UNIT 1: REAL NUMBERS.

Transcription

1 UNIT 1: REAL NUMBERS. Sets of Numbers: Natural Numbers: N={0,1,,, 4,, 6,...} In English books of Maths, the set of natural numbers is {1,,,4,...}, while the set of whole numbers is {0,1,,,4,...}, so there is a tiny difference between both sets, including or not the number zero. Integers: Z={...,,, 1,0,1,,,...} The set of integers is formed by the natural numbers including zero and the negatives, the opposites of the natural numbers. Rational Numbers: Q= { a b, where a, b Z, b 0 } Rational numbers are the fractions. All number that can be written as a fraction is a rational number. Every fraction, that is every rational number, has a decimal expression ( we can get dividing the numerator by the denominator). The different types of the decimal expression of the rational numbers are: An exact or terminating number is one which does not go on forever, so you can write down all its digits. For example: =0,4 A recurring, periodic or repeating decimal is a decimal number which does go on forever, but where some of the digits are repeated over and over again. For example: 0,1...=0,1 We can distinguish: - Decimals that the period starts just after the decimal point (pure periodic or recurring decimal). For example: =0, =0, 6 - Decimals that the period does not start just after the decimal point (mixed periodic or 1 recurring decimal). For example: 6 =0, =0,1 6 Irrational Numbers: I. Irrational numbers have decimal expressions that neither terminate or nor become periodic. Examples:,, π=,1419 1, ,

2 Real Numbers: R=Q I The real numbers include both rational numbers, such as 4 or 4/17, and irrational numbers such as π or 7. Your Turn Classify the following numbers into the corresponding set: 7-6 1, , , Natural Numbers N Integers Z Rational Numbers Q Irrational Numbers I Real Numbers R

3 Rational Numbers: As we know, the set of rational numbers is formed by all the numbers that can be written as a a fraction, where a and b are integers and b is not 0. b Converting a fraction into a decimal: To calculate the decimal expression of a fraction, we divide numerator by denominator. We can obtain an integer (if the numerator is a multiple of the denominator) or a decimal number of the following types: An exact or terminating decimal, if after the simplification of the fraction the numerator only has as factors either or. A pure periodic decimal, if after the simplification of the fraction, and are not factors of the denominator. A mixed periodic decimal, if after the simplification of the fraction, and/or are factors of the denominator, and it has other factors. Your Turn Without doing the division, try to say what kind of decimal the following fractions generate: a) 0 b) 7 c) 9 d) 1 e) 1 f) 16 0 g) 4 h) 7 i) 1

4 Converting decimal into a fraction: Exact or terminating decimals: Write in the numerator the number without decimal point and in the denominator the unit followed by as many zeros as decimal digits the number has. Examples: 0,017= ,46= =1 0 Pure periodic decimals: Write in the numerator the number without decimal point and subtract the whole part of the number. Then, write in the denominator as many nines as decimal digit the repeating part of the numbers has. Examples: 1,...=1, = 1 1 = ,...=0, = 0 99 = 99 Mixed periodic decimals: Write in the denominator the number without decimal point and subtract the whole part of the number followed by the non repeating part. Then, write in the denominator as many nines as decimal digit the repeating part of the number has followed by as many zeros as decimal digits the non repeating part has. Examples: 1,0...=1,0 = = ,1...=0,1 = 1 1 = = 7 0 Your Turn Convert into fractions the following decimal: a) 1, b) 0, 7 c) 0,66 d) 1,1 6 e) 1, 7 f) 0,4 g) 0,1 6 h), 79 i) 0,00 Be careful with the numbers whose repeating part is 9:, 9 10, 9 4

5 Irrational numbers: A number is irrational if it cannot be expressed as a fraction. Its decimal expression has an infinite number of digits that are not regularly repeated. Example: Calculate the diagonal of a square whose side is 1 cm long. d 1 cm 1 cm Example: Which of the following numbers are either rational or irrational? a) 1,... b) 1,... c) 1,... d) 1,... e) 1, f) 0,1... Real Numbers: The set of real numbers R is formed by the set of rational numbers Q and the set of irrational numbers I. R=Q I numbers Q { Real numbers R{Racional Integers Z Natural NumbersN { Negative Integers Terminating and Periodic decimals Irrational Numbers I Order in R : Given two different real numbers, a and b: a is less than b, and we write a b, when b a is positive. a is greater than b, and we write a b, when b a is negative.

6 Properties of real numbers: Properties Addiction Multiplication Associative a b c=a b c a b c=a b c Identity element a 0=0 a=a a 1=1 a=a Opposite/Inverse element a a =0 a 1 a =1 Commutative a b=b a a b=b a Distributive Extracting common factor: a b c =a b a c Using the distributivity of real numbers: a b a c=a b c, we can extract common factor to do easily some operations: Examples: Extract common factor and operate these expressions: a) 6+7 b) c) Your Turn 1. Order each set of decimals from lowest to highest: a) 1, 1, 1, 1, 1, 1, b) 0,0 1 0,0 1 0, 01 0,01 c) 7, 7, 7, 7, 7, 7, 6

7 . Order each set of decimals from greatest to least: a),47,447,744,444,47 b) 0,9 0,9 0, 9 0, 0, 9 c),0 17,9 71,97 1,91 7, 917. Extract common factor and operate these expressions: a) b) c) d) Real Line: The real numbers may be thought of as points on an infinitely long number line. Each point of the real line corresponds to a real number, and each real number corresponds to a point of the real line. Representing rational numbers on the real line: Examples:

8 Representing irrational numbers on the real line: Examples: Intervals: A real interval is a subset of real numbers that corresponds to the points of a segment or a half-line on the real line. INTERVAL NOTATION SET NOTATION GEOMETRIC PICTURE CLASSIFICATION a,b {x R :a x b} Finite; open a b [a,b] {x R : a x b} Finite; closed a b a,b] {x R : a x b} Finite; half-open a b [ a,b {x R : a x b} Finite; half-open a b a, {x R : a x} Infinite; open a [ a, {x R : a x} Infinite; closed a,b {x R : x b} Infinite, open b,b ] {x R : x b} Infinite; closed b, R Infinite; open and closed 8

9 Examples: Write as intervals and represent on the real line the following set of numbers: a) Greatest than - and less than 1 b) Greatest or equal than 4. c) Less than 0. d) Less or equal than -1. e) Greatest than -1. f) Greatest or equal than - and less than. Your Turn 1. Represent on the real line and write as intervals the following set: a) {x R : x } b) {x R : x 6} c) {x R : x } d) {x R :0 x }. Write as intervals and represent on the real line: a) x b) x c) x 9

10 Approximations: An approximation of a number is a representation of this number that is not exact, but still close enough to be useful. Approximation Methods: Truncation: We only consider the digits until an order. Examples: Approximate to the hundredths the following numbers: 1,4 0,0968 1, 8 0,0 Rounding: The figures are deleted from a order considered, and the last figure is increased by one unit if the following digit is greater than or equal to. Examples: Rounding off to the nearest thousandths the following numbers: 1,4 0,0968 1, 8 0,0 Absolute Error and Relative Error: The approximation error in some data is the discrepancy between an exact value and some approximation to it; an approximation error can occur because 1. The measurement of the data is not precise (due to the instruments), or. approximations are used instead of the real data (,14 instead of π). One commonly distinguishes between the absolute error and the relative error. The absolute error is the magnitude of the difference between the exact value and the approximation. The relative error is the absolute error divided by the magnitude of the exact value. Absolute Error: E a = V Exact V Approx. Relative Error: E r = E a V Exact Example: a) The height of a house is 4,7 m. If we say the height of the house is m, calculate the absolute error and the relative error of this approximation. b) The height of a skyscraper is 11, m. If we say the height of the skyscraper is 11 m, calculate the absolute error and the relative error of this approximation. 10

11 Your Turn 1. Skeeter, the dog, weighs exactly 6, pounds. When weighed on a defective scale, he weighed 8 pounds. a) What is the absolute error and the relative error in measurement of the defective scale? b) If Millie, the cat, weighs 14 pounds on the same defective scale, what is Millie's actual weight?. The actual length of this field is 00 feet. A measurement instrument shows the length to be 08 feet. Find: a) The absolute error in the measured length of the field. b) The relative error in the measured length of the field. 0 feet c) The percentage error on the measured length of the field. 00 feet. Find the absolute ans relative error of the approximation,14 to the value π. 11

12 Keywords: addition / sum = adición, suma subtraction / difference = resta, diferencia multiplication / product = multiplicación, producto division / quotient = división, cociente set of numbers = conjunto numérico Natural numbers = Números Naturales Integers = Números Enteros Rational Numbers = Números Racionales fraction = fración exact or terminating decimal = decimal exacto pure periodic decimal= decimal periódico puro mixed periodic decimal= decimal periódico mixto Irrational numbers = Números Irracionales Real numbers = Números Reales to be included = estar incluido factor / divisor = factor, divisor multiple = múltiplo Highest Common Factor (UK) / Greatest Common Factor (USA) = Máximo Común Divisor Lowest Common Multiple (UK) / Least Common Factor (USA) = Mínimo Común Múltiplo diagonal = diagonal square = cuadrado height = altura weight = peso to weigh = pesar measurement = medida to measure = medir rectangle = rectángulo triangle = triángulo a is less than b a b = a es menor que b a is less or equal than b a b = a es menor o igual que b a is greatest than b a b = a es mayor que b a is less or equal than b a b = a es mayor o igual que b to order from lowest to highest (UK) = ordenar de menor a mayor to order from least to greatest(usa) = ordenar de menor a mayor 1

13 Associative property = Propiedad asociativa Commutative property = Propiedad conmutativa Distributive property = Propiedad distributiva Identity element = elemento neutro Opposite element = elemento opuesto Inverse element = elemento inverso common factor = factor común Real line = Recta real Pythagoras' Theorem = Teorema de Pitágoras Square root = Raiz cuadrada Interval = intervalo segment = segmento half-line = semirrecta Open interval = intervalo abierto Closed interval = intervalo cerrado Half-open interval = intervalo semiabierto Half-closed interval = intervalo semicerrado Infinite = infinito Absolute value = valor absoluto approximation = aproximación Truncation = truncamiento to round off = redondear absolute error = error absoluto relative error = error relativo percentage = porcentaje place value of a digit / figure = valor posicional de una cifra ones or units = unidades tens = decenas hundreds = centenas thousands = millares, unidades de mil ten thousands = decenas de mil... tenths = décimas hundredths= centésimas thousandths = milésimas ten thousandths = diezmilésimas... 1