M01_manual PDF - Fundamentals of Mathematics Manual

Cat. B1 - Table2021 of Contents Module 1 FUNDAMENTALS OF MATEMATICS pag. 1  Cat. B1 -...

Cat. B1 - Table2021 of Contents Module 1 FUNDAMENTALS OF MATEMATICS pag. 1  Cat. B1 - Table of Contents Table of Contents 1.1 Arithmetic.............................................................................................................................. 4 1.1 Arithmetical terms and signs.............................................................................................. 4 1.2 Methods of multiplication and division.............................................................................. 7 1.3 Fractions and decimals........................................................................................................ 9 1.4 Factors and multiples........................................................................................................ 12 1.5 Measures and conversion factor...................................................................................... 15 1.6 Weights............................................................................................................................. 17 1.7 Scale, latitude and longitude............................................................................................ 18 1.8 Ratio and proportion......................................................................................................... 21 1.9 Averages and percentages................................................................................................ 23 1.10 Squares............................................................................................................................ 25 1.11 Cubes............................................................................................................................... 26 1.12 Square and cube roots.................................................................................................... 27 2 Algebra......................................................................................................................................... 2.1 Evaluating simple algebraic expressions........................................................................... 28 2.2 Addition............................................................................................................................. 29 2.3 Subtraction........................................................................................................................ 29 2.4 Multiplication and division................................................................................................ 30 2.5 Use of brackets................................................................................................................. 30 2.6 Simple algebraic fractions................................................................................................. 32 2.7 Linear equations and their solutions................................................................................ 35 2.8 second degree equations.................................................................................................. 39 2.9 Simultaneous equations and second degree equations with one unknown…..………………43 2.10 Indexes and powers, negative and fractional indexes.................................................... 45 2.11 Binary and other applicable numberingsystems………………………………………………………...48 2.12 Logarithms...................................................................................................................... 51 2.13 complex number............................................................................................................ 53 pag. 2  Cat. B1 - 1 Arithmetic 3 Geometry................................................................................................................................. 57 3.2 Isometry, Homothety and similarity................................................................................. 57 3.3 Areas and volumes…..………….……………………………………………………………………………………….60 3.4 Nature and uses of graphs................................................................................................ 67 3.5 Graphs of equations/functions......................................................................................... 70 3.6 Simple trigonometry......................................................................................................... 76 3.7 trigonometric relationships.............................................................................................. 75 3.8 Rectabgular and polar coordinates................................................................................... 76 pag. 3  Cat. B1 - 1 Arithmetic 1 Arithmetic 1.1 Arithmetical 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 nonnegative integers. The set of natural numbers is indicated by the symbol “ℕ ”. Example: 0,1,2,3, 600…∞ 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 and the so created set of 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” [ −∞..-2,-1,0,+1+2…+∞] In case of division, it is also possible to introduce fractional numbers and the so created set of integers plus fractional numbers, is called set of rational numbers. The set of rational numbers is indicated by the symbol “ℚ”. [ −∞..-2/3,-1,0,+1/2+2/3…+∞] An irrational number is a number that cannot be expressed as a fraction (a/b), for any integers a and b with b different from zero. The number set that includes integers, rational and irrational numbers represents the set of real numbers. The set of real numbers is indicated by the symbol “ℝ”. The main difference respect to rational number that fraction results is endless √2, π, e, The four basic math operations with numbers are: addition, subtraction, multiplication, and division. They are indicated by the following symbols: addition symbol plus, subtraction symbol minus, multiplication symbol multiplied by, division symbol divided or over. But the results of some operations are not always acceptable or possible in some sets. As instance if you are performing a calculation counting the people working in your company it should be a solution belonging the natural numbers (positive integers). If we refer to certain percentage of colleagues we have to deal with the other numerical sets. But somehow all these numerical sets are linked together, because the real numbers include all the other like subgroups of the main numerical set pag. 4  Cat. B1 - 1 Arithmetic Addition 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: 1. Commutative property: changing the order of addends the sum doesn’t change Ex: 7+5 = 12 =5+7 10+2+5= 17 = 2+5+10 2. Associative property: replacing two or more addends with their sum the result doesn’t change. This property is very simple and useful when we want to manage expressions in the manner and the form to check out what we are interested to. We can apply it to constant numbers and unknown variables as clarified later on - 30+3+1 = 34 = 30 + 4 - 5X + 6X + 3 + 5 = 11X+8 =X + 10X + 1.5+ 6.5 3. Dissociative property: replacing one addend with one or more addend whose sum is the replaced addend the result of the addition doesn’t change. The sum can be represented by the symbol ∑ pag. 5  Cat. B1 - 1 Arithmetic 𝑓 =1+2+3+4+5+6 ̇ The sum of two numbers, one addend positive and the second one negative, the result sign is imposed by the biggest absolute values of the addends. When addend numbers of sum increase this rule should be applied in order to establish the final sign of the result. Natural numbers, rational, irrationals can ben summed as discussed before, we can sum all possible number that human brain can think about, excluding some pure theoretical concept like endless numbers -∞ or +∞. 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 doesn’t change. The invariance in obvious way which let the calculation results unchanged- The following examples can be used as tips to figure out or null out some values in the calculation: - 5+ Y = 5 – 2 + Y-2 - 6 + X = 0 → 6-6 + X = 0 -6 - (A+5) -1= 0 → (A+5) -1+1 = 0+1 Example: Apply invariance property to the following equalities - 5x+1= 2x+1 - -1/2+1=2x - 1+ (3a+5b+2)+2-(3a+5b+2) 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. We can reach the same results by summing the first multiplicand N times as the multiplier. This mean for you man brain and computer the operation save time to solve complex problem. Well, the multiplication has the following properties. pag. 6  Cat. B1 - 1 Arithmetic Commutative property: means that the terms order of multiplication operation cannot change the results. When we refer to number multiplied by letter, conventionally is the number first followed by the letter omitting the multiplication symbol. You can the multiplication symbol marked by a dot. Examples:  2 x 30.5 = 30.5 x 3  2a(b+2).(5-Y) = (5-Y)2a(b+2) Note The multiplier and multiplicand sign determine the results sign as the following table. Multiplicand Sign Multip. Multiplier sign Sign of result - x - + - x + - + x + + + x - - Division The number that is divided is called dividend, while the other number that divides it is called divisor and 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. 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. Examples (3+5) x 2+1 =8x2 +1 = 16+1 = 17 pag. 7  Cat. B1 - 1 Arithmetic -( 2+5) x 2 +3 x (-1) = - 14 -3 = -17 -(1+x) (1+2) x 2 +1 = (-3-3x) x 2 + 1 = -6 -6x+1 = 6x - 5 1. 2 Methods of multiplication and division Multiplication is characterized by commutative, associative, dissociative, and distributive properties. 1. Commutative property: changing the order of factors the product doesn’t change. 𝑎⋅𝑏 =𝑏⋅𝑎 Examples:  6 ⋅ 3 = 3 ⋅ 6 = 18  5.3 = 3.5 = 15  2. Associative property: replacing two or more factors with their product the result does not changes. If 𝑏 ⋅ 𝐶 = 𝐴 than 𝑏⋅𝑐⋅𝐷 = 𝐴⋅𝐷 Examples  10 ⋅ 8 = 80 = 2 ⋅ 5 ⋅ 4 ⋅ 2  = 6 ⋅ 5 ⋅ 6 ⋅ 4 = 30 × 24 3. Dissociative property: replacing one factor with one or more factors whose product is the replaced factor the result of the multiplication doesn’t change. If 𝐴 = 𝑏⋅𝐶 Then: 𝐴⋅𝐷 = 𝑏⋅𝑐⋅𝐷 10 ⋅ 8 = 80 = 2 ⋅ 5 ⋅ 4 ⋅ 2 pag. 8  Cat. B1 - 1 Arithmetic 4. 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 If we have 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 We can proceed in revers way that it is very useful in different cases. 𝑎𝑏 + 𝑎𝑐 = 𝑎(𝑏 + 𝑐) Examples  5(3 + 2) = 15 + 10 = 25  25 + 50 = 5(5 + 10) Division is characterized by invariance and distributive property: 1. Invariance: dividing or multiplying by the same number the two terms of a division the result doesn’t change. Therefor we have: ⋅ If 𝑘 ≠ 0 then = = ⋅ Examples  = =   = = = 2. 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. If 𝑘 ≠ 0 then (𝑎 + 𝑏) = + Definitions pag. 9  Cat. B1 - 1 Arithmetic The least common multiple or lowest common multiple two integers a and b, usually denoted is the smallest positive o integer that is divisible by both a and b. If we consider the following numbers 3,6,12 their lowest common multiple is 12. Examples:  (6 + 12) = + =2+4=6  Note that we can reach the same results if proceed like: 1 18 (18) = =6 3 3 All those tools are useful to reduce the calculation load or to manage a long mathematic expression in order to simplify it, therefore by filtering the operation terms we conduct the result into a simple operation and calculating the final results. Test yourself  [18 + 3 · 2 − (2 + 3 + 4) · (3 · 2 − 6)] − 5 · 4  4 · 25 − {3 · 7– [50 + (16 − 12) − (8 + 6 − 12) – 7 · 5] + 19} − 7 · 11  [(84 + 36 · 3): 8 + 8 · 15]: 12 + (78– 90: 5): 6  (0,01 ⋅ 10) ⋅ 2 − 1 − (6 ⋅ 0,5)  apply the commutative propriety to 50, in three different ways.  Apply the invariance propriety to  Which number set could represent The following numbers: 2,√2, , −3, −  Find the lowest common multiple of the following series (4,8,24) ; (2,5,4) pag. 10  Cat. B1 - 1 Arithmetic 1.3 Fractions 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. 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 example 3/5. 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. When numerator and denominator are the same, the quotient of the fraction is one. 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. Examples  + = =2  + = 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. To convert a mixed number into an improper fraction it is necessary to multiply the integer by the denominator of the proper fraction and add the product obtained as numerator. The sum of these two numbers becomes the numerator of the improper fraction. When adding mixed numbers, either to themselves or to proper fractions, the first operation to do is to transform the mixed numbers into improper fractions. After that, determine the lowest common denominator and sum the two fractions using the same method used for proper fractions. Note that adding improper fractions the sum will be an improper fraction therefore, it is advisable, after having obtained the result, to transform it back into a mixed number. To determine the integer, divide the numerator by the denominator. If there is a remainder, leave it as a fraction. Examples:  + = = =2 pag. 11  Cat. B1 - 1 Arithmetic  1+ = =  𝜋+ 𝜋= = 𝜋 The subtraction of fractions also requires the determination of the lowest common denominator. After having determined this term, numerators are subtracted and the difference is placed on the lowest common denominator reducing the fraction to its lowest terms. 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. Examples  − + = =−  − − = =− ,  − − + = =  The division of common fractions is done inverting the divisor that is exchanging the numerator and the denominator between them and then multiplying the fractions. After having inverted the divisor, numerators are multiplied to obtain the quotient numerator, then the denominators are multiplied to obtain the quotient denominator and, finally, quotient fractions are reduced to their lowest terms. The value of a fraction doesn’t 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. Note that if “a” is present both in the numerator and in the denominator, since the two values are the same, they can be simplified from the fraction. 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 doesn’t change. This property allows simplifying fractions made of great numbers, thanks to the so-called reduction to lowest terms. To reduce a fraction to its lowest terms, the two terms are divided by their greatest common divisor. Examples:  ⋅ = = =  ⋅ ⋅ = = = pag. 12  Cat. B1 - 1 Arithmetic  = ⋅ = = Mixed numbers are numbers made of both integers and proper fractions. Before solving operations with mixed numbers it is necessary to convert them into improper fractions. 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. As an example, the fraction 3/4 can be converted into a decimal number, dividing the number 3 by the number 4. The decimal number equivalent to 3/4 is 0.75. The first number after the decimal point denotes tens, the second hundreds and the third thousands. 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. As in the case of addition, the subtraction of decimal numbers follows the same laws of integers. Even in the case of subtraction, it is important to maintain on the same vertical line the points indicating the decimal number. 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. The division of decimal numbers follows the same laws of integers division. The two following laws establish the position of the decimal point of the quotient. If the divisor is an integer, the decimal point of the quotient must be aligned on the vertical line of the decimal point of the dividend; while doing the division. When the divisor is a decimal fraction, first convert it into an integer, moving the decimal point to the right. Remember that, when moving the decimal point of the divisor, the decimal point of the dividend must be moved accordingly to the same number of figures and in the same direction. Often, decimal numbers are expressed with many figures after the decimal point. For practical reasons, this precision is not always necessary; therefor it is advisable to limit the number of decimal figures. The process of keeping a certain number of figures, eliminating the remaining ones, is called rounding. In other terms, the number obtained is an approximation of the original number. Rounding is done observing the figures immediately to the right of the last one to be kept. If this figure is 5 or higher, the last figure to be kept is increased by one. When the figure to the right of the last one to be kept is lower than 5 the last figure to be kept remains the same. The use of decimal numbers is easier that the one of fraction, but in some cases fractions simplify calculations. To transform a decimal number into a fraction is sufficient to take as numerator of the fraction the decimal number without the point and as denominator a multiple of ten with as many zero pag. 13  Cat. B1 - 1 Arithmetic as the number of figures after the point of the decimal number. Finally, the fraction is reduced to its lowest terms. Test yourself - + - − ( ) - −4 + − - + + - + + − - − - ⋅ + + pag. 14  Cat. B1 - 1 Arithmetic 1.4 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 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟 = 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. There are some useful rules to understand the divisors of a number looking at its figures. A number is divisible by 2 when the figure of its units it is divisible by 2, that is when a number ends with 0, 2, 4, 6, 8. Remember that a number divisible by 2is called even number. A number is divisible by 3 if the sum of its figures is divisible by 3; for example: 132, 1+2+3 = 6, since 6 is divisible by 3 also 132 is divisible by 3. There are also: 1. 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; 2. A number is divisible by 5 if its last figure is 0 or 5; 3. A number is divisible by 6 if it is divisible by 2 and 3 at the same time; 4. A number is divisible by 10 if its last figure is 0. 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 shared by them is called greatest common divisor. In order to find the least common multiple (l.c.m.) of two or more numbers, it is necessary to multiply the prime factors, common and non-common, obtained factorizing all numbers, each one taken with the highest exponent. pag. 15  Cat. B1 - 1 Arithmetic In order to find the largest common divisor (G.C.D.) of two or more numbers, after having factorized all numbers, it is necessary to multiply all the common factors each one taken with the smallest exponent. 1.5 Measures and conversion factors Systems of measurement Units of measurement are a standard for the measuring of physical quantities. It’s necessary to define standard systems of measurement with the aim to facilitate the measuring: however in the world there are different units of measurement. A measure is a numerical value obtained from the comparison between a quantity and a sample quantity, called standard of measurement. For example, the value of quantity Q is expresses as the product between the unit (q) and a number factor (n): 𝑄 =𝑛∙𝑞 There are different systems of measurement that are officially accepted; these systems are based on diverse set of fundamental units. The International System is the most used and it has 7 main units: 1. Meter: length. 2. Kilogram: mass. 3. Second: time. 4. Ampere: intensity of current. 5. Kelvin: temperature. 6. Mole: quantity of material. 7. Candle: luminous intensity. Conversion factor A conversion of the units of different systems is the comparison between all standards values. The following tables show main conversion factors. pag. 16  Cat. B1 - 1 Arithmetic 1.6 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). Its multiples are: decagram (1 dag = 10 g), hundred grams (1 hg = 100 g), kilogram (1 kg = 1,000 g), quintal (1 q = 100 kg) and ton (1 ton = 1,000 kg). Units smaller than gram are: decigram (1 dg = 0.1 g), centigram (1cg=0.01 g) and milligram (1 mg = 0.001 g). In the British system the pound is the main unit to measure the mass (1 pound = 0,453 kg). pag. 17  Cat. B1 - 1 Arithmetic Note: the mass is constant and it does not change in the entire universe, instead the weight represents an attraction force to which all masses are subjected to. In fact the forces felt on different planet is due the mass of the planet itself. The following table shows the conversion factor among different measurement unit and metric one. In aeronautics is convenient to use the right conversion factor because several countries use one or two measurements system. Weights can be obtained doing a multiplication: 𝑤𝑒𝑖𝑔ℎ𝑡 = 𝑚𝑎𝑠𝑠 ∙ 𝑔 𝑤𝑖𝑡ℎ 𝑔 = 9.81 𝑚/𝑠 the following table shows some time measures transformations. pag. 18  Cat. B1 - 1 Arithmetic 1.7 Scale, latitude and longitude The scale is the representation of a determined quantity or measure in terms of another measure. The scale is the ratio between the actual distances represented on a map, that is the actual kilometers, and the material distances on the map, that is the centimeters or millimeters on the sheet. Other important measures are the latitude and the 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. The latitude and the longitude of whatever place is calculated according to the position of the sun, the hour and the day. The observation of the position of the sun is done using a special tool, called sextant or using an octant. The differences of latitude and of longitude between two places are calculated according to the following rules: 1. If the latitudes of the two places are both northward and southward, the lower latitude is deducted from the higher one. While if one place is northward and the other southward, the two latitudes are summed up 2. If the longitudes of the two places are both east and west, the lower longitude is deducted from the higher one. While if one place is east and other west, the two longitudes are summed up. If the sum of the two longitudes exceeds 180 degrees, the sum must be deducted from 360 degrees to obtain the correct difference in longitude. The daily revolution of the Earth around its axis makes it look as if the sun is turning around the Earth, from east to west, along the 360 degrees of the Earth longitude in 24 hours. Starting from these data, the sun covers in 1 hour an arc corresponding to one twenty-fourth of 360 degrees, that is 15 degrees longitude. Therefore, in one minute it covers one sixtieth of 15 degrees, that is 15 minutes of longitude, while in 1 second it covers one sixtieth of 15 minutes, that is 15 seconds of longitude. pag. 19  Cat. B1 - 1 Arithmetic For this reason, 15 degrees of longitude correspond to 1 sun hour; 15 minutes of longitude correspond to one minute of sun time, while 15 seconds of longitude correspond to one second of sun time. Sun time in any place depends on the position of the Sun observed form that place. The clock indicates exactly 12 zero or midday in that place when the sun crosses the local meridian. As a consequence, if it is midday in the meridian where we are, it is afternoon post meridiem in all the places to the east, while it is morning ante meridiem in all the places to the west. Everywhere in the world the Standard Time is used, that is Time Ranges. As an example, the United States is divided into four ranges, each one of which is approximately 15- longitude degree wide. Each place within a time range uses the same Standard Time, that is the same hour, independently from the local hour, that is the local meridian. At the beginning of springtime, until the half of autumn, the clock hour hand is moved one hour forward to best exploit sun light during working hours. This mechanism is called summer hour. At the end of autumn, the hour hand is set one hour back to winter hour. To find the difference in sun hours between two places, it's necessary to divide by 15 the difference in longitude, expressed in degrees, minutes and seconds. The quotient is the difference in sun hours, expressed in hours, minutes and seconds. Since time is calculated both to the east and to the west of the meridian of reference, the time on the one hundred eighty eighth meridian can be considered 12 hours in advance or 12 hours late compared to the Greenwich meridian. Instead of considering the one hundred eighty eighth meridian as the division line between the time to the east and to the west of the meridian, an international agreement has set the 180th meridian as the pag. 20  Cat. B1 - 1 Arithmetic international line for changing date. The line follows the meridian with a zigzag course. This has been done to make all the Pacific islands have the same time. When it is midday in Greenwich, midnight has just passed and it is morning the same day in all the places slightly east of the international line of date, while it is almost midnight the same day in all the places slightly west of the line. When is one in the afternoon in Greenwich, it is about one in the morning the same day in the first places above, while it is one in the morning the day after in the other places above. Military, naval and aeronautic services express time referring to 24 hours. Starting from midnight, indicated by 00:00 time is indicated by the hundreds of the number. For example 8 anti-meridian is indicated by 08:00; 12 is indicated by 12:00; one in the afternoon, 1 post meridian, is indicated by 13:00, while 11 post meridian, that is 11 in the evening, are indicated by 23:00. Minutes are indicated by the units preceded by 0 if they are less than 10, because the number must always have 4 digits. Therefore, 8.10 anti-meridian are indicated by 08:10; zero, zero, zero, five pos- meridian by 00:05 and 11.59 post meridian by 23:59. 1.8 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. 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 pag. 21  Cat. B1 - 1 Arithmetic 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. From the fundamental property it is possible to derive other useful properties for determining an unknown term in a proportion: 1. Property of permuting: in a proportion, exchanging the means between them, or the extremes, the result is a new proportion 2. Property of inverting: in a proportion, exchanging each antecedent with its consequent the result is a new proportion 3. Property of combining: in each proportion, the sum of the first two terms is to the first (or to the second) as the sum of the other two terms is to the third (or to the fourth) 4. Property of factorizing: in each proportion the difference of the first two terms is to the first (or to the second) as the difference of the other two terms is to the third. Know let’s see how do we write ratios trough orange juice preparation:  The ratio of orange juice to sugar is a part-to-part ratio. It compares the amount of two ingredients.  The ratio of orange juice to whole orange juice is a part-to-whole ratio. It compares the amount of one ingredient to the sum of all ingredients.  Part : whole = part : all parts So, the use of proportion can be summarized as the any unknown quantities implicit in the relationship of the proportion. Examples  A rope's beam and weight are in proportion. When 20cm of rope weighs 1kg, then: - 40cm of that beam weighs 2kg - 200cm of that beam weighs 10kg  An aircraft crosses 5 km in 10 minutes. How long would it take to cross 20 km? Let's set up a proportion as follows: pag. 22  Cat. B1 - 1 Arithmetic 5𝐾𝑚 20𝐾𝑚 = 10𝑚𝑖𝑛 𝑥𝑚𝑖𝑛 5𝑥 = 10 ⋅ 20 → 5𝑥 = 200 therefor: 𝑥 = 40  A picture on a wall has length 44 inches and width 22 inches. Write the ratio of length to width in simplest from. 44 =2 22 That means the law governing the relation between the length and the width. This ration obviously can be compared to any new unknown side. What is Ratio and Proportion? A comparison by division is termed as ratio and the equality of two ratios is called proportion. When it comes to measuring the speed or the climbing ratio of an airplane, it is km per hour or feet/min. However, this is called a rate which is a kind of ratio. On the other hand, a proportion is an equation that says that two ratios are equivalent. A proportion is read as x is to y as z is to w. x/y = z/w where w & y are not equal to 0. And thus, x : y = z : w If you think about is something very intuitive, we do use proportion every day on our life, when we refer to different mathematical similarity, when we refer to a car travelling in space in time units it’s a fraction that could be used as fundamental scale to compare with other new distances. 1.9 Averages and 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 and inversely proportional quantities. Two variables depending on each other are directly proportional if multiplying the first one by two, three, etc and also the second one is multiplied by two, three, etc; and if dividing the first one by two, three, etc, the second one is also divided by two, three, etc. If two quantities are directly proportional, the ratio of two values of the first one is equal to the ratio of the corresponding values of the second one. Two variables depending on each other are inversely proportional if multiplying one of them by two, three, etc, the other is divided by two, three, etc. If two quantities are inversely proportional, the ratio of two values of the first one is equal to the inverse ratio of the corresponding values of the second one. pag. 23  Cat. B1 - 1 Arithmetic Using the properties of directly proportional quantities it is possible to solve problems including percentages. We can use proportions to solve questions involving percentages. We can put all the terms into this form: 𝑃𝑎𝑟𝑡 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 = 𝑊ℎ𝑜𝑙𝑒 100 Example The percent of a group is 20%, the whole group has 180 elements. Find the number of the group part: 𝑃𝑎𝑟𝑡 20 = 180 100 We can conclude that 20% of 180 elements is 36. The arithmetic mean of a set of statistical data corresponds to the sum of these data divided by their number. We can take into account the following example; by measuring the hight of five students we found Student Hight A 170 cm B 185 cm C 160 cm D 165 cm The average can is calculated as ; (170 + 185 + 160 +165) : 4 = 170 cm Note: The average can be expressed in different mathematical and statistics context, we can consider three main types of average: mean, median and mode. Each of these techniques works slightly differently and often results in slightly different typical values. The mean is the most commonly used average. To get the mean value, you add up all the values and divide this total by the number of values. pag. 24  Cat. B1 - 1 Arithmetic  The median, which places all your values in order from smallest to highest and finds the one in the middle. For example, the median of the values 2, 3, 3, 5, 8 10 and 16 is 5.  The mode is the most commonly occurring value. For example, the modal value of 1, 3, 4, 4, 6, 4, 7, 7, 14, 14 and 24 is 4 because it appears several times Test yourself:  Find the average of the following numbers: 3, 5, 7,  What is the percent of group of 50 students from a group of 200.  We have 3 red 5 black balls and if we add other three black balls, what should be the number the red balls in order to keep the same proportion? 𝟑  write 2 other fractions having the same ration of 𝟕  what is the travel time to cross 200 km of an airplane if it crosses 300 km in half hour?  Write the following fraction as percentages: 3/2, ½, 4/2,  If 35% of 80 all balls is black, what is the total balls number?  Write the following numbers as fractions: 0.1, 0. 002, 10  Write the following number as percentages: 2; 0.012; 0.3; 0.1. 10 pag. 25  Cat. B1 - 1 Arithmetic 1.10 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: 𝑛 =𝑛∙𝑛 For example: 3 =9 The square of a number has some properties: 1. The square of a real number is always greater (or equal) than 0; for example the square of 2 is 4, but also the square of -2 is 4. 2. The square of any integer (n) can be represented by the sum: 1 + 1 + 2 + 2+.. +(𝑛 − 1) + (𝑛 − 1) + 𝑛 For example: 4 = 1 + 1 + 2 + 2 + 3 + 3 + 4 = 16 3. The square of any integer (n) is equal to the sum of n prime odd numbers. For example: 4 = 1 + 3 + 5 + 7 = 16 The square returns back a positive number Test yourself:  Find the square of two square two  Find the double of two square two  (𝑎 + 𝑏)(𝑎 + 𝑏) = ; (𝑎 + 𝑏)(𝑎 − 𝑏) = ; (𝑎 − 𝑏)(𝑎 − 𝑏) = pag. 26  Cat. B1 - 1 Arithmetic 1.11 Cubes 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: 𝑛 =𝑛∙𝑛∙𝑛 For example: 2 =8 the cube conserves the multiplication sign, if the base is negative the result remains negative and vice versa. −2 = −8 1.12 Square and cube roots The symbol used for root extraction is √. The number under the root symbol is called radicand, while the number that represents the order of the root is called exponent. The function of the root is to find the number that, multiplied by itself for a certain number of times equal to the value of the exponent, has the radicand as result. The exponent of the root must always be indicated unless it is two. For example: √4 = 2 2 =4 Examples - Find the volume of the cylinder contained and tangent to the square of the cube volume having a side of 1m? - Find the pyramid volume contained in a cube of 1 𝑚 having one of its faces as base - √𝟒 + 𝟏𝟐 = 𝟑 - √−𝟗 = - A rhombus having diagonals d1 = 10 cm and d2= 20 cm, find the area of one triangle tracked by the diagonal interception pag. 27  Cat. B1 - 1 Arithmetic PAGE INTENTIONALLY LEFT BLANK pag. 28  Cat. B1 - 2 Algebra 2 Algebra 2.1 Evaluating simple algebraic expressions 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. Remember that plus and minus signs have double meanings. They can be used for operations, namely for addition and subtraction, and they can be used to indicate the sign of a relative number. Like Terms are terms it is a useful term used to refer same algebraic quantities which contain a commune term, for example 8𝑥 ; −𝑥𝑦, 3𝑥, are all terms are linked by x. To calculate the value of an algebraic expression it is necessary to use the following laws: if the expression doesn’t 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. Also in the brackets, the correct order in doing the different operations is the following: 1. Powers; 2. Multiplications and divisions. Operations of multiplication and division must be done from the left to the right after the calculation of exponents; 3. Additions and Subtractions. After multiplications and divisions, additions and subtractions must be done from the left to the right. pag. 29  Cat. B1 - 2 Algebra 2.2 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. As general rule summing two numbers having the same sign the result will have the same sign and the numbers sum. When the terms have opposite signs must be subtracted each other the sign will as the highest number. Example: 5𝑥 + 2𝑥 = 7𝑥 5𝑥 + 2𝑥 = 3𝑥 2.3 Subtraction To subtract numbers with different signs, the operation is turned into an addition changing at the same time the sign of the subtrahend. After that, the method is the same of addition. −8𝑥 + 3𝑥 = −5𝑥 −3𝑥 − 4𝑥 = −7𝑥 Remember that if we have sign “ – “ before the bracket, the terms inside it must change sign when they are bring out from the brackets. Example: −(3𝑥 + 4𝑦 − 𝑧 + 1) = −3𝑥 − 4𝑦 + 𝑧 − 1 pag. 30  Cat. B1 - 2 Algebra 2.4 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: 1. The product of two positive numbers is always positive 2. The product of two negative numbers is always positive 3. 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. 1. The quotient of two positive numbers is always positive 2. The quotient of two negative numbers is always positive 3. The quotient of a negative and a positive number is always negative. 2.5 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, square brackets, and braces. The first operations to be done are those indicated between the inner brackets. There is a tip used in order to proceed in the correct way, the term BODMAS indicates the hierarchy in which we perform any calculations: B stands for Brackets O stands for "Of" D stands for Division M stands for Multiplication A stands for Addition S stands for Subtraction An example of classical expression that could be simplified by removing the brackets hierarchically. Examples:  (3 + 2)x = 5x  (2𝑥 + 3𝑥)(5𝑥 + 𝑥) = (5𝑥)(6𝑥) = 30𝑥 Polynomial expressions rules: pag. 31  Cat. B1 - 2 Algebra How to multiply the brackets terms? (a+b) (c-d) = (a+b) (c-d) = 𝑎𝑐 − 𝑎𝑑 + 𝑏𝑐 − 𝑏𝑑 Important polynomial rules  (𝑎 + 𝑏) = 𝑎 + 𝑏 + 2𝑎𝑏  (a − b) = a + b − 2ab  (𝑎 − 𝑏)(𝑎 + 𝑏) = 𝑎 − 𝑏  The sum of two similar monomial is possible only when they have the same power: - 𝑎 + 𝑎 = 2𝑎 - 2𝑎 + 𝑎 𝑐𝑎𝑛𝑛𝑜𝑡 𝑏𝑒 𝑠𝑢𝑚𝑚𝑒𝑑 - 1 + 𝑎 + 𝑏 + 2𝑎 + 𝑎 − 3𝑏 = 3𝑎 − 2𝑏 + 𝑎 + 1 Examples:  {𝑎[(𝑎 + 𝑏) ⋅ (𝑎 − 𝑏)]}5𝑎𝑏 Leading to: {𝑎[𝑎 − 𝑏 ]}5𝑎𝑏2 → {𝑎 − 𝑎𝑏 } ⋅ 5𝑎𝑏 = 5𝑎 𝑏 − 5𝑎 𝑏  {[(𝑛 − 2) ⋅ (3 + 2)] ⋅ 2}2 −1 = {(𝑛 − 2) ⋅ 10} ⋅ 2 − 1−= 20𝑛 − 40 − 1 It gives: ⇒ 20𝑛 − 41  3𝑎 4 − + 2(3𝑎 + 5) = 12 − 3 + 6𝑎 + 5 = 6𝑎 + 14  (3𝑎 + 3)(3 + 3) = 9𝑎 + 9 + 9𝑎 + 9 = 18𝑎 + 18  √2𝑥 √2 + 2 ⋅ 3 + 2𝑥 = 2𝑥 + 2√2 ⋅ 3 + 2𝑥 6𝑥 + 6√2 + 3𝑥 = 9𝑥 + 6√2 pag. 32  Cat. B1 - 2 Algebra 2.6 Simple algebraic fractions Algebraic fractions are fractions with polynomials at the denominator and at the numerator. As polynomials comprise numbers, it must be possible to do with polynomials the same operations as with numbers. In fact the polynomial is defined as an expression with constant and variables, combined only with additions, subtractions and multiplications. When it has a fraction with a polynomial at the numerator and at the denominator, it must: 1. Break out numerator and denominator into factors 2. Control if there are two equal factors between numerator and denominator; if there are, they must be eliminated 3. Write the fraction with the left terms. Attention: between the numerator and the denominator you can only simplify factors terms of multiplications (or divisions). It can’t simplify addends or terms of subtractions. It has to follow the same process used for simple fractions to do additions and subtractions between algebraic fractions: 1. To break out numerator and denominator in factors 2. To calculate the l.c.m, and to put it as the denominator 3. To calculate the numerator 4. If possible, to make simplifications between the numerator and the denominator. Attention when it does the subtraction: the symbol minus, before a fraction, changes the sign of all numerator’s terms. When it does multiplications of algebraic fractions, it must proceed: 1. Breaking out numerator and denominator into factors 2. Eliminating the same terms between the numerator and the denominator 3. Multiplying the numerator to the numerator, and the denominator to the denominator. When it does the quotient of two algebraic fractions it has to: pag. 33  Cat. B1 - 2 Algebra 1. Multiply the first fraction by the inverse of the second fraction 2. Break out numerator and denominator into factors 3. Eliminate equal terms between the numerator and the denominator 4. Multiply the numerator to the numerator, and the denominator to the denominator. Basic fractions laws:  The simplification of a fraction rules regards multiplication and division only, like the following example: 6𝑎𝑏 𝑐 = 6𝑎𝑏 𝑏𝑐 And cannot be done when the simplified terms of the numerator and/or denominator are expressed in terms of sum or subtraction. 6𝑎𝑏 + 𝑐 = 6𝑎𝑏 𝑏𝑐  Dividing a fraction by another fraction is equal to the numerator fraction multiplied by the revers of the denominator fraction 𝑎 𝑏 =𝑎⋅𝑑 𝑐 𝑏 𝑐 𝑑  If a/b = c Then 𝑏 ≠ 0. The denominator cannot be zero, this because if you imagine to divide a very small number too close to 0, this kind of fraction leads to huge numbers or infinite ∞.  When it does the power of an algebraic fraction, it must raise to the power both the numerator and the denominator. 𝑎 𝑎 = 𝑏 𝑏 pag. 34  Cat. B1 - 2 Algebra Examples of fractions:  = =  =  + = =  + = = = 𝑎 ( )( )  + 𝑦= + 𝑦 = 2𝑦 + 1 + 𝑦 = +1 = +1 ( )  + + =  ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ =𝑎  − ⋅ + 𝑎= − + 𝑎 = Test yourself (5𝑥, +3)𝑥 + 3𝑥 −2𝑥 + 1 + √1 7/2 𝑎[(3𝑎 + 1)𝑥𝑎] 1 (7𝑦 + 3𝑦) − 𝑦 + 𝑦 3 (𝑥 − 1)(4𝑥 + 3𝑦) 4 + 2(4𝑥 + 3𝑦) 3 pag. 35  Cat. B1 - 2 Algebra 4 3 2 1 3𝑥 + + 𝑥 :8 𝑥 2 𝜋 +3𝑥 ( ) [(3𝑥 − 1)(3𝑦 + 1)] (𝑥 + (𝑦 ) ) + (3𝑥 + 5𝑥 + 𝑦) × 2 √2 √2𝑥 + √2 + 1 (3 + 1) + 3 −(−3 + 5) 7 + : 2 6−2 2.7 Linear equations and their solutions 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: 1. 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 2. The set of solutions contains a finite number of elements that is the solutions of the equation are in a finite number. In this case the equation is determinate 3. The set of solutions contains an infinite number of elements that is the solutions of the equation are infinite. In this case the equation is indeterminate Solving linear equations  The general rule for all equations is: Whatever you do to one side of the equation, you must do the same to the other side. By convention we name each side of the equation Left Hand Side (LHS) or Right Hand Side (RHS)  Equations Requiring Multiplication or Division pag. 36  Cat. B1 - 2 Algebra To solve division equation, we multiply both sides by the number we want to bring in the opposite side. Ex 6𝑥 =4 7 6𝑥 → ⋅ 7 = 4 ⋅ 76𝑥 = 28 7 → 𝑥= We have to proceed in opposite way when we are dealing with multiplications. 𝜋√2(1 + 3𝑥) = 0 𝜋√2(1 + 3𝑥) 0 = 𝜋 ⋅ √2 𝜋√2 1 + 3𝑥 = 0 𝑥 = −1 Let consider another example: 𝑥 +1=0 This solution seems simple, in fact it is, by solving it we can found 𝑥 = −1 The result shows that the square of x is negative, obviously the result cannot be found in all sets we saw at the beginning of the book. The only set which the solution can be found is the set of complex numbers ℂ. This logical conclusion derives that the square of positive or negative numbers the result will be always positive. That means this equation has an empty solution. If we rewrite the example as 𝑥 − 1 = 0 the result conduct to 𝑥 = 1 , this first step sound logical because the square of whatever number is always positive. Due the fact we do not know the sign of the value contained in X we have to consider both cases when x is negative and when it is positive. This consideration leads the equation accepts both values (positive and negative) because of their square will result always positive. Therefor the equation has two determinate values ± 1. pag. 37  Cat. B1 - 2 Algebra In the third case the equation could accept infinite solutions independently, example: 𝑥 = , the denominator is null such number lead to a huge variety of numbers that actually represents a divergent solution leading to false equality. Then the solution number lead to infinite, or just indeterminate. Examples:  An easy equation can be represented by: (2𝑥 + 3) ⋅ 2 = 6𝑥 − 1 We proceed collecting all unknown variables in one side of the equality sign and all other values in the opposite side. It is important to remember that every term we move from one side to another we have to change its sign. The previous equation becomes: 4𝑥 + 6 = 6𝑥 − 1 → 4𝑥 − 6𝑥 = −1 − 7 Which leads to: −2𝑥 = −8 → 𝑥 = 4  Another example can be represented by the following fractional equation: 2(𝑥 + 3) 6𝑥 + =0 3 9 We proceed by calculating the entire fraction: 6(𝑥 + 3) + 6𝑥 =0 9 Now we can proceed by multiplying both sides of the equality by 9: 9 × [6(𝑥 + 3) + 6𝑥] = 0×9 9 At that point it turns in a simple equation like the previous example, we proceed to clear all calculation and to rewrite the equation in useful way: pag. 38  Cat. B1 - 2 Algebra 6𝑥 + 18 + 6𝑥 = 0 𝑎𝑛𝑑 𝑡ℎ𝑒𝑛 12𝑥 = −18 𝑥=− =−  (𝑥 + 1) = 0 In this example the are squared but the value of x is determinate and one only, if the square of the bracket is null means that bracket itself is null (𝑥 + 1)(𝑥 + 1) = 0 → (𝑥 + 1) = 0 Therefore: 𝑥 = −1  3𝑥(𝑥 + 1) = 0 Even in this example we can apply the previous considerations, leading directly to two determinate solutions: 𝑥 = 0 𝑎𝑛𝑑 𝑥 = −1  Let’s see the following example: √𝑥 + 1 = 0 → √𝑥 = −1 In this case the set of solutions is empty, this because the route square term that should be always positive ( 𝑥 ≥ 0) Test yourself:  (4𝑥 − 1) + 2𝑥 +3 =0  + (𝑥 + 3) = 1  (𝑥 + 2) + 2[6(3𝑥 + 2)] = 0 pag. 39  Cat. B1 - 2 Algebra  √2𝑥 + 1 = 0  3𝑥 + 2 +5 =0 [( )⋅( )]⋅  =1 ( )  𝑥 − 1 = 2𝑥 + 2𝜋  (3𝑥) ⋅ + 3 = 0  16𝑥 + 3(2𝑥 − 1) = 2𝑥 + 2  3 ⋅ (𝑥 + 3) + 2 = 0 ( )( )( )  =0 ( )( )  =0 ( )( ) 2.8 Second degree equations with one unknown In mathematics a quadratic equation is an algebraic equation with one unknown x, which is present with two as the maximum degree (x2). In fact a quadratic equation is a polynomial equation of the second degree. The generic form is: 𝑎𝑥 + 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. An equation of this type will produce a curve called a parabola. The actual value for coefficients a, b and c will determine the exact shape and position of the curve. pag. 40  Cat. B1 - 2 Algebra It will be noted that one of the curves cuts the x-axis at points 𝑃 and 𝑃 , they are are known as the roots of the equation. Alternatively, 𝑃 and 𝑃 are the values of x which satisfy the condition y = ax2 + bx + c = o. According to the fundamental law of algebra, the solutions of a quadratic equation are always 2. It can distinguish two cases: 1. In the set of real numbers the equation admits 2 solutions, which can be coincident, or none 2. In the set of complex numbers the equation admits always 2 solutions, which can be coincident. 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 symbol = 2. It must write the generic form of the equation. 3. it must apply the resolving formula: pag. 41  Cat. B1 - 2 Algebra −𝑏 ± √𝑏 − 4𝑎𝑐 𝑥= 2𝑎 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. Note that the term 𝑏 − 4𝑎𝑐 ≥ 0 should be positive, the term under the square root it always positive in the real numbering set. Example  Find the roots of y = 2𝑥 − 2𝑥 − 4=0 where (a = 2, b = -2, c = -4) −(−2) ± (−2) − 4(2)(−4) 𝑥= 2(2) +2 ± √4 + 32 +2 ± 6 = 4.4 𝑥 = 2 𝑜𝑟 𝑥 = −1 Now let’s take the same equation replacing c = 4, the equation then is:  2𝑥 − 2𝑥 + 4=0 That means we have this situation: (−2) − 4(2)(4) = √4 − 32 Where 4 − 32 < 0. This kind of equation cannot be solved in ℝ set numbers. Later in this book we will refer to the basic principles of complex numbers. There are other forms of equations of second degree, those cases that do not require to use the previous method but can sole somehow directly manipulating the equation in order to simplify it and to reach the results. The procedure is like first degree equations or linear equations, because the do represent a line in which each point on “y” axes is related to a unique value on “x” axes. As shown on the previous graph, the quadratic equations are parabolic that means some points of “y” axes have two relations with “x” axes which are the solutions of the quadratic equation. Other point are not represented on the graph (as linear equations) therefor are not belonging the solution points domain. If a second degree does not assume the standard form, where all coefficients are different from zero. If the coefficient b or c are null, the equation can be solved as standard equation but two possible results. we can have the following cases: pag. 42  Cat. B1 - 2 Algebra 1. ax2 + bx = 0 type 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 This equation where c = 0 is represented as a product of two terms, and one of theme is null because the equality is null. Example  2𝑥 + 3𝑥 = 0 Using the associative propriety of multiplication, we collect x from each term: 𝑥(2𝑥 + 3) = 0 The logic thinking of this equality that one of the product terms is null that leads to: 3 𝑥 = 0 𝑜𝑟 𝑥 = − 2  − 𝑥 + 𝑥=0 We apply the same procedure, because the equation is presented as the case under study where c = 0. 1 𝑥 −3𝑥 + =0 2 We obtain: 1 𝑥 = 0 𝑜𝑟 𝑥 = 6 pag. 43  Cat. B1 - 2 Algebra 2. ax2 + c = 0 type In this case the coefficient b = 0. The term c must be put at the second member and then it can be divided by a: 𝑥 =− The solutions can be found doing the square root: 𝑐 𝑥=± − 𝑎 Examples Let’s take this simple example: x −1=0 We can have easily 𝑥 = ±√1 = ±1 Now let take the second case where x +1=0 It leads to: 𝑥 = −1 This result has not solutions in the real number set. Because the square of a real number is always positive. But in the complex set the square can be negative.  𝑥 − √2 𝑥 + √2 = 0 𝑥 − √2 =0 And then 𝑥 = ±2  𝑥 −3 =0 𝑥 = 3 ⇒ 𝑥 = ±√3 pag. 44  Cat. B1 - 2 Algebra 2.9 Simultaneous equations and second degree equations with one unknown Simultaneous equations 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 below steps. There are generally two methods to solving simultaneous equations: 1. By substitution 2. By elimination. Both of them are like the equation procedures of first degree and we have more than one unknown variable. What we have to solve here is each one of the unknown values using one of methods mentioned above. It may be better to use one method over the other for certain type of simultaneous equations question. For example: 2𝑥 + 𝑦 = 5 𝑎𝑛𝑑 𝑥 + 2𝑦 = 7 Remember that it first has to represent the equations in a clear form: 2𝑥 + 𝑦 = 5 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 + 2𝑦 = 7 𝑥 = 7 − 2𝑦 The next step consists of replacing the new of “x” expressed in in terms of “y” into the other equation 2(7 − 2𝑦) + 𝑦 = 5 𝑦=3 Now, substituting the value of y in the second equation, it obtains the value of x: pag. 45  Cat. B1 - 2 Algebra 𝑥 + 2𝑦 = 7 𝑥 + 2(3) = 7 𝑥=1 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; it cancels x by subtraction. This leaves y: 𝑥 + 2𝑦 = 7 → (∙ 2) → 2𝑥 + 4𝑦 = 14 2𝑥 + 𝑦 = 5 (−) 2𝑥 + 4𝑦 = 14 0𝑥 − 3𝑦 = −9 → 𝑦 = 3 Now substituting the value of y in the second equation, it arrives at the same conclusion: 𝑥=1 Another tip to solve the simultaneous equations is to find the two terms by finding the equality of one unknown variables in both equations in order to induce the equality of one unknown value. Let’s take the previous example: 2𝑥 + 𝑦 = 5 𝑥 + 2𝑦 = 7 We can find “y” in terms of “x” in both equations: 𝑦 = 5 − 2𝑋 7−𝑋 𝑦= 2 We can deduce that the two terms of the right of the equality are equal because both of them are equal to “y”. 7−𝑥 5 − 2𝑥 = 2 And then we can proceed solving as an equation of first degree: 10 − 4𝑥 = 7 − 𝑥 ⇒ −3𝑥 = −3 And then 𝑥 = 1. We can find the value of “y” by replacing “x” value in one of the equations: 𝑦 = 5 − 2(1) = 3 pag. 46  Cat. B1 - 2 Algebra 2.10 Indexes and powers, negative and fractional indexes It is defined as power of a number A raised to the exponent n the product of n factors equal to A: 𝐴 = 𝐴 ∙ 𝐴 ∙ 𝐴 ∙ 𝐴 … 𝑓𝑜𝑟 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: 1. 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. In other terms: If 𝑎 ≠ 0 then 𝑎 = 1 2. 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: 𝑎 ⋅ 𝑎 = 𝑎( ) Examples:  10 ⋅ 10 = 10( ) = 10  𝑎 ⋅ 𝑎 ⋅ 𝑎 = 𝑎( ) =𝑎  𝑎 ⋅𝑎 = = 𝑎( ) =𝑎 3. The quotient of two powers having the same base, is a power with the same base and as exponents the difference of exponents. The previous relation is applicable even if x and/or y is/are negative. This means that the sign of algebraic sum of the powers does not matter and depends only on the power values and signs. Examples: Ex: 𝟑 𝟏 𝟏  𝟏𝟎 = = = 𝟎. 𝟎𝟎𝟏 𝟏𝟎𝟑 𝟏𝟎𝟎𝟎 𝟏 𝟏  = 𝟏 → 𝟐𝟓 𝟓 𝟐 𝟐𝟓 𝟏 𝟏𝟎𝟑  =𝟏⋅ = 𝟏𝟎𝟎𝟎 𝟏𝟎 𝟑 𝟏  −𝟐𝒂 ⋅ 𝒂 = −𝟐𝒂𝟔 𝟐 𝟒 pag. 47  Cat. B1 - 2 Algebra 4. The power of a power is a power having the same base and product of the exponents as exponent. This rule can be summarized as following: (𝑎 ) = 𝑎 ⋅ Examples  (10 ) = 10  (𝑎 ) = 𝑎  (10 ) = 1𝑂 There are also powers with negative exponents (-n): 1 𝑎 = 𝑎 The definition of a power permits to use fractional exponents (x/y), where x and y are prime number between them. 𝑎 = √𝑎 Examples : 2 = 2 = √4 Test yourself  𝑎 + 3𝑎 + 3𝑏 + 2𝑏 + 3𝑏 = ? + 𝑘 =?  𝑎 ⋅ 𝑎 + 2𝑎 =?  (10 ) ⋅ 10 = ? pag. 48  Cat. B1 - 2 Algebra  + 10 =?  𝑘10 + 10  −10 ⋅ 10 + 𝑘 ⋅ 𝑘 =?  10 ⋅ 10 =?  𝑘 × 𝑘 =?  (𝑘 ) ⋅ 𝑘 =?  8 = 3√8 =?  10 ⋅ 10 = 10 =?  𝑎⋅𝑎⋅𝑎 =?  + 𝑘 =?  𝑎 ⋅ 𝑎 + 2𝑎 =?  (10 ) ⋅ 10 = ?  + 10 =?  𝑘10 + 10  −10 ⋅ 10 + 𝑘 ⋅ 𝑘 =?  𝑘 (𝑘 + 2𝑘 ) =? pag. 49  Cat. B1 - 2 Algebra 2.11 Binary and other applicable numbering systems 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. Generally, positional notations are used, that is a different meaning is assigned to the figures used to represent a number according to their position inside the number itself. The fundamental equation to represent a number N is: 𝑁 = 𝐶 ∙𝑏 +𝐶 ∙𝑏 +.. +𝐶 ∙ 𝑏 + 𝐶 ∙ 𝑏 Where the number b indicates the radix of the notation and the numbers C indicate the figures that must multiply the different multiples. Using the fundamental equation for representing numbers written before, it is possible, after having chosen an adequate radix that is a set of figures to create any kind of notation. Now, we will analyze two notations: decimal notation and binary notation. The decimal notation is a positional notation that uses ten figures, from 0 to 9, to represent numbers. For example the number 735 can be written as: 735 = 7 ∙ 10 + 30 ∙ 10 + 5 ∙ 10 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. Let’s try and count using the binary notation: 0, 1, 10, 11, 100, 101, 110, 111 … 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. Look at the example: 101 = 1 ∙ 2 + 0 ∙ 2 + 1 ∙ 2 = 1 ∙ 4 + 0 ∙ 2 + 1 ∙ 1 = 5 Note: In real mathematical applications, a decimal integer or dyadic fractional value converted to binary and then back to decimal matches the original decimal value; a non-dyadic value converts back only to an approximation of its original decimal value. For example, 0.1 in decimal — to 20 bits — is pag. 50  Cat. B1 - 2 Algebra 0.00011001100110011001 in binary; 0.00011001100110011001 in binary is 0.09999942779541015625 in decimal. Increasing the number of bits of precision will make the converted number closer to the original. CONVERSION TO OTHER BASES To convert a from decimal to any other base can be done by dividing the decimal number repeatedly by the new base and keeping the remainder. The remainder gives the number in the new base and should be read from bottom to top. Example: Convert 32 in decimal base to binary base. 32 2 rem 0 16 2 rem 0 8 2 rem 0 4 2 rem 0 2 2 rem 0 1 2 rem 1 0 We write the binary conversion from the bottom 32(10) = 100000(2) Scientific calculations include very large and very small numbers. To simplify operations and to reduce the percentage of error, scientific notation is used. Scientific notation is a way of indicating a value as the product of a number between 1 and 9.999 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. pag. 51  Cat. B1 - 2 Algebra 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 doesn’t change the value of the number, but only the way in which it is written. The multiplication of very large or very small numbers becomes easier using the scientific notation. The first operation is writing down the numbers to multiply in scientific notation. After that, the product of the numbers is calculated and the algebraic sum of exponents is found. For example: 0.275 = 2.75 ∙ 10 30000 = 3 ∙ 10 0.275 ∙ 30000 = (2.75 ∙ 10 ) ∙ (3 ∙ 10 ) = 8.25 ∙ 10 Division in scientific notation is similar to multiplication. The first operation is converting the numbers in scientific notation, then dividing them and finally finding the power of ten subtracting the exponents. For example: 5280 = 5.28 ∙ 10 0.25 = 2.5 ∙ 10 5280 ÷ 0.25 = (5.28 ∙ 10 ) ÷ (2.5 ∙ 10 ) = 2.112 ∙ 10 The Octal Numbering System is very similar in principle to the previous hexadecimal numbering system except that in Octal, a binary number is divided up into groups of only 3 bits, with each group or set of bits having a distinct value of between 000 (0) and 111 ( 5+2+1 = 7 ). Test yourself:  Convert the following number to binary base: 9; 16; 27  Convert the following binary numbers to decimal numbers: - 10010 - 110101 - 0100110 - 10000 - 00001 pag. 52  Cat. B1 - 2 Algebra 2.12 Logarithms Logarithms are a mathematical tip that was studied to simplify multiplication and division of large numbers. That means multiplication and division to be calculated using addition and subtraction and this turns a in good instrument in electronics and engineering. The 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 𝑥 → 𝑥 = 𝑧 Examples  log ( ) 100 = log ( ) 10 =2  𝑙𝑜𝑔( ) 8 = 𝑙𝑜𝑔( ) 2 = 3  𝑙𝑜𝑔( ) 64 = 𝑙𝑜𝑔( ) 8 = 2 The main propreties of logarithms are: 1. The logarithm of the product of two numbers is equal to the sum of the logarithms of the same numbers: 𝑙𝑜𝑔 (𝑎 × 𝑏) = 𝑙𝑜𝑔 𝑎 + 𝑙𝑜𝑔 𝑏 Examples  𝑙𝑜𝑔( ) (200) = 𝑙𝑜𝑔( ) (2 × 100) = log ( )2+ log ( ) 100 = log ( )2+ 2  𝑙𝑜𝑔( ) 24 = 𝑙𝑜𝑔( )(8 × 3) = 3 + 𝑙𝑜𝑔( ) 3  𝑙𝑜𝑔( ) 100 = 𝑙𝑜𝑔( ) (10 × 10) = 𝑙𝑜𝑔( ) 10 + 𝑙𝑜𝑔( ) 10 =2 2. The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the same numbers: 𝑎 𝑙𝑜𝑔 = 𝑙𝑜𝑔 𝑎 − 𝑙𝑜𝑔 𝑏 𝑏 pag. 53  Cat. B1 - 2 Algebra Examples:  𝑙𝑜𝑔( ) 100 = 𝑙𝑜𝑔( ) = 𝑙𝑜𝑔( ) 1000 − log ( ) (10) =3−1=2  𝑙𝑜𝑔( ) = 𝑙𝑜𝑔( ) − 𝑙𝑜𝑔( ) 2 3. The logarithms of the inverse of a is the opposite of the logarithm of a: 1 = −𝑙𝑜𝑔 𝑎 𝑙𝑜𝑔 𝑎 This propriety can be deduced from the previous one, in fact the denominator of the logarithm turns into a negative logarithm. Examples:  𝑙𝑜𝑔( ) = − 𝑙𝑜𝑔( )2  − 𝑙𝑜𝑔( ) = − − 𝑙𝑜𝑔( ) 10 = +1  − log ( ) =, − log ( )3 4. 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 𝑎 = 𝑘 × 𝑙𝑜𝑔 𝑎 Examples  log ( ) 100 = log ( ) 10 = 2 log ( ) 10 = 2⋅1=2  log ( ) 8 = log ( ) 3 = 3 log ( ) 2 = 3 ⋅ 1 = 3 5. The logarithm of k-root of a is equal to the quotient between the logarithm and k: 1 𝑙𝑜𝑔 √𝑎 = × 𝑙𝑜𝑔 𝑎 𝑘 This propriety is similar to the previous one. pag. 54  Cat. B1 - 2 Algebra Examples:  𝑙𝑜𝑔( ) 100 = 𝑙𝑜𝑔( ) 10 = 𝑙𝑜𝑔( ) 10 = 2⋅1=2  𝑙𝑜𝑔( ) √2 = 𝑙𝑜𝑔( )2 6. The logarithm of a, with also a as base, is 1: 𝑙𝑜𝑔 𝑎 = 1 7. The logarithm of 1, with any base, is 0: 𝑙𝑜𝑔 1 = 0 8. The following identity is true: 𝑎 = 𝑙𝑜𝑔 𝑎 =𝑥 Logarithms can be calculated with any positive base (different from 1); the bases generally used are: 1. Base 10 (decimal logarithms): log10, log, Log 2. Base e (natural logarithms): ln 3. Base 2 (binary logarithms): log2. It’s also possible to do the base change: 𝑙𝑜𝑔 𝑏 × 𝑙𝑜𝑔 𝑥 = 𝑙𝑜𝑔 𝑥 pag. 55  Cat. B1 - 2 Algebra 1.13 Complex numbers If we consider the previous equation of second degree that present: 𝐛𝟐 − 𝟒𝐚𝐜 ≤ 𝟎 Example