The document discusses the Pythagorean theorem and square roots. It defines a right triangle as having one 90 degree angle. The Pythagorean theorem states that for a right triangle with sides a, b, c, where c is the hypotenuse opposite the right angle, a^2 + b^2 = c^2. An example uses the theorem to calculate the height of a wall given the length of a ladder leaning against it. Square roots are then introduced, with the square root of a number x defined as the positive number that produces x when squared.
The document discusses basic geometric shapes and formulas for calculating their perimeters. It defines a loop or polygon as a shape formed by connecting line segments end to end. Triangles have three sides and their perimeter is calculated as the sum of the three side lengths. Specific types of triangles like equilateral triangles are discussed. Rectangles are four-sided polygons with right angles, and squares are rectangles with four equal sides. Formulas are provided for calculating the perimeters of squares and rectangles based on their side lengths. Some example problems demonstrate applying these concepts and formulas to calculate perimeters of fenced or roped areas composed of multiple shapes.
The document discusses right triangles and the Pythagorean theorem. It defines a right triangle as one with a 90-degree angle, and labels the sides as the hypotenuse (the side opposite the right angle) and the two legs. It presents the Pythagorean theorem, which states that for any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. The document provides an example of using the theorem to calculate the height of a wall given the length of a leaning ladder. It also defines the square root and how it relates to finding the length of a side of a triangle based on the Pythagorean theorem.
The document discusses the Pythagorean theorem and square roots. It begins by defining a right triangle and its components. It then states the Pythagorean theorem, which relates the lengths of the sides of a right triangle. An example problem demonstrates using the theorem to find the height of a wall. The document concludes by defining the square root, explaining that the square root of a number is its positive root and how to calculate and approximate square roots.
“Conic section” is a fundamental of the Mathematics. This
report is made from my studying about the conic section in the
Mathematics books and on the internet. This report contains
topics that involve with conic section such as: The history of Conic
section studying, Parabola, Ellipse, Hyperbola and their
applications with figures may help you to understand easily.
This report is may use to refer for next time and its can be
usefulness for the readers.
27 triple integrals in spherical and cylindrical coordinatesmath267
The document discusses cylindrical and spherical coordinate systems. It defines cylindrical coordinates as using polar coordinates in the xy-plane with z as the third coordinate. It provides an example of converting between rectangular and cylindrical coordinates. Spherical coordinates represent a point as (ρ, θ, φ) where ρ is the distance from the origin and θ and φ specify the direction. Conversion rules between the different systems are given.
This document discusses conic sections, which are plane curves formed by the intersection of a cone with a plane. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas. It provides examples of how to determine the equation of a conic section based on given properties, such as all points being equidistant from a fixed point and line. Key aspects covered include using the distance formula and completing the square to put equations in standard form.
- The conic sections (circles, ellipses, parabolas, and hyperbolas) have been studied for over 2000 years, with Apollonius of Perga making major contributions in the 2nd century BC by rigorously studying them and applying the work to astronomy.
- Apollonius was the first to note that conic sections can be constructed by cutting a circular cone with a plane in different ways: a circle from a perpendicular cut, a parabola from a parallel cut, a hyperbola from a cut through both parts of the cone, and an ellipse from a cut through one part of the cone not parallel to its side.
- Conic sections can be represented by a general second
The document discusses basic geometric shapes and formulas for calculating their perimeters. It defines a loop or polygon as a shape formed by connecting line segments end to end. Triangles have three sides and their perimeter is calculated as the sum of the three side lengths. Specific types of triangles like equilateral triangles are discussed. Rectangles are four-sided polygons with right angles, and squares are rectangles with four equal sides. Formulas are provided for calculating the perimeters of squares and rectangles based on their side lengths. Some example problems demonstrate applying these concepts and formulas to calculate perimeters of fenced or roped areas composed of multiple shapes.
The document discusses right triangles and the Pythagorean theorem. It defines a right triangle as one with a 90-degree angle, and labels the sides as the hypotenuse (the side opposite the right angle) and the two legs. It presents the Pythagorean theorem, which states that for any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. The document provides an example of using the theorem to calculate the height of a wall given the length of a leaning ladder. It also defines the square root and how it relates to finding the length of a side of a triangle based on the Pythagorean theorem.
The document discusses the Pythagorean theorem and square roots. It begins by defining a right triangle and its components. It then states the Pythagorean theorem, which relates the lengths of the sides of a right triangle. An example problem demonstrates using the theorem to find the height of a wall. The document concludes by defining the square root, explaining that the square root of a number is its positive root and how to calculate and approximate square roots.
“Conic section” is a fundamental of the Mathematics. This
report is made from my studying about the conic section in the
Mathematics books and on the internet. This report contains
topics that involve with conic section such as: The history of Conic
section studying, Parabola, Ellipse, Hyperbola and their
applications with figures may help you to understand easily.
This report is may use to refer for next time and its can be
usefulness for the readers.
27 triple integrals in spherical and cylindrical coordinatesmath267
The document discusses cylindrical and spherical coordinate systems. It defines cylindrical coordinates as using polar coordinates in the xy-plane with z as the third coordinate. It provides an example of converting between rectangular and cylindrical coordinates. Spherical coordinates represent a point as (ρ, θ, φ) where ρ is the distance from the origin and θ and φ specify the direction. Conversion rules between the different systems are given.
This document discusses conic sections, which are plane curves formed by the intersection of a cone with a plane. The four types of conic sections are circles, ellipses, parabolas, and hyperbolas. It provides examples of how to determine the equation of a conic section based on given properties, such as all points being equidistant from a fixed point and line. Key aspects covered include using the distance formula and completing the square to put equations in standard form.
- The conic sections (circles, ellipses, parabolas, and hyperbolas) have been studied for over 2000 years, with Apollonius of Perga making major contributions in the 2nd century BC by rigorously studying them and applying the work to astronomy.
- Apollonius was the first to note that conic sections can be constructed by cutting a circular cone with a plane in different ways: a circle from a perpendicular cut, a parabola from a parallel cut, a hyperbola from a cut through both parts of the cone, and an ellipse from a cut through one part of the cone not parallel to its side.
- Conic sections can be represented by a general second
This document discusses conic sections, including circles, ellipses, parabolas, and hyperbolas. It provides:
1) The definitions and standard equations of each conic section, describing how they are formed from the intersection of a plane with a double cone.
2) Examples of different forms the equations can take and the geometric properties of each conic section, such as foci, axes, vertices, and asymptotes.
3) Methods for writing the equations of tangents to conics and using parametric equations to represent loci.
In less than 3 sentences, it summarizes the key information about conic sections provided in the document.
This summary combines slides from Melanie Tomlinson and Morrobea on the topic of parabolas. The key points covered include:
- The geometric definition of a parabola as the set of all points equidistant from a fixed point (the focus) and fixed line (the directrix).
- Parabolas can be represented using various equation forms including vertex form, standard form, and general form.
- Methods for graphing parabolas by identifying features like the vertex, axis of symmetry, x-intercepts, focus, and directrix.
- Applications of parabolas to model real-world situations like searchlights and radio telescopes.
The document defines conic sections and describes parabolas. It provides specific objectives related to defining conic sections, identifying different types, describing parabolas, and converting between general and standard forms of parabola equations. It then gives details on the focus, directrix, vertex, latus rectum, and eccentricity of parabolas. Examples of problems involving finding parabola equations and properties from conditions are also provided.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance from the y-axis and y is the distance from the x-axis. The four quadrants (I, II, III, IV) are defined by the intersection of the positive and negative sides of the x and y axes. Examples are given of labeling points and finding coordinates on the grid.
The document discusses parabolas, including their key properties and equations. It defines a parabola as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The document derives the standard equation of a parabola from this definition and discusses how to graph parabolas based on their equations. It also covers transformations, latus rectum, and other geometric properties of parabolas.
This document defines and discusses parabolas. It begins by listing 4 learning outcomes related to understanding parabolas, their standard form equations, graphing them, and solving problems involving parabolas. It then defines a parabola as the set of all points that are the same distance from both a fixed focus point and directrix line. The standard form of the equation for a parabola is derived and explained to be x^2 = 4cy, where c is the distance between the focus and directrix. Key features of parabolas like the vertex, directrix, focus, and axis of symmetry are identified. Examples of determining standard equations and graphing parabolas are provided.
The document describes the rectangular coordinate system. It defines the x-axis and y-axis which intersect at the origin point (0,0). Each point in the plane is assigned an ordered pair (x,y) where x is the distance from the y-axis and y is the distance from the x-axis. The plane is divided into four quadrants based on whether x and y are positive or negative. Reflections of points across the axes are also described. Examples are provided to demonstrate labeling points and finding point coordinates.
This document discusses key concepts related to coordinate geometry including:
- The Cartesian coordinate system consisting of perpendicular x and y axes intersecting at the origin.
- Formulas for finding the distance between two points and the midpoint of a line segment.
- Equations and graphs of circles, parabolas, ellipses, and hyperbolas. Definitions involve relationships between distances to foci, vertices, and directrix elements.
- General forms of equations for each conic section depending on the orientation of the focal axis.
The document discusses finding areas and perimeters of various shapes such as parallelograms, triangles, trapezoids, and rhombi. It provides definitions for key terms used to calculate these measurements, such as base and height. Several examples are shown calculating perimeters and areas of specific shapes by applying the appropriate formulas. Methods for finding missing values like side lengths using properties of shapes are demonstrated.
The document discusses parabolas and their key properties. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex, axis of symmetry, and focus-directrix distance determine the shape and position of the parabola. Examples are provided to demonstrate how to find the equation of a parabola given properties like the vertex and focus.
This document discusses graphing parabolas using squares. It explains that parabolas can be graphed by comparing their equations to standard forms and using squares of length 2p, where p is determined by the equation. Examples are given of graphing parabolas from equations x^2=6y and y^2=8x. The focus, vertex, directrix, and latus rectum are identified for each parabola by comparing the equations to the standard forms.
This document provides an overview of parabolas including their key characteristics and equations. It defines a parabola as the set of all points that are an equal distance from both a fixed point called the focus and a fixed line called the directrix. The standard equation of a parabola is provided and examples are given of writing the equation of a parabola given its focus and directrix. The document also discusses graphing parabolas by identifying the vertex, axis of symmetry, focus, and directrix. Real-world applications of parabolas to reflect light and sound are briefly described.
The document describes the rectangular coordinate system. It defines a coordinate system as assigning positions in a plane or space with addresses. The rectangular coordinate system uses a grid with two perpendicular axes (x and y) intersecting at the origin (0,0). Any point in the plane is located by its coordinates (x,y), where x is the distance right or left of the origin and y is the distance up or down. The four quadrants divided by the axes are labeled based on the signs of the x and y coordinates.
The document discusses the standard form and identifying features of the equation of a vertical parabola, noting that it can be written as y = ax^2 + bx + c, the highest degree of y is 1 and x is 2, and the sign of the x^2 term determines whether the graph is right side up or upside down, with the vertex being the highest or lowest point respectively.
The rectangular coordinate system represents points in a plane using perpendicular axes (x-axis and y-axis) that intersect at the origin (0,0). Each point is assigned an ordered pair (x,y) where x is the distance from the origin on the x-axis and y is the distance from the origin on the y-axis. The system divides the plane into four quadrants based on whether the x and y values are positive or negative. The rectangular coordinate system allows any point in the plane to be uniquely addressed using its x and y coordinates.
This document discusses the four basic conic sections - circles, parabolas, ellipses, and hyperbolas. It provides the standard form equations and key characteristics for each conic section with varying positions of the vertex. Circles are defined by a center point and radius. Parabolas are defined by a focus, directrix, and vertex. Ellipses are defined by two foci and the sum of distances to these points. Hyperbolas are defined by two foci and the difference of distances to these points. Examples of each conic section in architecture and acoustics are also given.
A parabola is a conic section defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a line called the directrix. The general equation of a parabola is y=ax^2, where a is a constant related to the distance between the focus and directrix. Given a focus point and directrix, every point on the parabola satisfies the property that its distance to the focus is equal to its distance to the directrix. Parabolas can be used to model projectile motion under gravity.
The document defines and describes ellipses. It states that an ellipse is the set of points whose sum of the distances to two fixed foci is a constant. An ellipse has a center, major axis, and minor axis. The standard form of the ellipse equation is given as (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center, a is the x-radius, and b is the y-radius. An example problem demonstrates how to identify these properties from a given ellipse equation and sketch the ellipse.
EQUATION OF A PARABOLA FROM THE VERTEX AND DIRECTRIXsumanmathews
What is a parabola? How is it derived from conics?
Watch this presentation to find out.
Here, we learn how a parabola is derived when a plane cuts a cone. We learn that, for a parabola, distance of a point from the focus = distance of the point from the directrix. We solve problems based on this principle and also learn how to calculate equation of the axis and the coordinates of the vertex.
This is useful for grade 11 maths students. This channel has videos for grades 11, 12, engineering maths, nata maths and the GRE QUANT section.
Consider subscribing to my channel for more videos. You can visit my page
https://www.mathmadeeasy.co/lessons
For further help, you can join my classes for grade 11 maths
The document discusses parabolas and their key properties. It begins by introducing the standard form of a quadratic function and the steps to find the vertex, line of symmetry, and maximum/minimum value. It then provides examples of completing the square to derive the standard form (y = (x - h)2 + k) from other forms. Using this standard form, the vertex is identified as (h, k), the line of symmetry is defined as x = h, and the maximum/minimum is determined by the sign of the parabola based on whether k represents the highest or lowest point.
The document provides information about the Pythagorean theorem:
1) It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2) It gives examples of right triangles that satisfy the theorem, such as ones with sides of 3, 4, 5 or 5, 12, 13.
3) It includes an animated proof of the theorem showing how the area of the square on the hypotenuse equals the combined areas of the squares on the other two sides.
This document discusses conic sections, including circles, ellipses, parabolas, and hyperbolas. It provides:
1) The definitions and standard equations of each conic section, describing how they are formed from the intersection of a plane with a double cone.
2) Examples of different forms the equations can take and the geometric properties of each conic section, such as foci, axes, vertices, and asymptotes.
3) Methods for writing the equations of tangents to conics and using parametric equations to represent loci.
In less than 3 sentences, it summarizes the key information about conic sections provided in the document.
This summary combines slides from Melanie Tomlinson and Morrobea on the topic of parabolas. The key points covered include:
- The geometric definition of a parabola as the set of all points equidistant from a fixed point (the focus) and fixed line (the directrix).
- Parabolas can be represented using various equation forms including vertex form, standard form, and general form.
- Methods for graphing parabolas by identifying features like the vertex, axis of symmetry, x-intercepts, focus, and directrix.
- Applications of parabolas to model real-world situations like searchlights and radio telescopes.
The document defines conic sections and describes parabolas. It provides specific objectives related to defining conic sections, identifying different types, describing parabolas, and converting between general and standard forms of parabola equations. It then gives details on the focus, directrix, vertex, latus rectum, and eccentricity of parabolas. Examples of problems involving finding parabola equations and properties from conditions are also provided.
The document describes the rectangular coordinate system. It defines the system as using a grid with two perpendicular axes (x and y) that intersect at the origin (0,0). Any point in the plane can be located using its coordinates (x,y), where x is the distance from the y-axis and y is the distance from the x-axis. The four quadrants (I, II, III, IV) are defined by the intersection of the positive and negative sides of the x and y axes. Examples are given of labeling points and finding coordinates on the grid.
The document discusses parabolas, including their key properties and equations. It defines a parabola as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The document derives the standard equation of a parabola from this definition and discusses how to graph parabolas based on their equations. It also covers transformations, latus rectum, and other geometric properties of parabolas.
This document defines and discusses parabolas. It begins by listing 4 learning outcomes related to understanding parabolas, their standard form equations, graphing them, and solving problems involving parabolas. It then defines a parabola as the set of all points that are the same distance from both a fixed focus point and directrix line. The standard form of the equation for a parabola is derived and explained to be x^2 = 4cy, where c is the distance between the focus and directrix. Key features of parabolas like the vertex, directrix, focus, and axis of symmetry are identified. Examples of determining standard equations and graphing parabolas are provided.
The document describes the rectangular coordinate system. It defines the x-axis and y-axis which intersect at the origin point (0,0). Each point in the plane is assigned an ordered pair (x,y) where x is the distance from the y-axis and y is the distance from the x-axis. The plane is divided into four quadrants based on whether x and y are positive or negative. Reflections of points across the axes are also described. Examples are provided to demonstrate labeling points and finding point coordinates.
This document discusses key concepts related to coordinate geometry including:
- The Cartesian coordinate system consisting of perpendicular x and y axes intersecting at the origin.
- Formulas for finding the distance between two points and the midpoint of a line segment.
- Equations and graphs of circles, parabolas, ellipses, and hyperbolas. Definitions involve relationships between distances to foci, vertices, and directrix elements.
- General forms of equations for each conic section depending on the orientation of the focal axis.
The document discusses finding areas and perimeters of various shapes such as parallelograms, triangles, trapezoids, and rhombi. It provides definitions for key terms used to calculate these measurements, such as base and height. Several examples are shown calculating perimeters and areas of specific shapes by applying the appropriate formulas. Methods for finding missing values like side lengths using properties of shapes are demonstrated.
The document discusses parabolas and their key properties. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). The vertex, axis of symmetry, and focus-directrix distance determine the shape and position of the parabola. Examples are provided to demonstrate how to find the equation of a parabola given properties like the vertex and focus.
This document discusses graphing parabolas using squares. It explains that parabolas can be graphed by comparing their equations to standard forms and using squares of length 2p, where p is determined by the equation. Examples are given of graphing parabolas from equations x^2=6y and y^2=8x. The focus, vertex, directrix, and latus rectum are identified for each parabola by comparing the equations to the standard forms.
This document provides an overview of parabolas including their key characteristics and equations. It defines a parabola as the set of all points that are an equal distance from both a fixed point called the focus and a fixed line called the directrix. The standard equation of a parabola is provided and examples are given of writing the equation of a parabola given its focus and directrix. The document also discusses graphing parabolas by identifying the vertex, axis of symmetry, focus, and directrix. Real-world applications of parabolas to reflect light and sound are briefly described.
The document describes the rectangular coordinate system. It defines a coordinate system as assigning positions in a plane or space with addresses. The rectangular coordinate system uses a grid with two perpendicular axes (x and y) intersecting at the origin (0,0). Any point in the plane is located by its coordinates (x,y), where x is the distance right or left of the origin and y is the distance up or down. The four quadrants divided by the axes are labeled based on the signs of the x and y coordinates.
The document discusses the standard form and identifying features of the equation of a vertical parabola, noting that it can be written as y = ax^2 + bx + c, the highest degree of y is 1 and x is 2, and the sign of the x^2 term determines whether the graph is right side up or upside down, with the vertex being the highest or lowest point respectively.
The rectangular coordinate system represents points in a plane using perpendicular axes (x-axis and y-axis) that intersect at the origin (0,0). Each point is assigned an ordered pair (x,y) where x is the distance from the origin on the x-axis and y is the distance from the origin on the y-axis. The system divides the plane into four quadrants based on whether the x and y values are positive or negative. The rectangular coordinate system allows any point in the plane to be uniquely addressed using its x and y coordinates.
This document discusses the four basic conic sections - circles, parabolas, ellipses, and hyperbolas. It provides the standard form equations and key characteristics for each conic section with varying positions of the vertex. Circles are defined by a center point and radius. Parabolas are defined by a focus, directrix, and vertex. Ellipses are defined by two foci and the sum of distances to these points. Hyperbolas are defined by two foci and the difference of distances to these points. Examples of each conic section in architecture and acoustics are also given.
A parabola is a conic section defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a line called the directrix. The general equation of a parabola is y=ax^2, where a is a constant related to the distance between the focus and directrix. Given a focus point and directrix, every point on the parabola satisfies the property that its distance to the focus is equal to its distance to the directrix. Parabolas can be used to model projectile motion under gravity.
The document defines and describes ellipses. It states that an ellipse is the set of points whose sum of the distances to two fixed foci is a constant. An ellipse has a center, major axis, and minor axis. The standard form of the ellipse equation is given as (x-h)2/a2 + (y-k)2/b2 = 1, where (h,k) is the center, a is the x-radius, and b is the y-radius. An example problem demonstrates how to identify these properties from a given ellipse equation and sketch the ellipse.
EQUATION OF A PARABOLA FROM THE VERTEX AND DIRECTRIXsumanmathews
What is a parabola? How is it derived from conics?
Watch this presentation to find out.
Here, we learn how a parabola is derived when a plane cuts a cone. We learn that, for a parabola, distance of a point from the focus = distance of the point from the directrix. We solve problems based on this principle and also learn how to calculate equation of the axis and the coordinates of the vertex.
This is useful for grade 11 maths students. This channel has videos for grades 11, 12, engineering maths, nata maths and the GRE QUANT section.
Consider subscribing to my channel for more videos. You can visit my page
https://www.mathmadeeasy.co/lessons
For further help, you can join my classes for grade 11 maths
The document discusses parabolas and their key properties. It begins by introducing the standard form of a quadratic function and the steps to find the vertex, line of symmetry, and maximum/minimum value. It then provides examples of completing the square to derive the standard form (y = (x - h)2 + k) from other forms. Using this standard form, the vertex is identified as (h, k), the line of symmetry is defined as x = h, and the maximum/minimum is determined by the sign of the parabola based on whether k represents the highest or lowest point.
The document provides information about the Pythagorean theorem:
1) It states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
2) It gives examples of right triangles that satisfy the theorem, such as ones with sides of 3, 4, 5 or 5, 12, 13.
3) It includes an animated proof of the theorem showing how the area of the square on the hypotenuse equals the combined areas of the squares on the other two sides.
The document discusses the history and development of the Pythagorean theorem. It explains that Pythagoras founded a philosophical school in Croton, Italy, where he and his followers studied mathematics and believed that numbers were the ultimate reality. The document then provides several proofs of the Pythagorean theorem, including using similar triangles, adding the areas of shapes, and the converse theorem. It also discusses Pythagoras' contributions to music and other areas of mathematics.
1. The document discusses parabolas and their key characteristics including focus, directrix, and standard equation forms.
2. A parabola is defined as the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix).
3. The standard equation forms for parabolas are provided depending on the orientation and location of the vertex.
Pythagoras was an ancient Greek philosopher and mathematician born on the island of Samos in around 570 BC. He is best known for the Pythagorean theorem, which states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. While Pythagoras likely did not discover this theorem himself, he is credited as being the first to prove why it is true. The Pythagorean theorem is one of the earliest and most important theorems in mathematics.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Pythagoras, a Greek mathematician, is credited with discovering this theorem. The theorem can be used to calculate an unknown side of a right triangle if the other two sides are known. Worked examples are provided to demonstrate finding an unknown hypotenuse or leg using the Pythagorean theorem.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Pythagoras, a Greek mathematician, is credited with discovering this theorem. The theorem can be used to calculate an unknown side of a right triangle if the other two sides are known. Worked examples are provided to demonstrate finding an unknown hypotenuse or leg using the Pythagorean theorem.
Pythagoras was an ancient Greek thinker, but he was not the founder of the Pythagorean theorem. That honor goes to his followers, known as the Pythagorean Brotherhood, who established the theorem over 100 years after Pythagoras' death. The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This relationship has many practical applications in fields like engineering, construction, physics, and astronomy that involve calculating distances.
The document discusses rhombi and squares. It defines a rhombus as a parallelogram with four congruent sides and gives its properties. A square is defined as a parallelogram with four right angles and four congruent sides. The document provides theorems for identifying rhombi and squares. It then gives examples of using the properties and theorems to determine if a shape is a rhombus, rectangle, or square.
The document defines various polygons based on their number of sides and interior angles. It also defines types of triangles based on their angles and sides. It provides formulas for finding missing angles of triangles and quadrilaterals. Additionally, it defines complementary, supplementary and trigonometric ratios. It concludes by explaining how to calculate the areas of various shapes including squares, rectangles, parallelograms, trapezoids and triangles. It also provides formulas for finding perimeters, circumferences, surface areas and volumes of various 3D shapes like cubes, prisms, cylinders, pyramids, cones, spheres.
How to calculate the area of a triangleChloeDaniel2
The space that is occupied by a flat shape or the surface of an object. The standard unit of measurement is mostly either ㎡ or cm2. We are going to discuss the Area of triangles here
The document explains the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. It provides examples of using the theorem to determine if a triangle is a right triangle, and to find the length of missing sides when the other two sides are known.
The coordinate plane is formed by intersecting two number lines, called the x-axis and y-axis, at their zero points. The point of intersection is called the origin. To graph an inequality in two variables, graph the boundary curve and shade the region where the inequality is true. The distance formula can be used to find the distance between two points by treating it as the hypotenuse of a right triangle formed by the differences in the x and y coordinates. The midpoint formula finds the point halfway between two points by averaging the x and y coordinates. A circle is defined as all points equidistant from a center point, where the distance from the center is the radius. The standard form of a circle equation relates the
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The document explains the Pythagorean theorem and provides examples of using it to determine if a triangle is a right triangle or to find the length of a missing side.
This document provides information about Pythagoras' theorem and how it can be used to solve problems involving right triangles. It begins with a brief history of Pythagoras and the times he lived in. It then explains Pythagoras' theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Examples are provided to demonstrate how to use the theorem to calculate unknown sides of right triangles. The document also notes that rearranging the theorem allows calculating the lengths of the shorter sides given the hypotenuse.
This document provides information about calculating the surface areas of pyramids and cones. It defines key terms like vertex, slant height, and altitude. It presents examples of calculating the lateral area and total surface area of regular and right pyramids and cones using given dimensions. It also explores how changing the dimensions affects the surface area calculations and provides an example of finding surface area for a composite 3D shape made of a cube and pyramid.
The document discusses key concepts related to the Cartesian plane including:
- The Cartesian plane is formed by two perpendicular number lines called the x-axis and y-axis intersecting at the origin point.
- Any point P on the plane can be located using its coordinates (x,y) which indicate the point's position along the x and y axes.
- The distance between two points P1(x1,y1) and P2(x2,y2) can be calculated using the distance formula.
- Key curves that can be represented on the Cartesian plane include lines, circles, parabolas, ellipses, and hyperbolas through their defining equations.
C
h
a
p
t
e
r
dug84356_ch09a.qxd 9/14/10 2:11 PM Page 557
’
9.1
9.2
9.3
9.4
9.5
9.6
Radicals
Rational Exponents
Adding, Subtracting, and
Multiplying Radicals
Quotients, Powers,
and Rationalizing
Denominators
Solving Equations with
Radicals and Exponents
Complex Numbers
9
Radicals and Rational
Exponents
Just how cold is it in Fargo, North Dakota, in winter? According to local meteorol
ogists, the mercury hit a low of –33°F on January 18, 1994. But air temperature
alone is not always a reliable indicator of how cold you feel. On the same date,
the average wind velocity was 13.8 miles per hour. This dramatically affected how
cold people felt when they stepped outside. High winds along with cold temper
atures make exposed skin feel colder because the wind significantly speeds up
the loss of body heat. Meteorologists use the terms “wind chill factor,”“wind chill
index,” and “wind chill temperature” to take into account both air temperature
and wind velocity.
Through experimentation in Ant
arctica, Paul A. Siple developed a
formula in the 1940s that measures the
wind chill from the velocity of the wind
and the air temperature. His complex
formula involving the square root of
the velocity of the wind is still used
today to calculate wind chill temper
atures. Siple’s formula is unlike most
scientific formulas in that it is not
based on theory. Siple experimented
with various formulas involving wind
velocity and temperature until he
found a formula that seemed to predict
how cold the air felt.
W
in
d
ch
ill
te
m
pe
ra
tu
re
(
F
)
fo
r
25
F
a
ir
te
m
pe
ra
tu
re
25
20
15
10
5
0
5
10
15
Wind velocity (mph)
5 10 15 20 25 30
Siple s formula is stated
and used in Exercises 111
and 112 of Section 9.1.
dug84356_ch09a.qxd 9/14/10 2:11 PM Page 558
→
→
→
558 Chapter 9 Radicals and Rational Exponents 9-2
9.1 Radicals
In Section 4.1, you learned the basic facts about powers. In this section, you will
study roots and see how powers and roots are related.
In This Section
U1V Roots
U2V Roots and Variables
U3V Product Rule for Radicals
U4V Quotient Rule for Radicals U1V Roots
U5V Domain of a Radical We use the idea of roots to reverse powers. Because 32 = 9 and (-3)2 = 9, both 3 andExpression or Function
-3 are square roots of 9. Because 24 = 16 and (-2)4 = 16, both 2 and -2 are fourth
roots of 16. Because 23 = 8 and (-2)3 = -8, there is only one real cube root of 8 and
only one real cube root of -8. The cube root of 8 is 2 and the cube root of -8 is -2.
nth Roots
If a = bn for a positive integer n, then b is an nth root of a. If a = b2, then b is a
square root of a. If a = b3, then b is the cube root of a.
If n is a positive even integer and a is positive, then there are two real nth roots of
a. We call these roots even roots. The positive even root of a positive number is called
the prin ...
The document discusses constructing a quadrilateral with the maximum area given the lengths of its four sides. It presents the following key points:
1) The maximum area quadrilateral can be constructed by arranging the sides in a circle, with the opposite angles summing to 180 degrees.
2) The relationships between the radius of the circle, the angles between sides, and the side lengths are described by formulas using trigonometry.
3) Given the four side lengths, the maximum area quadrilateral can be constructed in a circle by first calculating the radius, then drawing the sides based on the radius and angle formulas.
This document provides examples and solutions for problems involving radicals. It begins by explaining how radicals are used in the Pythagorean theorem and distance formula. Several worked examples are then shown that apply these concepts to find lengths, areas, velocities, and unknown numbers. The examples demonstrate how to set up and solve equations using radicals to model real-world scenarios like finding the distance between objects or the speed of a falling object.
Similar to 22 addition and subtraction of signed numbers (20)
This document contains a sample algebra test with 24 multiple choice questions and solutions. Each question is followed by an answer option and a link to the solution. The solutions provide step-by-step workings to arrive at the correct answer. The test covers topics such as fractions, operations with fractions, percentages, word problems involving percentages, and solving simple equations. An interactive PowerPoint file is also included but viewing the slides is not interactive when using Slideshare.
The document describes the number line and how it assigns numbers to points on a line. It explains that 0 is assigned to the center or origin of the line. Positive numbers are assigned to the right of 0 and negative numbers to the left. Any number to the right of another number on the number line is defined as greater than the number to its left. Intervals on the number line, denoted by a < x < b, represent all the numbers between a and b. Examples are provided to illustrate drawing intervals on the number line.
The document discusses ratios, proportions, and how to solve proportional equations. It defines a ratio as two related quantities stated side by side, and gives the example of a 3:4 ratio of eggs to flour in a recipe. Proportions are defined as equal ratios. The key steps to solve proportional equations are: 1) write the ratios as fractions set equal to each other, 2) use cross-multiplication to convert the proportions into regular equations, and 3) solve the resulting equation using algebraic techniques. An example problem demonstrates these steps to solve a proportional equation for the variable x.
The document discusses exponents and rules for working with them. It defines exponents as the number of times a base is used as a factor in a repetitive multiplication. The main rules covered are:
- The multiply-add rule, which states that ANAK = AN+K
- The divide-subtract rule, which states that AN/AK = AN-K
Examples are provided to demonstrate calculating exponents and applying the rules.
The document discusses solving linear equations using examples of ordering pizzas. It explains that a linear equation contains linear expressions on both sides, such as 3x + 10 = 34, and can be solved by manipulating the equation through steps like subtraction to find the value of x that makes both sides equal. For example, in the equation 3x + 10 = 34, subtracting 10 from both sides and dividing both sides by 3 reveals that x = 8 is the solution.
The document discusses mathematical expressions and how to combine them. It defines an expression as a calculation procedure written with numbers, variables, and operations. Expressions have terms, with the x-term being the variable term and the number term being the constant. To combine expressions, like terms are combined in the same way numbers are, while unlike terms cannot be combined. The simplest expressions are linear expressions of the form ax + b.
34 conversion between decimals, fractions and percentagesalg-ready-review
The document discusses the conversion between decimals, fractions, and percentages. It states that fractions, decimals, and percentages are different ways to express quantities. Fractions provide instructions to divide a whole into parts, while decimals standardize fractions to powers of 10 to make addition and subtraction easier. Percentages express a quantity as a ratio out of 100. The document then provides examples and steps for converting between these representations.
The document defines percentages as expressing "how many out of 100" and can be written as a fraction with the percentage symbol or as a decimal. It provides examples of common percentages like 1% = 1/100, 5% = 1/20, 10% = 1/10, 25% = 1/4, and 50% = 1/2. The document also works through examples of calculating percentages of a total amount, like finding 3/4 of $100 is $75, or that 45% of 60 pieces of candy is 27 pieces. Finally, it lists some important percentages that relate to coins: 5% = 1/20 for nickels, 10% = 1/10 for dimes, and 25%
The document discusses how to multiply multi-digit decimal numbers. It explains that multiplication of multi-digit numbers is done by multiplying the digits in place value, starting from the ones place. The results are recorded and carried over as needed. It provides a step-by-step example of 47 x 6, showing how each digit is multiplied and the results carried to the next place value. It notes that the same process is followed for decimals, but the decimal point is placed in the final product so that the total number of decimal places is correct.
31 decimals, addition and subtraction of decimalsalg-ready-review
This document introduces decimals by using an analogy of a cash register holding coins of various values. It explains that decimals allow tracking of smaller quantities by including base-10 fractions in the number system. It assumes the US Treasury makes fictional smaller value coins like "itties" and "bitties", then demonstrates writing decimal numbers as representations of coins in different slots of a cash register. Finally, it provides steps for comparing decimal values by lining up numbers at the decimal point and determining the largest by the digits from left to right.
The document discusses variables, expressions, and evaluation in mathematics. It explains that variables like x, y, and z are used to represent numbers, and their values can change depending on the situation. Expressions are made using variables and mathematical operations, and evaluation involves replacing the variables in an expression with input values and calculating the output. The input values replace the variables within parentheses, and the process of evaluation finds the output. Several examples are provided to demonstrate evaluating different expressions by replacing variables with given input values.
The document discusses order of operations and how to correctly evaluate mathematical expressions. It provides examples of evaluating expressions involving multiplication, division, addition, subtraction, grouping symbols and exponents. The key steps are to perform operations within grouping symbols from the innermost out, then multiplication and division from left to right, followed by addition and subtraction from left to right. Setting clear rules for order of operations ensures the correct solution is obtained.
23 multiplication and division of signed numbersalg-ready-review
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use the following rules for the sign of the product: two numbers with the same sign yield a positive product; two numbers with opposite signs yield a negative product. It also discusses how multiplication is implied in algebra without an explicit operation symbol between terms.
The document discusses the procedure of cross multiplication. It explains that cross multiplication can be used to rewrite ratios involving fractions as ratios of whole numbers. This is done by writing the fractions as ratios, then multiplying the denominators diagonally to obtain two new numbers. The ratio between these new numbers represents the original fractional ratio using whole numbers. An example demonstrates taking a ratio of 3/4 cups sugar to 2/3 cups flour and rewriting it as 9:8 cups sugar to flour using cross multiplication. The document also notes cross multiplication can be used to compare two fractions, with the larger product corresponding to the larger fraction.
The document discusses finding the least common multiple (LCM) of numbers. It defines the LCM as the smallest number that is a multiple of all the given numbers. It provides examples of finding the LCM by listing multiples and by constructing it from prime factorizations. The preferred method when the LCM is large is to construct it by fully factorizing each number into prime factors and taking the highest power of each prime factor.
The document discusses rules for multiplying fractions. It states that to multiply fractions, one should multiply the numerators and multiply the denominators, canceling terms when possible. It then provides examples, such as multiplying 12/25 * 15/8, simplifying to 9/10. It also notes that word problems involving fractions of a quantity can often be solved by translating them into fraction multiplications.
The document defines fractions as numbers of the form p/q where p and q are whole numbers not equal to 0. Fractions represent parts of a whole, for example 3/6 of a pizza. The top number is the numerator and represents the number of parts, while the bottom number is the denominator and represents the total number of equal parts the whole was divided into. Calculations with fractions involve dividing the whole into the number of parts in the denominator and taking the number of parts indicated by the numerator. Whole numbers can be viewed as having a denominator of 1. Dividing by 0 is undefined in mathematics.
The document defines and explains prime factorization. It states that natural numbers can be divided into multiples and factors. A prime number is only divisible by 1 and itself. To find the prime factorization of a number, it is broken down into prime number factors. For example, the prime factorization of 12 is 2 * 2 * 3, as these are all prime numbers that multiply to 12. Finding the complete prime factorization involves writing the number as a product of only prime numbers.
The document discusses addition and subtraction. It defines addition as combining two quantities A and B to obtain a sum S, where A and B are the addends. To add two numbers, one lines them up vertically according to place value and adds the digits from right to left, carrying when necessary. Subtraction is defined as taking away or undoing an addition. To subtract, one lines numbers up vertically and subtracts the digits from right to left, borrowing when needed. Examples of adding and subtracting multi-digit numbers are provided.
This document provides an overview of the topics covered in an Algebra-Ready placement test, including numbers and arithmetic operations with fractions, decimals, percentages, and signed numbers; algebra basics like expressions, equations, exponents and variables; and geometry concepts involving basic shapes, the Pythagorean theorem, and coordinate systems. Fractions covered include arithmetic operations, LCM, LCD, addition/subtraction, and cross-multiplication. Signed numbers include addition, subtraction, multiplication, division and order of operations. Decimals cover addition, subtraction, multiplication, division and conversions between decimals and fractions.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Training: ISO/IEC 27001 Information Security Management System - EN | PECB
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Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
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 ...
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.
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.
2. A right triangle is a triangle with a right angle as one of its
angle.
Pythagorean Theorem and Square Roots
3. A right triangle is a triangle with a right angle as one of its
angles. The longest side C of a right triangle is called the
hypotenuse,
Pythagorean Theorem and Square Roots
hypotenuse
C
4. A right triangle is a triangle with a right angle as one of its
angles. The longest side C of a right triangle is called the
hypotenuse, the two sides A and B forming the right angle
are called the legs.
Pythagorean Theorem and Square Roots
hypotenuse
legs
A
B
C
5. A right triangle is a triangle with a right angle as one of its
angles. The longest side C of a right triangle is called the
hypotenuse, the two sides A and B forming the right angle
are called the legs.
Pythagorean Theorem
Given a right triangle as shown and A, B, and C
be the length of the sides, then A2 + B2 = C2.
Pythagorean Theorem and Square Roots
hypotenuse
legs
A
B
C
6. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
7. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
?
8. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.
9. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.
The wall and the ground form a right triangle,
hence by the Pythagorean Theorem
we have that h2 + 32 = 52
10. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.
The wall and the ground form a right triangle,
hence by the Pythagorean Theorem
we have that h2 + 32 = 52
h2 + 9 = 25
11. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.
The wall and the ground form a right triangle,
hence by the Pythagorean Theorem
we have that h2 + 32 = 52
h2 + 9 = 25
–9 –9
subtract 9
from both sides
12. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.
The wall and the ground form a right triangle,
hence by the Pythagorean Theorem
we have that h2 + 32 = 52
h2 + 9 = 25
–9 –9
h2 = 16
subtract 9
from both sides
13. Pythagorean Theorem
Given a right triangle
with labeling as shown,
then A2 + B2 = C2
Pythagorean Theorem and Square Roots
Pythagorean Theorem allows us
to compute a length, i.e. a distance,
without measuring it directly.
Example A. A 5–meter ladder leans
against a wall as shown. Its base is
3 meters from the wall. How high is
the wall?
5 m
3 m
? = h
Let h be the height of the wall.
The wall and the ground form a right triangle,
hence by the Pythagorean Theorem
we have that h2 + 32 = 52
h2 + 9 = 25
–9 –9
h2 = 16
By trying different numbers for h, we find that 42 = 16
so h = 4 or that the wall is 4–meter high.
subtract 9
from both sides
15. Pythagorean Theorem and Square Roots
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”,
Pythagorean Theorem and Square Roots
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
17. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
18. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
#
19. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Note that both +4 and –4, when squared, give 16.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
#
20. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
#
21. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
#
22. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x.
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
#
23. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
#
24. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Example A.
a. Sqrt(16) =
c.3 =
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 =
d. –3 =
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
#
25. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Example A.
a. Sqrt(16) = 4
c.3 =
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 =
d. –3 =
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
#
26. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Example A.
a. Sqrt(16) = 4
c.3 =
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. –3 =
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
#
27. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Example A.
a. Sqrt(16) = 4
c.3 = 1.732.. by calculator
or that 3 ≈ 1.7 (approx.)
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. –3 =
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
#
28. We also state this relation as “the square–root of 16 is 4”,
i.e. 4 is the source for output “16”, and it’s written as 16 = 4:
Example A.
a. Sqrt(16) = 4
c.3 = 1.732.. by calculator
or that 3 ≈ 1.7 (approx.)
Pythagorean Theorem and Square Roots
Definition: If a2 → x and a is not negative, then a is called the
square root of x. This is written as sqrt(x) = a, or x = a.
b. 1/9 = 1/3
d. –3 = doesn’t exist (why?),
and the calculator returns “Error”.
Note that both +4 and –4, when squared, give 16. But we
designate the “square root of 16” i.e. 16 or sqrt(16) to be +4.
We refer “–4” as the “negative of the square root of 16”.
Square Root
From example A, we encountered that “the square of 4 is16”:
4 16
(#)2
16 = 4 16
#
30. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
31. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table.
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
32. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
33. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
34. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
35. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Since 30 is about half way
between 25 and 36,
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
36. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Since 30 is about half way
between 25 and 36,
so we estimate that30 5.5.
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
37. 0 02 = 0 0 = 0
1 12 = 1 1 = 1
2 22 = 4 4 = 2
3 32 = 9 9 = 3
4 42 = 16 16 = 4
5 52 = 25 25 = 5
6 62 = 36 36 = 6
7 72 = 49 49 = 7
8 82 = 64 64 = 8
9 92 = 81 81 = 9
10 102 = 100 100 = 10
11 112 = 121 121 = 11
We may estimate the sqrt
of other small numbers using
this table. For example,
25 < 30 < 36
hence
25 < 30 <36
or 5 < 30 < 6
Since 30 is about half way
between 25 and 36,
so we estimate that30 5.5.
In fact 30 5.47722….
Pythagorean Theorem and Square Roots
Following are the square numbers and square-roots that one
needs to memorize. These numbers are special because
many mathematics exercises utilize square numbers.
38. Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
39. Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
40. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
we are to find the hypotenuse,
41. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
we are to find the hypotenuse,
so 122 + 52 = c2
42. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
43. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
44. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.
45. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
Example B.
Find the missing side of the following right triangles.
b. a = 5, c = 12,
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.
46. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
so 52 + b2 = 122
Example B.
Find the missing side of the following right triangles.
b. a = 5, c = 12, we are to find a leg,
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.
47. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
so 52 + b2 = 122
25 + b2 = 144
Example B.
Find the missing side of the following right triangles.
b. a = 5, c = 12, we are to find a leg,
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.
48. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
so 52 + b2 = 122
25 + b2 = 144
b2 = 144 – 25 = 119
Example B.
Find the missing side of the following right triangles.
b. a = 5, c = 12, we are to find a leg,
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.
49. a. We have the legs a = 5, b = 12,
Pythagorean Theorem and Square Roots
Depending on which is the missing side, there are two versions
of calculation based on the Pythagorean Theorem –
finding the hypotenuse versus finding a leg.
so 52 + b2 = 122
25 + b2 = 144
b2 = 144 – 25 = 119
Hence b = 119 10.9.
Example B.
Find the missing side of the following right triangles.
b. a = 5, c = 12, we are to find a leg,
we are to find the hypotenuse,
so 122 + 52 = c2
144 + 25 = c2
169 = c2
Hence c = 169 = 13.
51. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
Rational and Irrational Numbers
2
1
1
52. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.
53. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.
Hence these numbers are called irrational (non–ratio)
numbers.
54. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.
Hence these numbers are called irrational (non–ratio)
numbers. Most real numbers are irrational, not fractions, i.e.
they can’t be represented as ratios of two integers.
55. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.
Hence these numbers are called irrational (non–ratio)
numbers. Most real numbers are irrational, not fractions, i.e.
they can’t be represented as ratios of two integers. The real
line is populated sparsely by fractional locations.
56. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.
Hence these numbers are called irrational (non–ratio)
numbers. Most real numbers are irrational, not fractions, i.e.
they can’t be represented as ratios of two integers. The real
line is populated sparsely by fractional locations. The
Pythagorean school of the ancient Greeks had believed that
all the measurable quantities in the universe are fractional
quantities. The “discovery” of these extra irrational numbers
caused a profound intellectual crisis.
57. The number 2 is the length of the
hypotenuse of the right triangle as shown.
Pythagorean Theorem and Square Roots
Rational and Irrational Numbers
2
1
1
It can be shown that 2 can not be
represented as a ratio of whole numbers i.e.
P/Q, where P and Q are integers.
Hence these numbers are called irrational (non–ratio)
numbers. Most real numbers are irrational, not fractions, i.e.
they can’t be represented as ratios of two integers. The real
line is populated sparsely by fractional locations. The
Pythagorean school of the ancient Greeks had believed that
all the measurable quantities in the universe are fractional
quantities. The “discovery” of these extra irrational numbers
caused a profound intellectual crisis. It wasn’t until the last two
centuries that mathematicians clarified the strange questions
“How many and what kind of numbers are there?”
58. Pythagorean Theorem and Square Roots
x
3
4
Exercise C. Solve for x. Give the square–root answer and
approximate answers to the tenth place using a calculator.
1.
4
3
x2. x
12
53.
x
1
14. 2
1
x5. 6
x
6.
10
1. sqrt(0) = 2. 1 =
Exercise A. find the following square–root (no calculator).
3. 25 3. 100
5. sqrt(1/9) = 6. sqrt(1/16) = 7. sqrt(4/49)
Exercise A. Give the approximate answers to the tenth place
using a calculator.
1. sqrt(2) = 2. 3 = 3. 10 3. 0.6