Module 01 Mathematics PDF

Module 01 MATHEMATICS Pag.  Module 1 – Mathematics Copyright © 2020 by Aviotrace Swiss SA All rights reserved. No part of this publication may be reproduced, distributed, or transmitt...

Module 01 MATHEMATICS Pag.  Module 1 – Mathematics Copyright © 2020 by Aviotrace Swiss SA All rights reserved. No part of this publication may be reproduced, distributed, or transmitted in any form or by any means, including photocopying, recording, or other electronic or mechanical methods, without the prior written permission of the publisher. 01.04.2020 Ed2 Pag. 2  Table of contents 1.1 Arithmetic 1.2 Algebra 1.3 Geometry 01.04.2020 Ed2 Pag. 3  Module 1 – Mathematics Chapter 01.01 ARITHMETIC 01.04.2020 Ed2 Pag. 4  Module 1 – Mathematics Arithmatical terms and signs The study of mathematics starts with arithmetic, which represents its foundations. Arithmetic allows us to know the properties of numbers and of the four main operations: addition, subtraction, multiplication, and division. Natural numbers are the most common numbers we know, those we use to count any set of objects. This set of numbers, including zero, contains all the non-negative integers. The set of natural numbers is indicated by the symbol N. Adding or multiplying two natural numbers the result is always a natural number. In case of subtraction, it is also possible to use negative numbers. The so created set that enclose integers positive numbers, negative numbers, plus zero, is called set of integers or set of relative numbers. The set of integers is indicated by the symbol Z. In case of division, it is also possible to introduce fractional numbers and the so created set that enclose integers plus fractional numbers, is called set of rational numbers. The set of rational numbers is indicated by the symbol Q. 01.04.2020 Ed2 Pag. 5  Module 1 – Mathematics Addition The four basic math operations with numbers are: addition, subtraction, multiplication, and division. They are indicated by the following symbols: plus (+) is the addition symbol, minus (-) is the subtraction symbol, multiplied by (x) is the multiplication symbol and divided (÷) or over (/) is the division symbol. Addition is the operation that combines two generic numbers, called addends, and associates to them a third number, called sum or total, obtained counting as many units as those indicated by the second addend after the first addend. The addition has commutative, associative, and dissociative properties: Commutative property : changing the order of the addends the sum does not change. 23 + 5 + 1700 = 1700 + 23 + 5 = 1728 01.04.2020 Ed2 Pag. 6  Module 1 – Mathematics Addition Associative property: replacing two or more addends with their sum the result does not change 127 + 3 + 40 + 20 = 127 + 3 + 40 + 20 = 130 + 60 = 190 Dissociative property: replacing one addend with one or more addend whose sum is the replaced addend the result of the addition does not change 57 + 22 = 50 + 7 + 20 + 2 = 79 01.04.2020 Ed2 Pag. 7  Module 1 – Mathematics Multiplication Multiplication is the operation by which two generic numbers, called factors or multiplicand and multiplier, are associated to a third one, called product, obtained adding the units of the first one as many times as the units of the second one. Multiplication has commutative, associative, dissociative and distributive properties : Commutative property: changing the order of factors the product does not change 5 ∗ 25 = 25 ∗ 5 = 125 01.04.2020 Ed2 Pag. 8  Module 1 – Mathematics Multiplication Associative property: replacing two or more factors with their product the result does not change 5ถ ∗ 2 ∗ 3ถ ∗ 9 = 5 ∗ 2 ∗ 3 ∗ 9 = 10 ∗ 27 = 270 Dissociative property: replacing one factor with one or more factors whose product is the replaced factor the result of the multiplication does not change 25 ∗ 14 = 25 ∗ 2 ∗ 7 = 350 01.04.2020 Ed2 Pag. 9  Module 1 – Mathematics Multiplication Distributive property: to multiply the terms of an addition or of a subtraction it is possible to calculate the final product of the given factor for each single term of the addition or subtraction, and then sum or subtract them 6 ∗ 10 + 4 = 6 ∗ 10 + 6 ∗ 4 = 84 4 ∗ 2 + 5 = 8 + 20 = 28 6+4 ∗ 3+5 = 6∗3 + 6∗5 + 4∗3 + 4∗5 = 18 + 30 + 12 + 20 = 80 01.04.2020 Ed2 Pag. 10  Module 1 – Mathematics Multiplication: graphical method 01.04.2020 Ed2 Pag. 11  Module 1 – Mathematics Multiplication: examples 165 ∗ 3 = 100 ∗ 3 + 60 ∗ 3 + 5 ∗ 3 = 300 + 180 + 15 = 495 247 ∗ 58 = 200 + 40 + 7 ∗ 50 + 8 = 200 ∗ 50 + 200 ∗ 8 + 40 ∗ 50 + 40 ∗ 8 + 7 ∗ 50 + 7 ∗ 8 = 10000 + 1600 + 2000 + 320 + 56 = 14326 01.04.2020 Ed2 Pag. 12  Module 1 – Mathematics Subtraction Subtraction is the operation of taking two generic numbers, called minuend and subtrahend, and associating a third one, called difference, obtained subtracting from the minuend as many units as those indicated by the subtrahend. Subtraction is characterized by invariance: adding or subtracting the same number from the two terms of a subtraction the result does not change. 148 − 18 = 130 ⟹ 148 + 2 − 18 + 2 = 130 01.04.2020 Ed2 Pag. 13  Module 1 – Mathematics Division The number that is divided is called dividend, while the other number that divides it is called divisor. The result of division is called quotient. It represents the number of times that the divisor is “contained” in the dividend. In some cases, the quotient can have a remainder. It represents the part of the dividend that cannot be divided by the divisor. Division is characterized by invariance and distributive property: Invariance: dividing or multiplying by the same number the two terms of a division the result does not change 150 ÷ 50 = 3 ⟹ 150 ÷ 10 ÷ 50 ÷ 10 = 3 Distributive property: to divide the terms of an addition or of a subtraction by a number it is possible to divide each single term of the addition or of the subtraction by the given divisor and then sum or subtract them. 150 + 2 ÷ 25 = 7 ⟹ 150 ÷ 25 + 25 ÷ 25 = 7 01.04.2020 Ed2 Pag. 14  Module 1 – Mathematics Arithmetic expression An arithmetic expression is a set of two or more numbers separated by operation signs and possibly by the necessary brackets. Every time you have to solve an expression it is necessary to do the different operations on the numbers in a specific order. The sequence of operations is: 1. Do the operations indicated inside the brackets. 2. Do multiplications and divisions. 3. Do additions and subtractions. 01.04.2020 Ed2 Pag. 15  Module 1 – Mathematics Fraction and decimals The result of a division of two integers is a fraction. A common fraction represents a portion or a part of a given quantity. A fraction is made of two numbers. The number above the fraction symbol is called numerator, while the number under it is called denominator. 𝑎 𝑥= 𝑏 01.04.2020 Ed2 Pag. 16  Module 1 – Mathematics Proper and improper fraction When the numerator of a fraction is smaller than the denominator, the fraction is defined as proper fraction. Therefore, the value of a proper fraction is always less than one. 3 𝑥= Example of a proper fraction 5 If the numerator is greater than the denominator, the fraction is defined as improper. In this case the value of the fraction is more than one. 7 𝑥= Example of an improper fraction 5 When numerator and denominator are the same, the quotient of the fraction is one. 01.04.2020 Ed2 Pag. 17  Module 1 – Mathematics Fraction properties The addition of two fractions can be done only when the two fractions have the same denominator. In this case it is sufficient to add the numerators to obtain the sum and then the fraction can be reduced to its lowest terms. When the fractions do not have the same denominator it is necessary to find the lowest common denominator. The lowest common denominator is the least common multiple of the factors of the denominator. The product of fractions can be calculated by multiplying the numerators of each fraction to obtain the product numerator and then multiplying denominators among them to obtain the product denominator. The value of a fraction does not change if the same operation (multiplication or division) is done both on the numerator and the denominator. This property can be used to simplify multiplications between fractions. 01.04.2020 Ed2 Pag. 18  Module 1 – Mathematics Fraction properties Applying the invariance property of division is possible to define a fundamental law of fractions: multiplying and or dividing by the same number both the numerator and the denominator of a fraction, its value does not change. This property allows simplifying fractions made of great numbers, thanks to the so-called reduction to lowest terms. 01.04.2020 Ed2 Pag. 19  Module 1 – Mathematics Decimal numbers To work with fractions is not always so easy; for this reason it is often better to replace them with decimal numbers. A common fraction can be converted into a decimal number, simply dividing the numerator by the denominator. 3 = 0,75 4 The first number after the decimal point denotes tens, the second hundreds and the third thousands. 01.04.2020 Ed2 Pag. 20  Module 1 – Mathematics Decimal numbers properties The addition of decimal numbers follows the same laws of integers after having aligned on the same vertical line the points indicating the decimal number. 5,5 + 4,4 = 9,9 Multiplying decimal numbers, the decimal point is ignored at first and the resulting integers are multiplied. After having calculated the product, the overall number of figures to the right of the point of the multiplier and of the multiplicand is counted: this number represents the number of figures to the left of which the decimal point must be put in the product. 0,5 ∗ 0,03 = 0,015 01.04.2020 Ed2 Pag. 21  Module 1 – Mathematics Factors and multiples Multiplication and division allow us to introduce the concepts of multiple and divisor of a number. An integer A is multiple of another integer B if there is a third integer C that multiplied by B gives as result A: 𝐴=𝐶∗𝐵 An integer A is divisor of a number B only if dividing B by A the quotient obtained is an integer C and the remainder is zero: 𝐵÷𝐴 =𝐶 with reminder = 0 Prime numbers are those numbers that cannot be divided by any number except themselves and one. For example: 1, 3, 5, 7… A number that is not a prime number is called composite. 01.04.2020 Ed2 Pag. 22  Module 1 – Mathematics Factors and multiples There are some useful rules to understand the divisors of a number looking at its figures A number is divisible by 2 when ends with 0,2,4,6,8. A number is divisible by 3 if the sum of its figures is divisible by 3. A number is divisible by 4 if the last two digits are 00, or if they form a number that is multiple of 4, or if the second last figure is odd and the last one is 2 or 6, or if the second last figure is even and the last one is 0,4,8. A number is divisible by 5 if its last figure is 0 or 5. 01.04.2020 Ed2 Pag. 23  Module 1 – Mathematics Factors and multiples Factorization, or prime decomposition, is the process by which the prime numbers that are divisors of a given number are searched. To decompose into prime numbers a number, it is necessary to divide it by its smallest prime divisor and to continue this way until you obtain 1 as quotient. Given two or more numbers a, b, c the smallest multiple in common is called least common multiple. Given two or more numbers a, b, c the largest common divisor share by them is called greatest common divisor. 01.04.2020 Ed2 Pag. 24  Module 1 – Mathematics Weights Mass measures are used to find the quantity of matter contained in bodies. The unit of measurement of International System is the gram (g). In the British system the pound is the main unit to measure the mass (1 pound = 0,453 kg). Weights can be obtained doing a multiplication: 𝑤𝑒𝑖𝑔ℎ𝑡 = 𝑚𝑎𝑠𝑠 ∗ 𝑔 Where “g” is the gravity acceleration = 9,81 m/s2 01.04.2020 Ed2 Pag. 25  Module 1 – Mathematics Measures and conversion factors Units of measurement are a standard for the measuring of physical quantities. It is necessary to define standard systems of measurement with the aim to facilitate the measuring: however in the world there are different units of measurement. There are different systems of measurement that are officially accepted; these systems are based on different set of fundamental units. 01.04.2020 Ed2 Pag. 26  Module 1 – Mathematics Measures and conversion factors The International System is the most used and it has 7 main units: Meter: length. Kilogram: mass. Second: time. Ampere: intensity of current. Kelvin: temperature. Mole: quantity of material. Candle: luminous intensity. Other systems, which are used in the aeronautical world, are the British and the American systems. A conversion of the units of different systems is the comparison between all standards values. 01.04.2020 Ed2 Pag. 27  Module 1 – Mathematics Measures and conversion factors The following tables show main conversion factors. 1𝑚 = 0,001𝑘𝑚 = 39,37𝑖𝑛 = 3,28𝑓𝑡 = 1,09𝑦𝑑 1𝑚2 = 10000𝑐𝑚2 1𝑃𝑎 = 0,01𝑏𝑎𝑟 1𝑖𝑛 = 0.0254𝑚 1𝑦𝑑 = 0.917𝑚 1𝑓𝑡 = 0,305𝑚 1𝑘𝑛𝑜𝑡 = 1,148𝑚𝑝ℎ The scale is the ratio between the actual distances represented on a map, that is the actual kilometres, and the material distances on the map, that is the centimetres or millimetres on the sheet 01.04.2020 Ed2 Pag. 28  Module 1 – Mathematics Scale, latitude and longitude The latitude of a place is its north or south distance from the Equator calculated in degrees, minutes and seconds. Starting from the Equator, the distance to the north is called north latitude, while the one to the south is called south latitude. The longitude of a place is its east or west distance from a meridian of reference calculated in degrees, minutes and seconds. The distance to the east, starting from the meridian of reference, is called east longitude, while the one to the west, always from the meridian of reference, is called west longitude. By international convention, the meridian of reference, that is 0 degrees of longitude, is the one crossing the Greenwich observatory, an area of the city of London in England. 01.04.2020 Ed2 Pag. 29  Module 1 – Mathematics Ratio and proportion Ratio is a method to compare one number to another one. The ratio of two numbers A and B, with B different from zero, is the quotient. A divided by B can also be expressed with the fraction A/B. In a ratio A/B, the numbers A and B are called terms of the ratio. Using again the properties of division it is possible to state that: multiplying or dividing both terms of a ratio by the same number, different from zero, the ratio remains the same. 01.04.2020 Ed2 Pag. 30  Module 1 – Mathematics Ratio and proportion A proportion is an equivalence relation between two ratios and is an easier method to solve problems with ratios. The first and the third term of the proportion are called antecedents, while the second and the fourth are called consequents. Moreover, the first and the last term of the proportion are called extremes, while the second and the third terms are called means. The fundamental property of proportions is the following: in a proportion, the product of the means is equal to the product of the extremes. 3 ∗ 80 3 ∶ 4 = 𝑥 ∶ 80 ⟹ 𝑥= = 60 4 01.04.2020 Ed2 Pag. 31  Module 1 – Mathematics Ratio and proportion properties From the fundamental property it is possible to derive other useful properties for determining an unknown term in a proportion: Property of permuting: in a proportion, exchanging the means between them, or the extremes, the result is a new proportion. Property of inverting: in a proportion, exchanging each antecedent with its consequent the result is a new proportion. 01.04.2020 Ed2 Pag. 32  Module 1 – Mathematics Percentages Percentages are often used to express a part of an integer in an easier way. The symbol that indicates percentages is %. Two fundamental definitions are useful to better understand the meaning of percent: Directly proportional quantity Inversely proportional quantities. 01.04.2020 Ed2 Pag. 33  Module 1 – Mathematics Proportionality Two variables x and y are said to be proportional (or directly proportional) if there is a functional relationship of the form: 𝑥 = 𝑘𝑦 where k is a not null numerical constant Two variables x and y are said to be inversely proportional if there is a functional relationship of the form: 𝑘 𝑦= where k is a not null numerical constant 𝑥 01.04.2020 Ed2 Pag. 34  Module 1 – Mathematics Perimeter, area and volumes The perimeter is the measure of the length of the contour of a plane figure. To find the perimeter of a polygon it is necessary to sum its sides. The area is the measure of the extension of a two-dimensional segment of space that is the measure of a surface. The area of a surface is a two- dimension number, height and width, and is often expressed by surface measures, like “square” meters. 01.04.2020 Ed2 Pag. 35  Module 1 – Mathematics Triangle A triangle is a polygon with three sides and three angles. The angle vertices are called vertices of the triangle. The fundamental property of triangles is the following: the sum of angles in a triangle is always 180 degrees. Each side of a triangle is a base (b) and the segment of orthogonal line from that base to the opposite vertex is a height (h). 01.04.2020 Ed2 Pag. 36  Module 1 – Mathematics Area of triangle The area of a triangle is given by the product of the base (a) by the height divided by 2 : ah A= 2 01.04.2020 Ed2 Pag. 37  Module 1 – Mathematics Area of triangle The main kinds of triangle are : 1. Scalene triangle, which has three unequal sides and three unequal angles. 2. Isosceles triangle, which has two equal sides and two equal angles. 3. Equilateral triangle, which has three equal sides and three equal angles. A special kind of triangle is the right triangle, which has an angle of 90 degrees, which is a right angle. An obtuse triangle is a triangle with an angle greater than 90 degrees. An acute triangle is a triangle in which all angles are smaller than 90 degrees 01.04.2020 Ed2 Pag. 38  Module 1 – Mathematics Triangle: Pitagora’s theorem In a right triangle the square of the hypotenuse (the side opposite the right angle)is equal to the sum of the squares of the others two sides. The theorem can be written as an equation relating the lengths of the side a, b, and c, often called the “Pythagorean equation”. 𝑎2 + 𝑏 2 = 𝑐 2 01.04.2020 Ed2 Pag. 39  Module 1 – Mathematics Triangle: Pitagora’s theorem Example: Given: 𝑎=3 b=4 Which is the value of c? Remembering the Pythagorean equation 𝑎2 + 𝑏 2 = 𝑐 2 𝑐= 𝑎2 + 𝑏 2 = 32 + 42 = 9 + 16 = 25 = 5 01.04.2020 Ed2 Pag. 40  Module 1 – Mathematics Parallelogram: rectangle A parallelogram is a polygon in which the opposite sides are parallel and equal in length. The area of the parallelogram is given by the multiplication of the length or base (a) of the parallelogram by the height (h). 𝐴 =𝑎∗ℎ A rectangle is a parallelogram having all right angles. It is a quadrilateral with opposite sides of equal lengths and four right angles. The area of the rectangle can be found multiplying its base by its height. 01.04.2020 Ed2 Pag. 41  Module 1 – Mathematics Parallelogram: square and rhombus A square is a special parallelogram with four sides of equal length. The area of the square is given by the same formula used for the parallelogram. Since all sides of the square have the same length, the area of the square is given by the square of the side (l): 𝐴 = 𝑙 ∗ 𝑙 = 𝑙2 The rhombus is a special parallelogram with four sides of the same length. This polygon has two orthogonal diagonals, a long diagonal (d1) and a short one (d2). The area of the rhombus is given by the product of the diagonals divided by 2: 𝑑1 ∗ 𝑑2 𝐴= 2 01.04.2020 Ed2 Pag. 42  Module 1 – Mathematics Parallelogram: Trapezium The trapezium is a quadrilateral with two parallel sides. The two parallel sides are also called bases of the trapezium: one is the greater base (B) and the other is the smaller base (b). The other two sides are called oblique sides or simply sides. If the two oblique sides are congruent, the trapezium is called isosceles trapezium. If one of the two non-parallel sides is perpendicular to the base, the trapezium is called right trapezium. Its area is given by : 𝐵+𝑏 ∗ℎ 𝐴= 2 01.04.2020 Ed2 Pag. 43  Module 1 – Mathematics Circle The polygon consisting of all the points of a circumference and of the internal points is called circle, whose circumference is the contour. It is a set of points in a plane equidistant from a given point from a circumference. The given point is called center and the distance from the center is the radius (r) of the circumference. Each segment passing through the center of a circumference having its extremes on the circumference is called diameter of the circumference. The ratio of the circumference and the diameter (d) of a circle are always equal to a fixed value indicated by the Greek character π (π = 3,14). 01.04.2020 Ed2 Pag. 44  Module 1 – Mathematics Circle The circumference of a circle can be calculated multiplying by π the diameter: 𝑐𝑖𝑟 = 𝜋 ∗ 𝑑 The area of the circle is given by the square of the radius multiplied by π: 𝐴 = 𝜋 ∗ 𝑟2 01.04.2020 Ed2 Pag. 45  Module 1 – Mathematics Volume: prism and cube The volume or capacity is the amount of space occupied by a body. The volume of a solid body is a numerical value used in describing the tridimensional space occupied by the body. A prism whose bases are two parallelograms is called parallelepiped. The volume of a parallelepiped with rectangular base is given by the product of the three dimensions, width (l), depth (p) and height (h): 𝑉 =𝜌∗𝑙∗ℎ A cube is a parallelepiped in which the bases are squares. The volume of a cube is given, as in the case of the rectangle parallelepiped, by the product of the three dimensions. The three dimensions of a cube are equal (l); therefore the volume corresponds to one cubed dimension: 𝑉 = 𝑙3 01.04.2020 Ed2 Pag. 46  Module 1 – Mathematics Volume: pyramid A solid delimitated by a polygon and laterally having as many triangles as the number of the sides of the base polygon is called pyramid. The polygon delimitating the pyramid is called base and the side triangles are the faces of the pyramid. All the triangles share the same vertex, called vertex of the pyramid. The height (h) is the distance between the vertex and the base. The volume of a pyramid is calculated multiplying the area of the base by the height of the pyramid and then the product is divided by 3: 𝐴∗ℎ 𝑉= 3 01.04.2020 Ed2 Pag. 47  Module 1 – Mathematics Volume: cylinder The cylinder is a solid created by the complete rotation of a rectangle around the straight line of one of its sides. This straight line is the rotation axis and the side taken into consideration for the revolution is the height (h) of the cylinder. The parallel side designs the surface of the cylinder and the other two sides are the radius of the cylinder and create the two base surfaces. The volume of a cylinder can be calculated multiplying the area of a base by the height of the cylinder: 𝑉 =𝐴∗ℎ 01.04.2020 Ed2 Pag. 48  Module 1 – Mathematics Volume: cone A cone is a solid created by the complete rotation of a right triangle around one leg whose straight line is the rotation axis. This leg is the height of the cone. The hypotenuse of this triangle designs the side surface. The other leg is the radius of the cone. The volume of a cone can be calculated multiplying the cone bases area by the height (h), the product is then divided by 3: 𝐴∗ℎ 𝑉= 3 01.04.2020 Ed2 Pag. 49  Module 1 – Mathematics Volume: sphere A sphere is a solid created by the rotation of a semicircle around its diameter. The semi-circumference that limits the semicircle creates the surface of the sphere. The surface of the sphere is where a set of point in space are equidistant from a given point called center. The distance of the center from whatever point of the sphere is called radius (r) and all the meridians of the surface are circumferences. The surface of a sphere can be calculated multiplying four times the square of its radius by π: 𝑆 = 4𝜋 ∗ 𝑟 2 The volume of the sphere is given by the cubed radius of the sphere multiplied by π. The result obtained is then multiplied by (4/3): 4 𝑉 = 𝜋 ∗ 𝑟3 3 01.04.2020 Ed2 Pag. 50  Module 1 – Mathematics Squares The square of a number n is the raise of the same number to the second power, which is a multiplication of the number by itself: 𝑛2 = 𝑛 ∗ 𝑛 ⟹ 32 = 9 The square of a number has some properties: The square of a real number is always greater (or equal) than 0 The square of any integer (n) can be represented by the sum: 1 + 1 + 2 + 2 + ⋯+ 𝑛 − 1 + 𝑛 − 1 + 𝑛 The square of any integer (n) is equal to the sum of n prime odd numbers: 42 = 1 + 3 + 5 + 7 = 16 01.04.2020 Ed2 Pag. 51  Module 1 – Mathematics Cubes and square roots The cube of a number n is the raise of the same number to the third power, which is the multiplication of the number for three times: 𝑛3 = 𝑛 ∗ 𝑛 ∗ 𝑛 ⟹ 23 = 2 ∗ 2 ∗ 2 = 8 𝑛 The symbol used for root extraction is 3. The number under the root symbol is called radicand, while the number that represents the order of the root is called exponent (n). The function of the root is to find the number that, multiplied by itself for a number of times equal to the value of the exponent, has the radicand as result: 4=2 The exponent of the root must always be indicated unless it is two. An irrational number is indicated when the number cannot be expressed as a ratio of integers. 01.04.2020 Ed2 Pag. 52  Module 1 – Mathematics Chapter 01.02 ALGEBRA 01.04.2020 Ed2 Pag. 53  Module 1 – Mathematics Algebra Using zero as starting number, a positive value is assigned to all numbers greater than zero and a negative value to those less than zero. In the line of relative numbers, negative values are indicated by the sign minus while positive values are indicated by the sign plus or by the absence of any sign. Each writing containing operations to be done on relative numbers is called algebraic expression. To calculate the value of an algebraic expression means finding the relative number that represents the result of the indicated operations. To calculate the value of an algebraic expression it is necessary to follow the following laws: if the expression does not contain any bracket, it is necessary to do powers first, then multiplications and divisions and finally additions and subtractions; if the expression contains brackets, first eliminate the inner ones, then the external ones. 01.04.2020 Ed2 Pag. 54  Module 1 – Mathematics Addition When summing two or more numbers having the same sign, the sign is ignored and the sum of values is calculated, then the sign common to the values is added before of the result. In other terms, adding two or more positive numbers the sum is a positive number, while adding two or more negative numbers their sum is always a negative number. Instead, when adding positive and negative numbers, the two numbers are subtracted and then the positive or negative sign of the greatest number is added. The result obtained adding or subtracting numbers with a sign, that is relative numbers, is called algebraic sum of the numbers. 01.04.2020 Ed2 Pag. 55  Module 1 – Mathematics Multiplication and division The multiplication of relative numbers follows the same laws of the multiplication of generic numbers. After having done the multiplication, the product takes the sign established by the following 3 laws: The product of two positive numbers is always positive The product of two negative numbers is always positive The product of a positive and a negative number is always negative. As in the case of multiplication, the division of relative numbers follows the same laws of the division of generic numbers. The sign of the quotient is determined by the same laws used for the multiplication: The quotient of two positive numbers is always positive The quotient of two negative numbers is always positive The quotient of a negative and a positive number is always negative. 01.04.2020 Ed2 Pag. 56  Module 1 – Mathematics Use of brackets Brackets are used in mathematics to group the terms on which the same operation must be done and to define priorities with reference to some operations. Brackets are always used in pairs of the same kind. Here is the increasing hierarchical order of the brackets used in arithmetic: round brackets 3 , square brackets 3 , and braces 3. The first operations to be done are those indicated between the inner brackets. 01.04.2020 Ed2 Pag. 57  Module 1 – Mathematics Powers It is defined as power of a number A raised to the exponent n, the product of n factors equal to A: 𝐴𝑛 = 𝐴 ∗ 𝐴 ∗ 𝐴 ∗ 𝐴 ∗ ⋯ 𝑛 − 𝑡𝑖𝑚𝑒𝑠 This power is indicated by the symbol A to the nth. The number A is called base of the power. Let’s define the main properties of powers: Whatever number raised to the power of zero, is always equal to one, with the exception of zero that remains zero (or more precisely undetermined). The multiplication of two powers having the same base is a power with the same base and having as exponent the sum of the exponents. The quotient of two powers having the same base, is a power with the same base and as exponents the difference of exponents. The power of a power is a power having the same base and product of the exponents as exponent. 01.04.2020 Ed2 Pag. 58  Module 1 – Mathematics Negative power There are also powers with negative exponents (-n): 1 𝑎−𝑛 = 𝑛 𝑎 The definition of a power permits to use fractional exponents (x/y), where x and y are prime number between them. 𝑥 𝑦 𝑎𝑦 = 𝑎𝑥 01.04.2020 Ed2 Pag. 59  Module 1 – Mathematics Negative power Examples: 22 ∗ 24 2 2+4 26 = = 5=2 25 25 2 2 𝑥5 ∗ 𝑥4 2 = 𝑥 5+4 = 𝑥9 2 = 𝑥 18 −2 2 2 5 52 25 = = 2= 5 2 2 4 2 22 ∗ 23 2 = 2 3+2 = 210 01.04.2020 Ed2 Pag. 60  Module 1 – Mathematics Linear equation Equality between two algebraic expressions, containing one or more characters, is called equation. The characters in an equation are called unknown. In an equation, like in any other equality, the expressions to the left and to the right of the sign equal are called first and second member of the equation. Solving an equation means finding the set of its solutions. The set of solutions of an equation is a set of real numbers. In general, the following cases can be found: The set of solutions is empty. This means that replacing the unknown with whatever number, the equation would transform into a false equality. In this case the equation is defined as impossible. The set of solutions contains a finite number of elements that are the solutions of the equation. In this case the equation is determinate. The set of solutions contains an infinite number of elements that are the solutions of the equation. In this case the equation is indeterminate. 01.04.2020 Ed2 Pag. 61  Module 1 – Mathematics Linear equation To solve the equations we have to move all the unknown at the left member. When the unknown cross the equal symbol (=), we have to change the sign of the unknown. Then we have to sum the similar terms on both member of the equation. At the end we have to divide both terms of the equation for the coefficient of the unknown. 3𝑥 + 5 = 𝑥 − 2 ⟹ 3𝑥 − 𝑥 = −2 − 5 ⟹ 7 ⟹ 2𝑥 = −7 ⟹ 𝑥=− 2 01.04.2020 Ed2 Pag. 62  Module 1 – Mathematics Simultaneous equation Simultaneous equations (also known as system of equations) are a set of equations which have more than one unknown values. Questions involving simultaneous equations require finding the unknowns. First, it has to represent the equations in a clear form. Then we proceed with the following steps. There are generally two methods to solving simultaneous equations: By substitution. By elimination. 01.04.2020 Ed2 Pag. 63  Module 1 – Mathematics Simultaneous equation: substitution method For example let’s consider: 2𝑥 + 𝑦 = 5 ቊ 𝑥 + 2𝑦 = 7 In the method of substitution, it expresses “x” in terms of “y” in one equation (in this case the second) and substitute it into the other: 𝑥 = 7 − 2𝑦 So we obtain: 𝑦=3 ቊ 𝑥=1 01.04.2020 Ed2 Pag. 64  Module 1 – Mathematics Simultaneous equation: elimination method In the method of elimination, it can choose to eliminate x. To eliminate x, it multiplies the second equation by the same number of x in the first equation; so it cancels x by subtraction. For example let’s consider: 2𝑥 + 𝑦 = 5 ൜ ⟶ 𝑥 + 2𝑦 = 7 ⟶ 2 ∗ 𝑥 + 2𝑦 = 2 ∗ 7 ⟹ 2𝑥 + 4𝑦 = 14 𝑥 + 2𝑦 = 7 Now let’s subtract the two equations: 2𝑥 + 𝑦 = 5 − 2𝑥 + 4𝑦 = 14 ⟶ 2𝑥 − 2𝑥 + 𝑦 − 4𝑦 = 5 − 14 ⟹ 𝑦=3 Now substituting the value of y in the second equation, it arrives at the same conclusion: 𝑥=1 01.04.2020 Ed2 Pag. 65  Module 1 – Mathematics Second degree equation In mathematics a quadratic equation is an algebraic equation with one unknown “x”, which is present with two as the maximum degree 𝑥 2. In fact a quadratic equation is a polynomial equation of the second degree. The generic form is: 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 𝑤𝑖𝑡ℎ 𝑎 ≠ 0 “a” is called the quadratic coefficient, “b” is called the linear coefficient, and “c” is the constant term or free term. According to the fundamental law of algebra, the solutions of a quadratic equation are always 2. It is possible to distinguish two cases: In the set of real numbers the equation admits 2 solutions, which can be coincident, or none. In the set of complex numbers the equation admits always 2 solutions, which can be coincident. 01.04.2020 Ed2 Pag. 66  Module 1 – Mathematics Second degree equation A quadratic equation is called complete equation when all its coefficients are different from 0. It can be solved as follows: 1. All terms must be written at the first member, which is the one at the left of the equal symbol (=). 2. It must write the generic form of the equation. 3. It must apply the resolving formula: −𝑏 ± 𝑏 2 − 4𝑎𝑐 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 ⟶ 𝑥= 2𝑎 01.04.2020 Ed2 Pag. 67  Module 1 – Mathematics Second degree equation In mathematics there are also the incomplete quadratic equations. Both these equations can be solved according to the generic process and/or using faster methods. 𝑎𝑥 2 + 𝑏𝑥 = 0 It can be solved breaking out the first member: 𝑥 𝑎𝑥 + 𝑏 = 0 The solutions can be found putting all terms equal to 0: 𝑥=0 ; 𝑎𝑥 + 𝑏 = 0 𝑎𝑥 2 + 𝑐 = 0 The term “c” must be put at the second member and then it can be divided by “a”: −𝑐 −𝑐 𝑥2 = 𝑡ℎ𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑐𝑎𝑛 𝑏𝑒 𝑓𝑜𝑢𝑛𝑑 𝑑𝑜𝑖𝑛𝑔 𝑡ℎ𝑒 𝑠𝑞𝑢𝑎𝑟𝑒 𝑟𝑜𝑜𝑡 𝑥 = ± 𝑎 𝑎 01.04.2020 Ed2 Pag. 68  Module 1 – Mathematics Binary and other applicable numbering system Mathematics is based on numbers. In order to count a certain number of items first of all it is necessary to choose a notation. A notation is a set of symbols and rules for representing numbers. The binary notation is a positional notation that uses two figures to represent numbers: 0 and 1. All digital electronic devices are based on the binary notation. In order to understand the value of a binary figure, that is “translate” it into decimal notation, you just have to use the fundamental equation, keeping in mind that the different 𝑏 𝑛 factors in this case represent powers of 2. 101 = 1 ∗ 22 + 0 ∗ 21 + 1 ∗ 20 = 5 1001000 = 1 ∗ 27 + 0 ∗ 26 + 0 ∗ 25 + 1 ∗ 24 + 0 ∗ 23 + 0 ∗ 22 + 0 ∗ 21 + 0 ∗ 20 = 128 + 16 = 144 Example of the opposite operation from decimal system into binary: 42=101010 01.04.2020 Ed2 Pag. 69  Module 1 – Mathematics Binary and other applicable numbering system Scientific notation is a way of indicating a value as the product of a number between 1 and 9. 99 by a base ten. To express a number using the scientific notation, the number that must be transformed is multiplied by the power of ten, so many times as the number of figures after which the decimal point must be moved. The choice of the power to use is connected to the kind of quantities in use, but generally it is better to reduce all numbers to the unit. This operation does not change the value of the number, but only the way in which it is written. The first operation is writing down the numbers to multiply in scientific notation. 0,275 = 2,75 ∗ 10−1 30000 = 3 ∗ 104 01.04.2020 Ed2 Pag. 70  Module 1 – Mathematics Monomial and polynomials A monomial is a number, a variable or a product of a number and a variable. 𝑎2 , 5𝑎2 A polynomial is a sum of monomials where each monomial is called a term. 𝑎2 + 5𝑎5 + 27𝑎3 01.04.2020 Ed2 Pag. 71  Module 1 – Mathematics Monomial and polynomials Multiplication of two binomials 𝑎𝑥 + 𝑏 𝑐𝑥 + 𝑑 = 𝑎𝑥 ∗ 𝑐𝑥 + 𝑎𝑥 ∗ 𝑑 + 𝑏 ∗ 𝑐𝑥 + 𝑏 ∗ 𝑑 = 𝑎𝑐𝑥 2 + 𝑎𝑐𝑥 + 𝑏𝑐𝑥 + 𝑏𝑑 There are particular cases reported hereafter: 𝑎 + 𝑏 𝑎 − 𝑏 = 𝑎2 − 𝑏 2 𝑎+𝑏 2 = 𝑎2 + 2𝑎𝑏 + 𝑏 2 01.04.2020 Ed2 Pag. 72  Module 1 – Mathematics Polynomial factorization Factorization of polynomials or polynomial factorization refers to factoring a polynomial with coefficients in a given field or in the integers into irreducible factors with coefficients in same domain. Polynomial factorization is one of the fundamental tools of the computer algebra systems. 𝑎𝑏 + 𝑎𝑐 = 𝑎 𝑏 + 𝑐 𝑎 + 𝑏 𝑎2 = 𝑎+𝑏 𝑎 𝑎 𝑎 + 𝑏 𝑎2 𝑎3 𝑎7 = 𝑎 + 𝑏 𝑎𝑏 2 𝑐 6 𝑎𝑏𝑐 01.04.2020 Ed2 Pag. 73  Module 1 – Mathematics Logarithms: definition and properties Logarithm with z base of a number (x) is the exponent to which the base of the logarithm must be raised in order to produce the number: 𝑦 = log 𝑧 𝑥 → 𝑥 = 𝑧𝑦 The main proprieties of logarithms are: The logarithm of the product of two numbers is equal to the sum of the logarithms of the same numbers: log 𝑚 𝑎 ∗ 𝑏 = log 𝑚 𝑎 + log 𝑚 𝑏 01.04.2020 Ed2 Pag. 74  Module 1 – Mathematics Logarithms: definition and properties The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the same numbers: 𝑎 log 𝑚 = log 𝑚 𝑎 − log 𝑚 𝑏 𝑏 The logarithm of 1, with any base, is 0: log 𝑚 1 = 0 The logarithms of the inverse of “a” is the opposite of the logarithm of “a”: 1 log 𝑚 = − log 𝑚 𝑎 𝑎 The logarithms of a number raised to the “k” power is equal to the product of the exponent (k) and the number’s logarithm: log 𝑚 𝑎𝑘 = 𝑘 log 𝑚 𝑎 01.04.2020 Ed2 Pag. 75  Module 1 – Mathematics Logarithms: definition and properties The logarithm of k-root of “a” is equal to the quotient between the logarithm and k: 𝑘 1 log 𝑚 𝑎 = log 𝑚 𝑎 𝑘 The logarithm of “a”, with also “a” as base, is 1: log 𝑎 𝑎 = 1 The following identity is true: 𝑎 log𝑎 𝑥 = log 𝑎 𝑎 𝑥 = 𝑥 01.04.2020 Ed2 Pag. 76  Module 1 – Mathematics Logarithms Logarithms can be calculated with any positive base (different from 1); the bases generally used are: Base 10 (decimal logarithms): log10, log, Log Base “e” (natural logarithms): ln Base 2 (binary logarithms): log2 The logarithms are valid only when the argument is above 0. “Log 0” has no meaning (and no value). Log 0 = No Value 01.04.2020 Ed2 Pag. 77  Module 1 – Mathematics Chapter 01.03 GEOMETRY 01.04.2020 Ed2 Pag. 78  Module 1 – Mathematics Simple geometrical construction: angle An angle is a portion of a plane between two half-lines having the same origin. The half-lines are called sides of the angles, while their origin is called vertex of the angle. The magnitude of an angle can be measured using several systems: Sexagesimal measure Centesimal measure. Radiant measure. 01.04.2020 Ed2 Pag. 79  Module 1 – Mathematics Simple geometrical construction: angle According to the sexagesimal measure the round angle is divided into 360 parts corresponding to the standard of measurement called sexagesimal degree, indicated by the symbol “°”. This name is due to the fact that the subunits of the degree, the minute and the second are divided by sixty. According to the centesimal measure, the degree, called grad, represents one hundredth part of a right angle. With this system the round angle is divided into 400 equal parts and the submultiples of the centesimal degree represent decimal fractions. A simpler measure for the magnitude of angles often used in physics is the radian. According to the radian measure the round angle measures 2π, as we will see the circumference with radius r is equal to 2πr and therefore the radian is the angle created in correspondence of an arc with a length equal to the radius of the circumference. 01.04.2020 Ed2 Pag. 80  Module 1 – Mathematics Simple geometrical construction: angle An angle con be defined in different way, according to the measure. An angle defined by one single half-line is called round angle An angle defined by two half-lines on the same straight line is called straight angle An angle defined by two orthogonal half-lines is called right angle An angle smaller than a right angle is called acute angle An angle greater than a right angle is called obtuse angle. Considering an angle, its complementary angle is the difference to 90°, instead its supplementary angle is the difference to 180 °of the considered angle. 01.04.2020 Ed2 Pag. 81  Module 1 – Mathematics Simple geometrical construction: angle Example: Consider the angle 16°14’ 16’’. Its complementary (to 90°) angle is: 89° 59’ 60’’ - 16°14’ 16’’ = 73°45’ 44’’ Instead its supplementary (to 180 °) angle is: 179° 59’ 60’’- 16°14’ 16’’ = 163°45’ 44’’ 01.04.2020 Ed2 Pag. 82  Module 1 – Mathematics Simple geometrical construction: isometry, homothety and similarity In geometry there are three main simple constructions: isometry, homothety, similarity An isometry is a “distance-preserving map” between two metric spaces. Each rigid movement in the plane or in the space that does not distort the object is an isometry. All translations, all rotations and all reflections are isometries A translation in the space is a transformation that moves, in the same direction, all points at a fixed distance. It can also think the translation as an addition of a vector(V) and a constant, for each point. 𝑇𝑝 𝑝 = 𝑃 + 𝑉 01.04.2020 Ed2 Pag. 83  Module 1 – Mathematics Simple geometrical construction A rotation is a transformation that moves the object with a rigid movement and that leaves one fixed point. A reflection is a transformation that “mirrors” all points in relation with another point called center of reflection. The center of reflection can be also a straight line or a plane. The homothety is a particular transformation, in the plane or in the space, which dilates or makes smaller the object and maintains constant both angles and shapes. The homothety is characterized by a value “c”, called ratio of homothety. The homothety multiplies distances by “c”, areas by “c2” and volumes by “c3”. 01.04.2020 Ed2 Pag. 84  Module 1 – Mathematics Simple geometrical construction In the plane, a similarity is a particular transformation that maintains ratios between distances. Each similarity can be obtained by the composition of a homothety and an isometry (or vice versa). These transformations keep the shape of the object, but change the position, the orientation and the dimension. It is important remember that two similar objects have the same shape. if you analyze triangles there are three particular laws about the similarity: Two triangles are similar only if they have three angles congruent. Two triangles are similar if they have two sides congruent and the angle, included between them, congruent. Two triangles are similar if they have neatly three sides congruent. 01.04.2020 Ed2 Pag. 85  Module 1 – Mathematics Graphical representation In mathematics, and in statistics, the graphical representation is often used. The graphical representation is based on the use of diagrams. A diagram is a curve showing the relationships between two or more quantities or elements. In mathematics, the diagrams are often used to represent a mathematical function, because the diagrams have the advantage to be of more immediate comprehension than the common data tables. In fact the diagrams are based on the visual perception of the user. The main elements of a diagram are two axes, called X and Y, on which the elements are represented. However, to build the diagram it is necessary to start from a table that identifies the relations between different data, which must be represented. 01.04.2020 Ed2 Pag. 86  Module 1 – Mathematics Nature and uses of graphs Diagrams are very used, besides in mathematics and statistics, also in other scientific and resource fields, because they present a simple comprehension and a simple use. There are different types of diagram: The line graphs. Bar graphs (also called histograms). Aerogram or circle graphs (generally used to represent the percentages of a set). Ideograms or picture graphs (generally used to present statistical information). In these graphs the measure is not the unitary distance represented on the sheet, as in the bar graphs, but the symbol of the represented quantity itself. 01.04.2020 Ed2 Pag. 87  Module 1 – Mathematics Graphs of equations/functions A graph similar to a diagram is used to represent quantities that are mathematically, or physically, connected among them. It is called Cartesian representation. A Cartesian graph is made of two orthogonal axes. This pair of axes is known as Cartesian or orthogonal axes. The horizontal axis is called x-axis or axis of abscissas, while the vertical axis is called y-axis or ordinate axis. The point where the two axes meet is called origin and is indicated by the symbol zero. By convention, the values along the x-axis, to the right of the origin, are positive values, while those to the left are negative. The convention also refers to the y-axis, because the values above the origin, upwards, are considered positive, while those under the origin, downwards, are negative. 01.04.2020 Ed2 Pag. 88  Module 1 – Mathematics Graphs of equations/functions To identify a point on the graph it is necessary to provide two values corresponding to x and y. Usually, this pair of values is written between brackets: the x value is written first and separated from the y value by a comma. 𝑥, 𝑦 The ratio of x and y is a constant and represents a straight line passing through the origin. The value of this constant represents the slope, the inclination of the straight line passing through the origin. A character, for example “m”, generally indicates the constant. We can therefore write that the ratio of x and y is equal to “m” that is: 𝑥 =𝑚 → 𝑦 = 𝑚𝑥 𝑦 01.04.2020 Ed2 Pag. 89  Module 1 – Mathematics Graphs of equations/functions If the straight line does not pass through the origin, the ratio becomes the following: 𝑦 = 𝑚𝑥 + 𝑐 Where “c” is the value measured on the y-axis, starting from the origin, until the point where the straight line of the graph intersects at the y-axis. If we introduce a value for “x” raised to a power, for example “x” raised to a power of two, we obtain a non-linear graph, more precisely a curve. The standard form of the second degree equation is the following: 𝑦 = 𝑚𝑥 2 01.04.2020 Ed2 Pag. 90  Module 1 – Mathematics Graphs of equations/functions The equation can be solved graphically, generating a curve when joining a series of points corresponding to the series of values of “x” and of the corresponding values of “y”, when “x” is replaced by a series of positive and negative values. Non-linear graphic representations are: The parabola 3 The hyperbola 𝐸𝑥: 𝑦=. 𝑥 The sinusoid, which describes the harmonic motion. 01.04.2020 Ed2 Pag. 91  Module 1 – Mathematics Graphs of equations/functions The formula of the parabola is 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐. When 𝑏 = 0 (Example 𝑦 = 5𝑥² + 7) the parabola has the vertex on the y axis. We can find the interception of any curve with the x axis imposing 𝑦 = 0 to the formula. We can find the interception of any curve with the “y” axis imposing 𝑥 = 0 to the formula. Example: the curve 4𝑦 = 𝑥 + 8 intercept “y” when 𝑥 = 0 and 𝑦 = 2. 01.04.2020 Ed2 Pag. 92  Module 1 – Mathematics Graphs of equations/functions Having two points 𝐴(𝑥1 , 𝑦1 ), and 𝐵(𝑥2 , 𝑦2 ) in order to find the line that cross the two points we have to find “m”: 𝑦2 − 𝑦1 𝑚= 𝑥2 − 𝑥1 In order to write the equation of the line that cross the two points we have to use the following low: 𝑦 − 𝑦_1 = 𝑚(𝑥 − 𝑥_1 ) 01.04.2020 Ed2 Pag. 93  Module 1 – Mathematics Graphs of equations/functions Two lines are parallel if they have the same angular coefficient: 𝑚1 = 𝑚2 Two lines are perpendicular if they have the angular coefficient opposite and mutual: 1 𝑚1 = − 𝑚2 Example: 𝑦1 = 𝑥 (bisecting of first quadrant) 𝑦2 = 𝑥 + 3 𝑚1 = 𝑚2 → parallel 01.04.2020 Ed2 Pag. 94  Module 1 – Mathematics Simple trigonometry The trigonometry is the branch of mathematics that studies the relationships between the sides and the angles of triangles. The main task of trigonometry consists of calculating the measures of the elements of triangles through special functions. The trigonometric functions are all angle functions. The main trigonometric elements are: sine, cosine, tangent, cotangent, secant and cosecant. The definition of these functions can be studied with the unit- circle analysis. 01.04.2020 Ed2 Pag. 95  Module 1 – Mathematics Simple trigonometry If on the unit-circle we draw a half line, from the center to a generic point P, forming with the abscissas axis an angle “x”, we can define the sine of angle (sin x) as the value of the coordinate y of the point P. The cosine (cos x) is defined as the value of the coordinate “x” of the point P. The tangent, instead, is defined as the ratio of sin x to cos x, while the cotangent is the ratio of cos x to sin x. 01.04.2020 Ed2 Pag. 96  Module 1 – Mathematics Trigonometric relationships The trigonometric relationships permit to join the different trigonometric functions. The first fundamental correlation is between the sine and the cosine: 𝑠𝑖𝑛2 𝑥 + 𝑐𝑜𝑠 2 𝑥 = 1 The second relation introduces the definition of the tangent: sin 𝑥 tan 𝑥 = cos 𝑥 A characteristic of sine and cosine is that they are always between -1 and 1 01.04.2020 Ed2 Pag. 97  Module 1 – Mathematics Simple trigonometry Values of sine, cosine and tangent are listed here below: 𝐶𝑜𝑠 0° = 1 𝑆𝑖𝑛 0 = 0 𝑇𝑎𝑛 0° = 0 𝐶𝑜𝑠 90 ° = 0 𝑆𝑖𝑛 90° = 1 𝑇𝑎𝑛 90° = ∞ 𝐶𝑜𝑠 180 ° = −1 𝑆𝑖𝑛 180° = 0 𝑇𝑎𝑛 180° = 0 𝐶𝑜𝑠 270° = 0 𝑆𝑖𝑛 270° = −1 𝑇𝑎𝑛 270° = ∞ 𝐶𝑜𝑠 360° = 1 𝑆𝑖𝑛 360° = 0 𝑇𝑎𝑛 360° = 0 01.04.2020 Ed2 Pag. 98 Thank you for your attention. Pag.