M01 - Mathematics PDF

Module 1 CAT. B1/B2 Mathematics Ed.0 Rev.1 M01 - Mathematics Ed.0 Rev.1 Contents 1. Arithmetic............................................................................................................................................... 3 Arithmetical terms and signs.................................................................................................... 3 1.1.1. Addition.................................................................................................................................................... 4 1.1.2. Subtraction............................................................................................................................................. 5 1.1.3. Multiplication........................................................................................................................................ 6 1.1.4. Division...................................................................................................................................................... 6 1.1.5. Arithmetic expression.................................................................................................................... 7 Methods of multiplication and division............................................................................. 7 Fractions and decimals................................................................................................................. 9 1.3.1. Fractions................................................................................................................................................... 9 1.3.2. Decimals..................................................................................................................................................12 Factors and multiples....................................................................................................................15 Weights................................................................................................................................................... 18 Measures and conversion factors......................................................................................... 19 1.6.1. Systems of measurement.......................................................................................................... 19 1.6.2. Scale, latitude, and longitude................................................................................................ 24 Ratio and proportion.................................................................................................................... 29 1.7.1. Ratio......................................................................................................................................................... 29 1.7.2. Proportion............................................................................................................................................ 29 Averages and percentages...................................................................................................... 30 Area and volumes............................................................................................................................ 32 1.9.1. Area............................................................................................................................................................ 33 1.9.2. Volume................................................................................................................................................... 39 Squares................................................................................................................................................... 42 Cubes....................................................................................................................................................... 42 Square and cube roots................................................................................................................ 43 2. Algebra................................................................................................................................................... 45 Evaluating simple algebraic expressions....................................................................... 45 2.1.1. Addition................................................................................................................................................. 45 M01 - Mathematics Ed.0 Rev.1 2.1.2. Subtraction.......................................................................................................................................... 46 2.1.3. Multiplication and division....................................................................................................... 46 2.1.4. Use of brackets................................................................................................................................. 46 2.1.5. Simple algebraic fractions........................................................................................................ 47 2.1.6. Linear equations and their solutions................................................................................ 48 2.1.7. Indices and powers, negative and fractional indices............................................ 48 2.1.8. Binary and other applicable numbering systems................................................... 49 2.1.9. Simultaneous equations and second-degree equations with one unknown.................................................................................................................................................51 2.1.10. Logarithms........................................................................................................................................... 53 3. Geometry............................................................................................................................................... 55 Simple geometrical constructions...................................................................................... 55 Graphical representation...........................................................................................................57 Nature and uses of graphs........................................................................................................57 3.3.1. Graphs of equations/functions.............................................................................................. 59 Simple trigonometry.................................................................................................................... 60 3.4.1. Trigonometric relationships..................................................................................................... 61 Use of tables and rectangular and polar coordinates.......................................... 62 M01 - Mathematics Ed.0 Rev.1 1. Arithmetic Arithmetical terms and signs Arithmetic has been drafted by Babylonian, Egyptian, and Chinese more than 2000 years ago and it is the foundation of mathematics. Arithmetic concerns the properties of numbers and the four main operations: 1. Addition. 2. Subtraction. 3. Multiplication. 4. Division. Considering main operations, we must consider sets of numbers. · The set of natural numbers is indicated by the symbol ܰ. · Natural numbers are commonly used to count any set of objects, like ͳǡ ʹǡ ͵ǡ Ͷǡ ǥ until infinity. · The set of whole numbers is indicated by the symbol ܹ. · Whole numbers are natural numbers (ܰ) including 0. · The set of integers is indicated by the symbol ܼ. · The set of integers (ܼ) includes negative numbers, like െͳǡ െʹǡ െ͵ǡ etc. · ܰ set includes zero and all the nonnegative integers. The result of an addition (൅) or multiplication (ൈ) of two or more natural numbers is always a natural number. The result of a subtraction of two or more natural numbers can be a natural number, a whole number, or an integer. Considering division (ൊ or Ȁ), we must consider fractional numbers, likeͳȀʹǡ ͵ȀͶǡ െͷȀʹ, etc. Fractional numbers belong to rational numbers. The set of rational numbers is indicated by ܳ. We have also to consider irrational numbers. Irrational numbers are those numbers that cannot be expressed as a fraction (ܽȀܾ), for any integers ܽ and ܾ with ܾ different from zero. Irrational numbers set is indicated by the symbol ‫ܫ‬. There is a last set of numbers we should consider, set that includes integers, rational and irrational numbers, it is real number set. It is indicated by the symbol ܴ. Figure 1 provide a very simple graphical representation of set of numbers. 1 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Figure 1 sets of number The four main math operations are indicated by a specific symbol as follow: 1. Addition (൅) 2. Subtraction (െ) 3. Multiplication (ൈ or ‫)כ‬ 4. Division (ൊ or Ȁ) 1.1.1. Addition The input of an addition are two generic numbers, addends. The output of the addition is a third number, called sum or total. The sum is obtained by counting as many units as those indicated by the second addend after the first addend, and so on according to the number of addends. ͳ ൅ ͵ ൌ ͳ ൅ ͳ ൅ ͳ ൅ ͳ ൌ Ͷ ʹ ൅ ʹ ൌ ʹ ൅ ͳ ൅ ͳ ൌ Ͷ Wefly Academy srl 4 2 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 The addition has three properties: 1. Commutative. 2. Associative. 3. Dissociative Commutative property Changing the order of addends , the sum doesn’t change. ͳ ൅ ʹ ൌ ʹ ൅ ͳ Associative property Replacing two or more addends with their sum the result doesn’t change. ͵ ൅ ʹ ൅ ͷ ൌ ͷ ൅ ͷ ൌ ͵ ൅ ͹ Dissociative property Replacing one addend with one or more addend s whose sum is the replaced addend , the result of the addition doesn’t change. ͳͲ ൅ ͷ ൅ ͸ ൌ ͷ ൅ ͷ ൅ ͷ ൅ ͵ ൅ ʹ ൅ ͳ 1.1.2. Subtraction The input of a subtraction two generic numbers, the minuend and the subtrahend. The output is a third one, called the difference, obtained by subtracting from the minuend as many units as those indicated by the subtrahend. ͵െʹൌ ͵െͳെͳൌ ͳ ͷെ͵ൌ ͷെͳെͳെͳൌ ʹ Wefly Academy srl 5 3 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Subtraction has only one property, invariance. Adding or subtracting the same number from the two terms of a subtraction the result doesn’t change. ͵ െ ʹ ൌ ͵ ൅ ͳȂ ሺʹ ൅ ͳሻ ൌ ͳ ͷ െ ͵ ൌ ͷ െ ͳ െ ሺ͵ െ ͳሻ ൌ ʹ 1.1.3. Multiplication The input of a multiplication are two generic numbers. These two generic numbers are called factors (multiplicand) and multipliers. The output is a third number, the product, obtained by adding the units of the first one as many times as the units of the second one. ʹൈ͵ൌʹ൅ʹ൅ʹ ͳͲ ‫ ʹ כ‬ൌ ͳͲ ൅ ͳͲ Multiplication can be considered as a sequence of addition. 1.1.4. Division The input of a division are two generic numbers, the dividend and the divisor. The output is a third number, the quotient. The quotient is the number of times that the divisor is “contained” within the dividend. Quotient can have a remainder; it is the part of the dividend that cannot be divided by the divisor. ͸Ȁʹ ൌ ͵ ͳͲ ൊ ͵ ൌ ͵ሺremainder isͳሻ Division can also be considered as a sequence of subtraction. Quotient is the number of times the divisor can be subtracted from the dividend. ͸െʹൌ Ͷ ͸Ȁʹ ൌ ͵ ՜ ൝Ͷ െ ʹ ൌ ʹ ʹെʹൌ Ͳ Quotient ൌ ͵, remainder ൌ Ͳ ͳͲ െ ͵ ൌ ͹ ͳͲ ൊ ͵ ൌ ͵ with remainder ൌ ͳ ՜ ൝͹െ͵ൌͶ Ͷെ͵ൌͳ Quotient ൌ ͵, remainder ൌ ͳ Wefly Academy srl 6 4 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 1.1.5. Arithmetic expression An arithmetic expression is an expression made by two or more numbers, separated by operation signs (൅ǡ െǡ‫ כ‬and ൊ) and possibly by brackets, like ሺሻǡ ሾሿǡ and ሼሽ. Solving an arithmetic expression is a matter of doing the different operations on the numbers in a specific order. 1. Do first the operations inside the brackets if any 2. Do first multiplications and divisions 3. Do additions and subtractions. ሺͷ െ ʹሻ ൅ ሺͻ ൅ ͳሻ ‫ ʹ כ‬െ Ͷ ൌ ͵ ൅ ͳͲ ‫ ʹ כ‬െ Ͷ ൌ ͵ ൅ ʹͲ െ Ͷ ൌ ͳͻ ሼ͵ ൅ ሾሺͶ ൈ ʹሻ െ Ͷሿሽ െ ʹ ൌ ൛͵ ൅ ሾͺ െ Ͷሽൟ െ ʹ ൌ ሼ͵ ൅ Ͷሽ െ ʹ ൌ ͹െʹ ൌ ͷ Methods of multiplication and division Multiplication has four properties. 1. Commutative Changing the order of factors, the product doesn’t change. ʹ ‫ ͵ כ‬ൌ ͵ ‫ʹ כ‬ 2. Associative. Replacing two or more factors with their product the result doesn’t change. ʹ ‫ כ ͵ כ‬Ͷ ൌ ͸ ‫ כ‬Ͷ ൌ ʹ ‫ ʹͳ כ‬ൌ ʹͶ 3. Dissociative. Replaci ng one factor with one or more factors whose produ ct is the replaced factor the result of the multiplication doesn’t change. Ͷ ‫ כ‬͸ ‫ ʹ כ‬ൌ ʹ ‫ כ ʹ כ‬͸ ‫ ʹ כ‬ൌ Ͷ ‫ ʹ כ ʹ כ ͵ כ‬ൌ Ͷͺ Wefly Academy srl 7 5 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 4. Distributive. 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 sing le term of the addition or subtraction, and then sum or subtract them. ሺʹ ൅ ͵ሻ ‫ כ‬Ͷ ൌ ͷ ‫ כ‬Ͷ ൌ ʹͲǢ ሺʹ ൅ ͵ሻ ‫ כ‬Ͷ ൌ ʹ ‫ כ‬Ͷ ൅ ͵ ‫ כ‬Ͷ ൌ ʹͲǢ ʹ ‫ כ‬ሺ͵ െ ͳሻ ൌ ʹ ‫ ʹ כ‬ൌ ͶǢ ʹ ‫ כ‬ሺ͵ െ ͳሻ ൌ ʹ ‫͵ כ‬Ȃ ʹ ‫ ͳ כ‬ൌ ͸Ȃ ʹ ൌ ͶǢ Division has two properties invariance and distributive. 1. Invariance. Dividing or multip lying by the same number the two terms of a division the result doesn’t change. Ͷ ൊ ʹ ൌ ሺͶ ‫ʹ כ‬ሻȀሺʹ ‫ʹ כ‬ሻ ൌ ʹ ሺͶ ൊ ʹሻȀሺʹ ൊ ʹሻ ൌ ʹȀͳ ൌ ʹ 2. Distributive To divide the terms of an addition or of a subtraction by a number it is possible to divide each single term of the add ition or of the subtraction by the given divisor and then sum or subtract them. ሺͶ ൅ ʹሻ ൊ ʹ ൌ ͸Ȁʹ ൌ ͵ ሺͶ ൅ ʹሻ ൊ ʹ ൌ ሺͶȀʹሻ ൅ ሺʹȀʹሻ ൌ ʹ ൅ ͳ ൌ ͵ Wefly Academy srl 8 6 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Fractions and decimals 1.3.1. Fractions A fraction is made of two numbers separated by the symbol ‘Ȁ’. ଵ ହ ଷ ǡ ǡ ǡ etc. ଶ ସ ଵ଴ Numerator is the number above the fraction symbol, denominator is the number under the fraction symbol ݊‫ݎ݋ݐܽݎ݁݉ݑ‬ ݀݁݊‫ݎ݋ݐܽ݊݅݉݋‬ A fraction is a portion of a given quantity and it is the result of a division of two integer numbers. Fractions can be: 1. Proper 2. Improper A fraction is proper when numerator is smaller than denominator. The value of a proper fraction is always less than 1. A fraction is improper when numerator is greater than denominator. The value of an improper function is always more than 1. If numerator is equal to the denominator, the quotient of the fraction is 1. Wefly Academy srl 9 7 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Addition Addition requires fractions to have the same denominator. 1. Find the Lowest Common Denominator (LCD). The lowest common denominator is the smallest number that is exactly divisible by each denominator of a set of fractions. It is sometimes known as the least common denominator. 2. Do the addition of numerators. 3. Reduce the fraction to its lowest terms, if possible. ͳ ͵ ሺͳ ‫ʹ כ‬ሻ ൅ ͵ ͷ ൅ ൌ ൌ ʹ Ͷ Ͷ Ͷ Subtraction Subtraction requires fractions to have the same denominator. 1. Find the Lowest Common Denominator (LCD). 2. Do the subtraction of numerators. 3. Reduce the fraction to its lowest terms, if possible. ͳ ͵ ሺͳ ‫ʹ כ‬ሻ െ ͵ െͳ െ ൌ ൌ ʹ Ͷ Ͷ Ͷ Product 1. Do the product of numerators 2. Do the product of denominator 3. Reduce the fraction to its lowest terms, if possible. ͵ ʹ ͵‫ʹכ‬ ͸ ͵ ͳ ‫ כ‬ൌ ൌ ൌ ൌ ͷ ͸ ͷ ‫ כ‬͸ ͵Ͳ ͳͷ ͷ Wefly Academy srl 10 8 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Division 1. Change numerator with denominator for divisor fraction. 2. Do the product of numerators 3. Do the product of denominator 4. Reduce the fraction to its lowest terms, if possible. ͵ ͸ ͵ ʹ ͵‫ʹכ‬ ͸ ͵ ͳ ൊ ൌ ‫ כ‬ൌ ൌ ൌ ൌ ͷ ʹ ͷ ͸ ͷ ‫ כ‬͸ ͵Ͳ ͳͷ ͷ Multiplying and or dividing both the numerator and the denominator of a fraction for the same number the fraction doesn’t change. By using the invariance property of division, we can easily simplify fraction to lowest terms. Mixed numbers Mixed numbers are those numbers made by integers and proper fractions. ͵ ͵൅ Ͷ Any integer is an improper fraction where the denominator is equal to 1. ͵ ͵ ൅ ͳ Ͷ Solving the operation is like to solve a standard operation between fractions. ͵ ͵ ሺ͵ ‫ כ‬Ͷሻ ൅ ͵ ͳͷ ൅ ൌ ൌ ͳ Ͷ Ͷ Ͷ 1 -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 ob tained as numerator. 2 - Adding improper fractions the sum is an improper fract ion. Wefly Academy srl 11 9 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 1.3.2. Decimals We have seen a fraction is like a division. Frequently working with decimal numbers can be better than working with fractions. We can easily convert any fraction into a decimal number just doing the division. ͹ͷ ൌ ͲǤ͹ͷ ͳͲͲ We have that ͹ͷȀͳͲͲ is equal to ͲǤ͹ͷ. Within a decimal number we can have several decimal figures. ͲǤ͹ͷ has two figures, ͲǤͲͳʹ͵ has four figures. Regardless the number of figures: the first figure refers to tens, the second to hundreds, the third to thousands, etc. Addition/Subtraction The addition/subtraction of decimal numbers uses the same laws with that of integers. We need only to align the points indicating the decimal number on the same vertical line. Ͳǡͳʹ͵ ൅ Ͳǡʹͳ ͲǤ ͳ ʹ ͵ ൅ ͲǤ ʹ ͳ ͲǤ ͵ ͵ ͵ ͲǤ͵͵͵ Product The product is done as a standard product between numbers without zero. Calculate the sum of the digits (multiplier and multiplicand). Move the point/comma left as for the sum of the digits. ͲǤͳʹ͵ ‫Ͳ כ‬Ǥʹͳ ՜ ͳʹ͵ ‫ ͳʹ כ‬ൌ ʹͷͺ͵ ͲǤͳʹ͵ has ͵ digits after the decimal point ͲǤʹͳ has ʹ digits after the decimal point Total digits are ͷ The result is ͲǤͲʹͷͺ͵ Wefly Academy srl 12 10 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Division We should consider: · Case 1 – dividend is an integer, and divisor is a decimal number. dividend is a decimal number and divisor is an integer. · Case 2 – dividend and divisor are both decimal numbers. · Case 3 – dividend is a decimal number and divisor is an integer. Case 1 We just need to convert the decimal number into an integer. We can do this by multiplying ͳͲ, ͳͲͲ, or ͳͲͲͲ, etc., according to the decimal digits. We can proceed in a similar fashion as for any standard division between integers. ͹ ͳͲ ͹ ൊ ͳǤͶ ൌ ‫כ‬ ͳǤͶ ͳͲ ͹Ͳ ൌ ͳͶ ൌ ͷ Case 2 We just need to convert dividend into integer or, convert both numbers into integer. ͳǤʹͷ ൊ ͲǤʹͷ ՜ ͳʹͷ ൊ ʹǤͷ or ͳʹͷͲ ൊ ʹͷ Case 3 It is similar to case 2, we need just to convert dividend into an integer number. ͺǤͻͷ ൊ ͷ ൌ ሺͺǤͻͷ ‫ͲͲͳ כ‬ሻ ൊ ͷ ൌ ͺͻͷ ൊ ͷ ൌ ͳǤ͹ͷ Wefly Academy srl 13 11 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Precision, significant figures and rounding Decimal numbers are expressed with many figures after the decimal point. ͲǤͳǡ ͲǤͲͲʹǡ ͲǤͳʹ͵Ͷͷ͸͹ͺͻǡ etc The higher is the numbers of figures, the higher the precision of the number becomes. Not all figures are required for practical reasons, figures can be limited/reduced. We can perform a rounding by keeping a certain number of figures only and then eliminating the remaining ones. In any case, the number obtained is only an approximation of the original number. We have to look at figures immediately to the right of the last one to be kept: 1. If figure is 5 or higher, last figure (to be kept) must be increased by 1. 2. If figure is lower than 5, last figure will be the same value. Let’s see two examples ͳǤʹ͵͹ to be rounded to the second figure The second figure is ͵, and the following is ͹. Since ͹ is higher than ͷ, the second figure will be rounded to Ͷ. The result is ͳǤʹͶ. ͳǤͻ͸͹Ͷ to be rounded to third figure The third figure is ͹, and the following one is Ͷ. Since Ͷ is lower than ͷ, the third figure will remain as the same value. The result is ͳǤͻ͸͹. From decimal to fraction Sometimes, it can be useful to transform a decimal into its fractional form. It can be done by: 1. Taking the numerator with the decimal number without the point. 2. Taking the denominator with a multiple of ten with as many 0 as the number of figures after the point. 3. Finally, reduce the fraction to its lowest terms, if possible. ͳʹͷ ͳ ͲǤͳʹͷ ൌ ൌ ͳͲͲͲ ͺ ͵ʹ ͳ͸ ͺ ͲǤ͵ʹ ൌ ൌ ൌ ͳͲͲ ͷͲ ʹͷ Wefly Academy srl 14 12 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Factors and multiples Concepts of multiple and divisor arise from multiplication and division. ‫ܣ‬, an integer is multiple of ‫ܤ‬, another integer, if there is a third integer ‫ ܥ‬that multiplied by ‫ ܤ‬gives as result ‫ܣ‬. ‫ ܣ‬ൌ ‫ ܥ כ ܤ‬ ͸ ൌ ʹ ‫͵ כ‬ ͳͲ ൌ ͷ ‫ʹ כ‬ At the same time, ‫ ܣ‬is divisor of a number ‫ ܤ‬only if dividing ‫ ܤ‬by ‫ ܣ‬the quotient is an integer ‫ ܥ‬and the remainder is zero: ‫ ܤ‬ൊ ‫ ܣ‬ൌ ‫ܥ‬ǡ ‫ ݎ‬ൌ Ͳ ͸ ൊ ʹ ൌ ͵ǡ ‫ ݎ‬ൌ Ͳ ͳͲ ൊ ͷ ൌ ʹǡ ‫ ݎ‬ൌ Ͳ Prime numbers Are those numbers that cannot be divided by any number except themselves and one. ͳǡ ͵ǡ ͷǡ ͹ ǥ Composite numbers Any number that is not a prime number. ͳʹǣ ሺ͸ ‫ʹ כ‬ሻǡ ሺ͵ ‫ כ‬Ͷሻǡetc. Divisors We can 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, A number divisible by 2 is cal led an even number. In other words when a number ends with 0, 2, 4, 6, 8. Wefly Academy srl 15 13 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 A number is divisible by 3 if the sum of its figures when it is divisibl e by 3. ʹͲ͹ ՜ ʹ ൅ Ͳ ൅ ͹ ൌ ͻ ʹͲ͹ is divisible by ͵ because ͻሺʹ ൅ Ͳ ൅ ͹ሻ is divisible by ͵. ʹͲ͹ ൊ ͵ ൌ ͸ͻ There are additional rules. A number is divisible by 4 if the last two di gits are 00 or if they f orm a number that is a multiple of 4, or if the second to 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. ͸ͲͲ ൊ Ͷ ൌ ͳͷͲ A number is divisible by 5 if its last figure is 0 or 5. ͵ͷ ൊ ͷ ൌ ͹ ͹ͲͲ ൊ ͷ ൌ ͳʹͲ A number is divisible by 6 if it is divisible by 2 and 3 at the same time. ͳʹ ൊ ͸ ൌ ʹ A number is divisible by 1 0 if its last figure is 0. ͳͶͲͲ ൊ ͳͲ ൌ ͳͶͲ Wefly Academy srl 16 14 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Factorization The process by which the prime numbers that are divisors of a given number are searched is called factorization, or prime decomposition. Factorization requires the division of the number by its smallest prime divisor until the quotient is 1. Let’s make an example ͳͺͲ We have to proceed step-by-step taking into consideration rules for divisors. It ends with Ͳ so it divisible by ͷ and ͳͲ. It is better to start with ͷ. ͳͺͲ ͷ ͵͸ ͵͸is divisible for ʹ and ͵ too. Let’s use ʹ. ͵͸ ʹ ͳͺ ͵ ͸ ʹ ͵ ͵ ͳ So, at the end ͳͺͲ ൌ ͷ ‫͵ כ ʹ כ ͵ כ ʹ כ‬ Any integer has infinite factorization Least Common Multiple (LCM) Given two or more numbers a, b, c it is the smallest multiple in common. Greatest Common Divisor (GCD) Given two or more numbers a, b, c it is the largest common divisor sha re by them. Let’s make some examples to better understand the above concepts. ͵͸Ͳ and ͵ͲͲ, LCM? First, we have to do factorization. ͵͸Ͳ ൌ ʹ ‫ כ ʹ כ ʹ כ‬ͷ ‫͵ כ ͵ כ‬ ͵ͲͲ ൌ ʹ ‫ כ ʹ כ‬ͷ ‫ כ‬ͷ ‫͵ כ‬ Wefly Academy srl 17 15 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Now we have to multiply the prime factors, common and non-common, each one taken with the highest exponent. ʹ ‫ כ ʹ כ ʹ כ‬ͷ ‫ כ‬ͷ ‫ ͵ כ ͵ כ‬ൌ ͳͺͲͲ Considering GCD. ͵͸ and ʹͶ, GCD? We have to consider their divisors ͵͸ ൌ ሼͳǡ ʹǡ ͵ǡ Ͷǡ ͸ǡ ͻǡ ͳʹǡ ͳͺǡ ͵͸ሽ and ʹͶ ൌ ሼͳǡʹǡ͵ǡͶǡ͸ǡͳʹሽ The greatest common divisor is ͳʹ. Weights The quantity of matter contained in a body is defined as mass. The unit of measurement of International System (SI) is the kilogram (݇݃). k letter is small because there is no Mr. Kilo Table 1 provides multiples and sub-multiples of SI weight units. Units of masses ͳͲ milligrams ሺ݉݃ሻ ൌ ͳ centigrams ሺܿ݃ሻ ͳͲ centigrams ൌ ͳ decigram ሺ݀݃ሻ ൌ ͳͲͲ milligrams ͳͲ decigrams ൌ ͳͲͲͲ milligrams ͳͲ grams ൌ ͳ dekagram ሺ݀ܽ݃ሻ ͳͲ dekagrams ൌ ͳ hectogram ሺ݄݃ሻ ͳͲ dekagrams ൌ ͳͲͲ grams ͳͲ hectograms ൌ ͳ kilogram ሺ݇݃ሻ ͳͲ hectograms ൌ ͳͲͲͲ grams ͳͲͲͲ kilograms ൌ ͳ megagram ሺ‫݃ܯ‬ሻ or metric ton ሺ‫ݐ‬ሻ Table 1 unit of masses multiples and sub -multiples Wefly Academy srl 18 16 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 In the British System (BS) the unit for mass is the pound; ͳ pound ൌ ͲǤͶͷ͵ kg. Here after the main units for mass according to the BS. Table 2 shows the BS weight units and their SI counterparts. Unit Pounds Grams Kilograms grain ሺ݃‫ݎ‬ሻ ͳ Τ ͹ͲͲͲ ͲǤͲ͸Ͷ͹ͻͺͻͳ drachm ሺ݀‫ݎ‬ሻ ͳ Τ ʹͷ͸ ͳ͹Ǥ͹ͳͺͶͷͳͻͷ͵ͳʹͷ ounce ሺ‫ݖ݋‬ሻ ͳ Τ ͳ͸ ʹͺǤ͵Ͷͻͷʹ͵ͳʹͷ pound ሺ݈ܾሻ ͳ Ͷͷ͵Ǥͷͻʹ͵͹ ͲǤͶͷ͵ͷͻʹ͵͹ stone ሺ‫ݐݏ‬ሻ ͳͶ ͸͵ͷͲǤʹͻ͵ͳͺ ͸Ǥ͵ͷͲʹͻ͵ͳͺ quarter ሺ‫ ݎݍ‬or‫ݎݐݍ‬ሻ ʹͺ ͳʹǤ͹ͲͲͷͺ͸͵͸ hundredweight ሺܿ‫ݐݓ‬ሻ ͳͳʹ ͷͲǤͺͲʹ͵ͶͷͶͶ ton ሺ‫ݐ‬ሻ ʹʹͶͲ ͳͲͳ͸ǤͲͶ͸ͻͲͺͺ Table 2 British system units and its SI counterparts We should take into consideration that the weight is a force, and it is the product of the mass for gravity acceleration. ܹ݄݁݅݃‫ ݐ‬ൌ ݉ܽ‫݃ כ ݏݏ‬ ݉ ݃ ൌ ͻǤͺͳ ‫ݏ‬ଶ Measures and conversion factors 1.6.1. Systems of measurement Physical quantities require units of measurement to be standardized. This is to provide a common and sharable way to take measurements. A measurement is a quantitative descriptio n of on e or more fundamental properties compared to a standard. The measurement of a quantity consists of two parts, the first gives how many times the measurement is compare to a standard unit and the second part gives the name of the unit. The value of any measurable quantity ܳ can be expressed by the product between the number factor ሺ݊ሻ and the unit ሺ‫ݑ‬ሻ. ܳ ൌ ݊ ‫ݑ ڄ‬ Wefly Academy srl 19 17 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Different systems of measurement are officially accepted; each system is based on diverse sets of standard/fundamental units. The International System (SI) has 7 main units as described by Table 3: Order Unit Unit symbol 1. Length Meter ݉ 2. Time Second ‫ݏ‬ 3. Quantity of substance Mole ݉‫݈݋‬ 4. Mass Kilogram ݇݃ 5. Temperature Kelvin ݇ 6. Intensity of current Ampere ‫ܣ‬ 7. Luminous intensity Candle ܿ݀ Table 3 The 7 main units of the International System (SI) Ampere, Kelvin units adopt capital letters as units ሺ‫ܣ‬ሻǡ ሺ‫ܭ‬ሻ because there were Mr. Ampere and Mr. Kelvin. At the same time kilograms adopt small letter as unit ሺ݇݃ሻ. There were not Mr. Kilo. Even if SI is a standard, frequently we can find British or American systems within aviation industry. Here after you can find some conversion listed in Table 4. Quantity U.S. / British Unit SI Unit Linear Measures 1 inch (in.) = 25.4 mm 1 foot (ft) 1 ft = 12 in. = 0.3048 m 1 Yard (yd) = 3 ft = 0.9144 m 1 English Mile = 1.6093 km 0.03970 in. = 1 millimeter (mm) 0.393701 in. = 1 centimeter (cm) 3.280840 ft = 1.093613 yd. = 1 meter (m) 0.6214 English Mile = 1 kilometer (km) Wefly Academy srl 20 18 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Square Measures 1 square inch (sq.in.) = 645.160 sq.mm 1 square inch (sq.in) = 6.45160 sq.cm 1 square foot (sq. ft) = 9.2903 sq.dm 1 square yard (sq.yd) = 0.836127 sq.m 1 sq.ft = 144 sq.in. = 0.092903 sq.m 0.001550 sq.in. = 1 square millimeter (sq.mm) 0.155000 sq.in. = 1 square centimeter (sq.cm) 10.763910 sq.ft = 1 square meter (sq.m) 1.195990 sq.yd = 1 square meter (sq.m) Volume 1 cubic inch (cu.in.) = 16.387064 cu.cm 1 cubic foot (cu.ft) = 28.316847 cu.dm 1 ft = 1728 cu.in. = 0.028317 cu.m 1 gallon (U.S.) = 3.784 cu.dm 1 gallon (U.K.) = 4.546 cu.dm 1 barrel (U.S.) = 158.987 cu.dm 0.061024 cu.in = 1 cubic centimeter (cu.cm) 0.035315 cu.ft = 1 cubic decimeter (cu.dm) 35.31467 cu.ft = 1 cubic meter (cu.m) Weights 1 ounce (oz) = 28.3495 g 1 pound (lb) = 16 ounces = 0.45359237 kg 1 long ton (l ton) = 2240 lb = 1016.04706 kg 1 short ton (sh ton) = 2000 lb = 907.185 kg 0.035274 oz = 1 gram (g) 2.204622 lb = 1 kilogram (kg) 0.984206 l ton = 1 metric ton (t) = 1000 1.10231 sh ton = 1 metric ton (t) Weights per Length 1 lb / ft = 1.488164 kg / m 1 lb / yd = 0.496054 kg/m 0.671969 lb / ft = 2.015907 lb / yd = 1 kg / m Wefly Academy srl 21 19 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Force* 1 pound-force (lbf) = 4.448222 Newton (N) 0.224809 lbf = 1N Pressure / Stress 1 pound-force per square inch 0.06895 bar = (psi) 1 lbf / sq.in. (psi) ؆ 1 lb / sq.in. = 0.006895 N / sq.mm (MPa) 1 lbf / sq.ft = 47.88 N / sq.mm 14.5038 lbf / sq.in. = 1 bar 145.038 lbf / sq.in. = 1 N / sq.mm (MPa) Density 1 lb / ft = 0.016018 kg / dm 62.427952 lb / ft = 1 kg / dm Torque 1 foot pound-force ft-lbf ؆ 1 ft-lb = 1.3558 Nm 0.7376 ft-lbf = 1 Nm Energy 1 ft-lbf ؆ 1 ft-lb = 1.355818 Joule (J) 0.737562 ft-lbf = 1J Speed 1 mile per hour (m.p.h.) = 1.609344 km / h 1 foot per second (ft / s) = 0.3048 m / s 0.621371 m.p.h. = 1 km / h 3.28084 ft /s = 1m/s Power 1 ft lbf / s = 1.35582 W; J / s; Nm / s 1000 ft lbf / s = 1.8182 hp = 1.28182 1.35582 kW = btu / s 737.562 ft lbf / s = 1 kW = 1.359621617 1 PS = 0.73549875 kW Wefly Academy srl 22 20 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Flow Rate 1 barrel per day = 0.158987 m / day 1 cubic foot per minute (ft / min) 0.02831685 m / min = 40.776192 = m / day Temperature Conversion formula (°F) to °C = 5 / 9 (°F – 32) 1 degree Fahrenheit (°F) = 0.5556 °C 32 °F = 0 °C 212 °F = 100 °C * 1 pound-force (lbf) ؆ 1 pound (lb) Table 4 Uni ts and their conversions Wefly Academy srl 23 21 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 1.6.2. Scale, latitude, and longitude Scale The scale is the relation between the real size of something and its size on a map, model, or diagram. In mathematics, a scale in graph can be defined as the system of marks at fixed intervals, which define the relation between the units being used and their representation on the graph. A typical mathematical scale is shown below in Figure 2. Figure 2 Example of a typical mathematic al scale One unit of the horizontal axis is equal to 1 cm. Wefly Academy srl 24 22 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Scales are also used with maps as shown in Figure 3. Figure 3 Scale example on a map In this case scale is on the lower right corner of the picture and it is 2 km. Latitude and longitude The latitude and the longitude are important measures. Latitude and longitude can determine and describe the position of any location or place on Earth’s surface. The la titude of a point/location on a globe or map is its north or south distance from the Equator. It is provided/ calculated in degrees, minutes, and seconds. Figure 4 Latitude on a globe Wefly Academy srl 25 23 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 The distance, from Equator, to the north is called north latitude, while the one to the south is called south latitude. The longitude of a poi nt/location on a globe or map is its east or west distance from a meridian of reference (Greenwich) calculated in degrees, minutes and seconds. Figure 5 Longitude on a gl obe 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. Greenwich observatory (London area) is, by international convention, the meridian of reference, that is 0 degrees of longitude. The latitude and the longitude of a 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, the sextant or the octant. The calculation about differences of latitude and of longitude between two places is based on standardized rules. Latitude 1. If both places are northward or southward, the lower latitude is deducted from the higher one. 2. If places are opposite side (one northward and the other southward) the two latitudes are summed. Wefly Academy srl 26 24 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Longitude 1. If both places are east or west, the lower longitude is deducted from the higher one. 2. If places are opposite side (one is east and other is west) the two longitudes are summed. 3. 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 Earth spins around its axis from east to west. It completes a 360° revolution in 24 hours. For the people on the earth surface, the revolution makes it look like the sun is turning around the Earth. The sun covers in 1 hour an arc of 1/24 of 360 degrees, that is equivalent to 15 degrees longitude. In one minute, it covers 1/60 of 15 degrees, that is 15 minutes of longitude, while in 1 second it covers 1/60 of 15 minutes, that is 15 seconds of longitude. 1 5 degrees of longitude correspond to 1 sun hour; 1 5 minutes of longitude correspond to one minute o f sun time, while 1 5 second s of longitude correspond to one second of sun time. Sun time depends on the position of the Sun observed form that place. The clock indicates exactly 12 0 0or midday in that place where the sun crosses the local meridian. If it is midday in the meridian where we are, it is afternoon post-meridiem in all the places to the east, and it is morning ante-meridiem in all the places to the west. Standard Time is adopted worldwide, and it is based on Time Ranges. For example, United States is divided into four time zones, each one of which is approximately 15-longitude degree wide. A place within a time zone uses the same Standard Time, that is the same hour, independently from the local hour, that is the local meridian. In countries like those within the EU and the U.S., at the beginning of springtime until the half of autumn, the clock’s hour hand is moved one hour forward to best exploit sun light during working hours. This is called summertime or Daylight-Saving Time (DST). At the end of autumn, the hour hand is set one hour back to wintertime/solar time. Wefly Academy srl 27 25 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 How to find the difference in sun hours between two places? it's necessary to divided 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. 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 international line for changing the date. The line follows the meridian with a zigzag course. This was done so that all the Pacific islands have the same time. When it is midday in Greenwich, midnight has just passed, and it is morning of the same day in all the places slightly east of the international line of date. However, it is almost midnight of the same day in all the places slightly west of the line. When it is one o’clock in the afternoon in Greenwich, it is about one o’clock 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 is indicated by 08:10; zero, zero, zero, five post meridian by 00:05 and 11.59 post meridian by 23:59. Wefly Academy srl 28 26 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Ratio and proportion 1.7.1. Ratio Ratio is a comparison between two numbers. ͳ‫ʹ׷‬ The ratio of two numbers A and B, with B different from zero, is the quotient. ‫ ܣ‬divided by ‫ ܤ‬can be expressed also as the fraction ‫ܣ‬Ȁ‫ܤ‬. In the ratio ‫ܣ‬Ȁ‫ܤ‬, the numbers ‫ ܣ‬and ‫ ܤ‬are called terms of the ratio. As for any division, it is possible to use its main property. Multiplying or divi ding both terms of a ratio by the same number, different from zero, the ratio remains the same. 1.7.2. Proportion A proportion is an equivalence relation between two ratios (division) ͳ ͷ ൌ ʹ ͳͲ ͳ ‫ ʹ ׷‬ൌ ͷ ‫Ͳͳ ׷‬ Here after some useful definitions. · The first and the third term of the proportion are called antecedents. · The second and the fourth are called consequents. · The first and the last term of the proportion are called extremes. · The second and the third terms are called means. The fundamental property of proportions is the following. The product of the means is equal t o the product of the extremes. From the fundamental property it is possible to get additional properties for determining an unknown term in a proportion. Wefly Academy srl 29 27 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 1. Property of inverting. In a propo rtion, exchanging each antecedent with its consequent the proport ion is still valid. If ͳ ‫ ʹ ׷‬ൌ ͵ ‫ ׷‬͸ Then, ʹ ‫ ͳ ׷‬ൌ ͸ ‫͵ ׷‬ 2. Property of permuting. In a proportion, exchanging the means between them, or the extremes, the resul t is a proportion still valid. If ͳ ‫ ʹ ׷‬ൌ ͵ ‫ ׷‬͸ Then ͸ ‫ ʹ ׷‬ൌ ͵ ‫ͳ ׷‬orͳ ‫ ͵ ׷‬ൌ ʹ ‫ ׷‬͸ 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). If ͳ ‫ ʹ ׷‬ൌ ͵ ‫ ׷‬͸ Then, ሺͳ ൅ ʹሻǣʹ ൌ ሺ͵ ൅ ͸ሻǣ͸ 4. Property of factorizing. In each proportion the differenc e of the first two terms is to the first (or to the second) as the difference of the other two terms is t o the third. If ͵ ‫ ʹ ׷‬ൌ ͻ ‫ ׷‬͸ Then, ሺ͵ െ ʹሻ ‫ ʹ ׷‬ൌ ሺͻ െ ͸ሻ ‫ ׷‬͸ Averages and percentages Percentages is a number or ratio expressed as a fraction of 100. It expresses a part of an integer. The symbol for percentages is Ψ. ͳͲΨ ՜ ͳͲȀͳͲͲ ͵ͲΨ ՜ ͵ͲȀͳͲͲ Wefly Academy srl 30 28 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Two definitions help us to better understand the meaning of percent. · Direct proportionality. · Inverse proportionality. Direct proportion is the relationship between two variables whose ratio is equal to a constant value. Direct proportion implies that an increase in one quantity causes a corresponding increase in the other quantity, or a decrease in one quantity results in a decrease in the other quantity. In contrast with direct proportion, in inverse proportion, an increase in one variable causes a decrease in the other variable, and vice versa. Average is it is a single number that is used to represent a collection of numbers. Let’s consider a collection of numbers (3): ʹǢ ͶǢ ͷǢ The mean value is the sum of the three above numbers ሺʹ ൅ Ͷ ൅ ͷሻ divided by the total number of numbers (3). ሺʹ ൅ Ͷ ൅ ͷሻ ͳͳ ൌ ൌ ͵Ǥ͸͸ ͵ ͵ Wefly Academy srl 31 29 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Area and volumes Before explaining the concept of area, it is useful to introduce the concept of the perimeter. The perimeter is the measure of the length of the contour of a plane figure. To find the perimeter of any 2D shape it is necessary to sum its sides. Here after some useful formulas for perimeters. Figure 6 Polygons Wefly Academy srl 32 30 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 1.9.1. Area The amount of space inside the boundary of a flat (2-dimensional) object such as a triangle or circle, etc. The unit of measurement of usually is square meter ሺ݉ଶ ሻ for SI. Triangle A polygon with three sides and three angles. The angle vertices are called vertices of the triangle. The main property of triangles is the following. The sum of the internal angles of a triangle is always 1 80 degrees. Each side of a triangle is a base (ܾ) and the segment of orthogonal line from that base to the opposite vertex is a height (݄). The area (‫ )ܣ‬is given by the product of the base (ܾ) by the height (݄) divided by 2. ܾ‫݄ڄ‬ ‫ܣ‬ൌ ʹ There are different types of triangle shown in Figure 7, Figure 8, Figure 9, and Figure 10 1. Scalene Triangle it has three unequal sides and three unequal angles. Figure 7 Scalen e triangle Wefly Academy srl 33 31 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 2. Isosceles Triangle it has two equal sides and two equal angles. Figure 8 Isosceles triangle 3. Equilateral Triangle it has three equal sides and three equal angles. Figure 9 Equilat eral triangle 4. Right Triangle It has a n angle of 90 degrees. Figure 1 0 Right triangle Here after some additional useful definition for triangles. · A triangle with an angle greater than 90 degrees., is an obtuse triangle. · A triangle in which all angles are smaller than 90 degrees, is an acute triangle. Wefly Academy srl 34 32 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Parallelogram A polygon in which the opposite sides are parallel and equal in length. The area (‫ )ܣ‬of the parallelogram is given by the multiplication of the length or base (ܾ) of the parallelogram by the height. The height (݄) is equal to the segment of the perpendicular line traced from the base to the opposite vertex of the consecutive side: ‫ ܣ‬ൌ ܾ ‫݄ ڄ‬ Figure 1 1 Parallelogram Rectangle A parallelogram having all right angles. The area of the rectangle base by height: ‫ ܣ‬ൌ ܾ ‫݄ ڄ‬ Figure 1 2 Rectangle Wefly Academy srl 35 33 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Square A parallelogram with four sides of equ al length and four right angles. 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 the square of the side (݈): ‫ ܣ‬ൌ ݈ଶ Figure 1 3 Square Rhombus A parallelogram with opposite equal acute angles, opposite equal obtus e angles, and four equal sides. This polygon has two orthogonal diagonals: a long diagonal (݀ଶ ) and a short one (݀ଵ ). The area of the rhombus is the product of the diagonals divided by 2: ݀ଵ ‫݀ ڄ‬ଶ ‫ܣ‬ൌ ʹ Figure 1 4 Rhombus Wefly Academy srl 36 34 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Trapezium A quadrilateral with two parallel sides. The two parallel sides are also called bases, one is the greater base (‫ )ܤ‬and the other is the smaller base (ܾ). 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 a right trapezium. The area of the trapezium is given by the formula sum of the bases (‫ ܤ‬൅ ܾ) multiplied by the height (݄) and the product is divided by 2 ሺܾ ൅ ‫ܤ‬ሻ ‫݄ ڄ‬ ‫ܣ‬ൌ ʹ Figure 1 5 Trapezium Wefly Academy srl 37 35 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Circumference It is a set of point s in a plane equidistant from a given point. The given point is called center and the distance from the center is the radius (‫)ݎ‬ 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 of a circumference are always equal to a fixed value indicated by the Greek character ߨ ൌ ͵ǤͳͶ. The circumference (perimeter) of a circle can be calculated multiplying by ߨ the diameter. Circ. ൌ ݀ ‫ ߨ ڄ‬ൌ ʹߨ‫ݎ‬ The area of the circle is given by the square of the radius multiplied by ߨ ‫ ܣ‬ൌ ߨ‫ ݎ‬ଶ Figure 1 6 Circle Wefly Academy srl 38 36 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 1.9.2. Volume 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 (݈), depth (‫ )݌‬and height (݄): ܸ ൌ݈‫݄ڄ݌ڄ‬ Figure 1 7 Prism In the computation of the volume, it is important that the three dimensions are indicated with the same measure. 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 (݈); therefore, the volume corresponds to one cubed dimension: ܸ ൌ ݈ଷ Figure 1 8 Cube Wefly Academy srl 39 37 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 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 (݄) is the distance between the vertex and the base. The volume of a pyramid is calculated multiplying the area of the base by its height; the product is divided by ͵: ‫݄ڄܣ‬ ܸൌ ͵ Figure 1 9 Pyramid 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 (݄) 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: ܸ ൌ‫݄ڄܣ‬ ൌ ߨ ‫ݎ ڄ‬ଶ ‫݄ ڄ‬ Figure 20 Cylinder Wefly Academy srl 40 38 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 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 (‫ )ܣ‬by the height (݄), the product is then divided by ͵: ‫݄ڄܣ‬ ܸൌ ͵ Figure 21 Cone 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 (‫ )ݎ‬and all the meridians of the surface are circumferences. The surface of a sphere can be calculated multiplying the square of its radius by ߨ by Ͷ: ܵ ൌ ‫ݎ‬ଶ ‫ ڄ ߨ ڄ‬Ͷ The volume of the sphere is given by the cubed radius of the sphere multiplied by ߨ. The result obtained is then multiplied by (ͶȀ͵): Ͷ ܸ ൌ ‫ݎ‬ଷ ‫ڄ ߨ ڄ‬ ͵ Figure 22 Sphere Wefly Academy srl 41 39 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Squares When a number is being squared, it means that the same number is being multiplied by itself. In this case when the number n is being squared, it will be raised to the second power: ݊ଶ ൌ ݊ ‫݊ ڄ‬ For example: ͵ଶ ൌ ͻ The are some important properties when squaring a number: · When a real number is being squared, it will always be greater than or equal to Ͳ; For instance, the square of ʹ is equal to Ͷ but the square of െʹ is also equal to Ͷ. · When any integer is being squared, its sum can be represented by: ͳ ൅ ͳ ൅ ʹ ൅ ʹ൅Ǥ Ǥ ൅ሺ݊ െ ͳሻ ൅ ሺ݊ െ ͳሻ ൅ ݊ As an example: Ͷଶ ൌ ͳ ൅ ͳ ൅ ʹ ൅ ʹ ൅ ͵ ൅ ͵ ൅ Ͷ ൌ ͳ͸ If an integer (݊) is being squared, it will be equal to the sum of ݊ prime odd numbers: Ͷଶ ൌ ͳ ൅ ͵ ൅ ͷ ൅ ͹ ൌ ͳ͸ Cubes When a number is being cubed, it is the same as multiplying the same number with itself three times: ݊ଷ ൌ ݊ ‫݊ ڄ ݊ ڄ‬ As an example: ʹଷ ൌ ͺ Wefly Academy srl 42 40 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 Square and cube roots The root extraction symbol is represented by this ξ. The number inside the root symbol is called a radicand while the number that represents the power of the number is called an exponent. The root function is used for finding the number that can be multiplied by itself for a certain number of times until it is equal to the radicand. As an example: ξͶ ൌ ʹ ʹଶ ൌ ʹ Wefly Academy srl 43 41 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 PAGE INTENTIONALLY LEFT BLANK Wefly Academy srl 44 42 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 2. Algebra Evaluating simple algebraic expressions Using zero as the starting number, all numbers greater than zero are assigned a positive value and numbers less than zero are assigned a negative value. In the relative number row, negative values are represented by a minus sign and positive values are represented by a plus or no sign. Each statement that contains an operation to be performed on a relative number is called an algebraic expression. Calculating the value of an algebraic expression means finding a relative number that represents the result of a specific operation. Remember, the plus and minus signs have a dual meaning. They can be used for calculations, namely addition and subtraction, and also for symbols that represent relative numbers. The calculation of the value of an algebraic expression must follow the following rules: if the expression does not contain parentheses, the exponential must be considered first, then multiply and divide, and finally add and subtract; if the expression contains square brackets, the inner ones must be removed first, and then followed by the outer ones. Also the numbers in parentheses, and the correct order to perform different operations is as follows: 1. Power 2. Multiplication and division: Multiplication and division operations must be performed from left to right after calculating the exponent. 3. Addition and subtraction: After multiplying and dividing, addition and subtraction should be done from left to right. 2.1.1. Addition When adding two or more numbers with the same sign, ignore the sign and calculate the sum of the values, and then add the common sign to the value before the result. In other words, when you add two or more positive numbers, the sum is positive, and when you add two or more negative numbers, the sum is always negative. Conversely, when adding positive and negative numbers, subtract the two numbers, and then add the positive or negative sign of the larger number. The result of adding or subtracting signed numbers, that is, relative numbers, is called the algebraic sum of numbers. Wefly Academy srl 45 43 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 2.1.2. Subtraction By subtracting numbers with different signs, this operation becomes a sum that changes the minus sign at the same time. After that, the method is the same as adding. 2.1.3. Multiplication and division The multiplication of relative numbers follows the law of multiplication of universal numbers. After doing the multiplication, the product takes the symbol established by the following 3 laws: 1. The product of two positive numbers is always a positive number 2. The product of two negative numbers is always a positive number 3. The product of a positive number and a negative number is always a negative number. In the case of multiplication, the division of relative numbers follows the same rules as the division of universal numbers. The sign of the quotient is determined by the same law used for 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 number and a positive number is always a negative number. 2.1.4. Use of brackets Square brackets are used in mathematics to group terms that must perform the same operation, and to define priority based on certain operations. Parentheses are always used in pairs and have the same type. This is the increasing hierarchical order of brackets used in arithmetic: brackets, square brackets, and curly brackets. The first operation to be performed is the operation indicated between the inner brackets. Wefly Academy srl 46 44 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 2.1.5. Simple algebraic fractions Algebraic fractions are fractions with polynomials in the denominator and numerator. Because polynomials contain numbers, it must be possible to perform the same operations with polynomials as with numbers. In fact, polynomials are defined as expressions that contains constants and variables, and can be combined with only addition, subtraction, and multiplication. When you have a fraction with polynomials in the numerator and denominator, you must: 1. Decompose the numerator and denominator into factors 2. Check if there are two equal factors between the numerator and denominator, if any, they must be eliminated 3. Use the term on the left to write the fraction. Note: Only multiplication factor (or division) terms can be simplified between the numerator and the denominator. You cannot simplify addition or subtraction terms. You must follow the same process used for simple fractions to add and subtract between algebraic fractions: 1. To break out denominator and numerator in factors 2. To calculate the LCM, and to put it as the denominator 3. To calculate the numerator 4. Try to make simplifications between the numerator and the denominator as best as possible. Be careful when doing subtraction: the minus sign, before a fraction, changes the sign of all terms of the numerator. When multiplying algebraic fractions, you must proceed with: 1. Dividing the numerator and denominator by the factor 2. Eliminate the same terms between the numerator and the denominator 3. Multiply the numerator by the numerator and the denominator by the denominator. Wefly Academy srl 47 45 M01 Module 01 -– Mathematics Mathematics Ed.0 Edition 24.11.2021 Rev.1 – Rev. 02 When there is

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