring is rational .
4. PYTHAGORAS
Pythagoras of Samos (Ancient Greek: Πυθαγόρας ὁ Σάμιος [Πυθαγόρης in Ionian Greek] Pythagóras ho
Sámios "Pythagoras the Samian", or simply Πυθαγόρας; b. about 570 – d. about 495 BC[1][2]) was an Ionian Greek
philosopher, mathematician, and founder of the religious movement called Pythagoreanism. Most of the information
about Pythagoras was written down centuries after he lived, so very little reliable information is known about him. He
was born on the island of Samos, and might have travelled widely in his youth, visiting Egypt and other places seeking
knowledge. Around 530 BC, he moved to Croton, a Greek colony in southern Italy, and there set up a religious sect.
His followers pursued the religious rites and practices developed by Pythagoras, and studied his philosophical
theories. The society took an active role in the politics of Croton, but this eventually led to their downfall. The
Pythagorean meeting-places were burned, and Pythagoras was forced to flee the city. He is said to have ended his
days in Metapontum.
Pythagoras made influential contributions to philosophy and religious teaching in the late 6th century BC. He is often
revered as a great mathematician, mystic and scientist, but he is best known for the Pythagorean theorem which
bears his name. However, because legend and obfuscation cloud his work even more than with the other pre-Socratic
philosophers, one can give account of his teachings to a little extent, and some have questioned whether he
contributed much to mathematics and natural philosophy. Many of the accomplishments credited to Pythagoras may
actually have been accomplishments of his colleagues and successors. Whether or not his disciples believed that
everything was related to mathematics and that numbers were the ultimate reality is unknown. It was said that he
was the first man to call himself a philosopher, or lover of wisdom,[3] and Pythagorean ideas exercised a marked
influence on Plato, and through him, all of Western philosophy.
5. Chapter 2
Polinomial
A polynomial have one term is called monomial.
A polynomial have 2 terms is called binomial.
A polynomial have 3 terms is called a trinomial.
A polynomial of degree one is called linear polynomial.
A polynomial of degree 2 is called a quadratic polynomial.
A polynomial of degree 3 is called a cubic polynomial.
A real number ‘a’ is a zero of polynomial p(x) if p(a) = 0. In this case , a is
also called a root of the equation p(x) = 0.
Remainder theorem : If p(x) is any polynomial of degree greater than or
equal to 1 and p(x) is divided by the linear polynomial x – a, than the
remainder is p(a).
Factor Theorem : x – a is a factor of the polynomial p(x), if p(a) = 0. Also, if
x –a is a factorof p(x), then p(a) = 0.
6. R.Dedekind
While teaching calculus for the first time at the Polytechnic, Dedekind came up with
the notion now called a Dedekind cut (German: Schnitt), now a standard definition of
the real numbers. The idea behind a cut is that an irrational number divides the
rational numbers into two classes (sets), with all the members of one class (upper)
being strictly greater than all the members of the other (lower) class. For example, the
square root of 2 puts all the negative numbers and the numbers whose squares are
less than 2 into the lower class, and the positive numbers whose squares are greater
than 2 into the upper class. Every location on the number line continuum contains
either a rational or an irrational number. Thus there are no empty locations, gaps, or
discontinuities. Dedekind published his thoughts on irrational numbers and Dedekind
cuts in his pamphlet "Stetigkeit und irrationale Zahlen" ("Continuity and irrational
numbers");[1] in modern terminology, Vollständigkeit, completeness.
In 1874, while on holiday in Interlaken, Dedekind met Cantor. Thus began an enduring
relationship of mutual respect, and Dedekind became one of the very first
mathematicians to admire Cantor's work on infinite sets, proving a valued ally in
Cantor's battles with Kronecker, who was philosophically opposed to Cantor's
transfinite numbers.
If there existed a one-to-one correspondence between two sets, Dedekind said that
the two sets were "similar." He invoked similarity to give the first precise definition of
an infinite set: a set is infinite when it is "similar to a proper part of itself," in modern
terminology, is equinumerous to one of its proper subsets. (This[clarification needed] is
known as Dedekind's theorem.[citation needed]) Thus the set N of natural numbers can be
shown to be similar to the subset of N whose members are the squares of every
member of N, (N → N2):
7. Georg Ferdinand Ludwig Philipp Cantor ( /ˈ kæntɔr/ KAN-tor; German: [ɡeˈ ɔʁk ˈfɛʁdinant
ˈ luˈtv ˈ ɪp ˈ
ɪç fiˈl kantɔʁ]; March 3 [O.S. February 19] 1845 – January 6, 1918[1]) was a German
mathematician, best known as the inventor of set theory, which has become a fundamental theory
in mathematics. Cantor established the importance of one-to-one correspondence between the
members of two sets, defined infinite and well-ordered sets, and proved that the real numbers are
"more numerous" than the natural numbers. In fact, Cantor's method of proof of this theorem
implies the existence of an "infinity of infinities". He defined the cardinal and ordinal numbers and
their arithmetic. Cantor's work is of great philosophical interest, a fact of which he was well aware. [2]
Cantor's theory of transfinite numbers was originally regarded as so counter-intuitive—even
shocking—that it encountered resistance from mathematical contemporaries such as Leopold
Kronecker and Henri Poincaré[3] and later from Hermann Weyl and L. E. J. Brouwer, while Ludwig
Wittgenstein raised philosophical objections. Some Christian theologians (particularly neo-
Scholastics) saw Cantor's work as a challenge to the uniqueness of the absolute infinity in the
nature of God - [4] on one occasion equating the theory of transfinite numbers with pantheism[5] - a
proposition which Cantor vigorously refuted. The objections to his work were occasionally fierce:
Poincaré referred to Cantor's ideas as a "grave disease" infecting the discipline of mathematics,[6]
and Kronecker's public opposition and personal attacks included describing Cantor as a "scientific
charlatan", a "renegade" and a "corrupter of youth."[7] Kronecker even objected to Cantor's proofs
that the algebraic numbers are countable, and that the transcendental numbers are uncountable,
results now included in a standard mathematics curriculum. Writing decades after Cantor's death,
Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms
of set theory," which he dismissed as "utter nonsense" that is "laughable" and "wrong". [8] Cantor's
recurring bouts of depression from 1884 to the end of his life have been blamed on the hostile
attitude of many of his contemporaries,[9] though some have explained these episodes as probable
manifestations of a bipolar disorder.[10]
The harsh criticism has been matched by later accolades. In 1904, the Royal Society awarded
Cantor its Sylvester Medal, the highest honor it can confer for work in mathematics.[11] It has been
suggested that Cantor believed his theory of transfinite numbers had been communicated to him by
God.[12] David Hilbert defended it from its critics by famously declaring: "No one shall expel us from
the Paradise that Cantor has created."[13]
8. Made by --- Blossom Shrivastava
Class --- 9 ‘ A ’
Roll no : 16
29. Definition of a Linear Equation
A linear equation in two variable x is
an equation that can be written in the
form ax + by + c = 0, where a ,b and c are
real numbers and a and b is not equal to
0.
An example of a linear equation in x
is .
30. Equations of the form ax + by = c are
called linear equations in two variables.
Equations of the form ax + by = c are (0,4)
called linear equations in two variables.
The point (0,4) is the y-intercept.
The point (6,0) is the x-intercept.
-2 2
31. Solution of an Equation in Two Variables
Example:
Given the equation 2x + 3y = 18, determine
if the ordered pair (3, 4) is a solution to the
equation.
We substitute 3 in for x and 4 in for y.
2(3) + 3 (4) ? 18
6 + 12 ? 18
18 = 18 True.
Therefore, the ordered pair (3, 4) is a
solution to the equation 2x + 3y = 18.
33. Finding Solutions of an Equation
Find five solutions to the equation y = 3x + 1.
Start by choosing some x values and then computing the
corresponding y values.
If x = -2, y = 3(-2) + 1 = -5. Ordered pair (-2, -5)
If x = -1, y = 3(-1) + 1 = -2. Ordered pair ( -1, -2)
If x =0, y = 3(0) + 1 = 1. Ordered pair (0, 1)
If x =1, y = 3(1) + 1 =4. Ordered pair (1, 4)
If x =2, y = 3(2) + 1 =7. Ordered pair (2, 7)