Lesson 1: Natural numbers

Transcription

1 Lesson 1: Natural numbers Contents: 1. Number systems. Positional notation. 2. Basic arithmetic. Algorithms and properties. 3. Algebraic language and abstract reasoning. 4. Divisibility. Prime numbers. Greatest common divisor. Least common multiple. 1

2 N = {1, 2, 3, 4, 5...} Origin: need to count. Natural numbers Problem: (word and number) representation of big numbers. 2

3 1. Aditive systems Types of number systems Number is obtained adding up the value of the symbols. 3 From jgodino/edumat-maestros/welcome.htm Greek number system: I = 1, Π = 5, = 10, H = 100, X = 1000 and M = (roman numerals come from it).

4 Types of number systems 2. Aditive-multiplicative systems Instead of repeating a symbol several times, an extra symbol is added to indicate that. An example: the chinese system 4

5 Types of number systems 3. Multiplicative systems (like ours) Origin: Hindu system. Symbols were (and some additional ones for powers of 10). Around 5h-8th centuries, symbols for powers of 10 are substituted by bars: 5

6 The symbol 0 (zero). Number systems The name comes from sanscrit word shunya (empty). Translated to arabic as sifr. (Origin of the Spanish word palabra cifra). The system arrives at Europe via muslims (hindu-arabic system). Al-Jwarizmi wrote the book The Book of Addition and Subtraction According to the Hindu Calculation around 825. With the introduction of the new number system, arithmetic develops very fast. 6

7 Two digit numbers in 1st Grade Traditional approach (in Spain): - Lesson 0: review of numbers from 0 to 9. - Lesson 1: numbers from 10 to Lesson 2: numbers from 20 to Drawback: Lack of number sense. 7

8 Tens and units in Spanish textbooks Usual approach, as in the figure: It is better, at least for some time, represent tens explicitely, as in the figure: (Example extracted from book 1B of Singapore). 8

9 A comparison Let us compare these two examples: From Singapore K-3 Spanish textbook, K-2 9 Pedro Ramos. Mathematics I. Grado de Educacio n Primaria (bilingual track). Alcala University. Year

10 Introducing 2-digit numbers Alternative approach: we count making groups of ten. There are two groups of ten and 6 (two tens and 6). Furthermore, examples for counting can be chosen in order to develop the number sense. 10

11 Introducing 2-digit numbers Once enough practice on counting by tens has been made, next step would be: - 3 tens (groups of ten) and 5 is written the group of ten is called decena. - introduce the symbol 0. After these steps, a student is ready to answer a question like: how many are ? This topic (and its relationship with the introduction of addition and substraction algorithms) will be revisited in more detail next year, in Didactics of Mathematics. 11

12 Base b Why do we count in base ten (making groups de ten )? If human beings had 8 fingers, how the number of dots in the figure would be represented? Because 26 = , in base 8 the number 26 is written as 32 (8. 12

13 Base b Expression of a number in base b: given a natural numberl b > 1 (the base), every number n can be written in a unique way as follows n = a k b k + a k 1 b k a 1 b + a 0. where digits a i are natural numbers from 0 to b 1. The expression of n in base b is a k a k 1 a 1 a 0(b. Why base b is interesting? - Didactics. - Conection with math applications: Computer Science. 13

14 Base b: exercises Exercices: 1. Write the first 5 numbers in base How to convert from base 10 to base b, and conversely? (a) Write 354 (7 in base 10. (b) Write 928 in base Write ten numbers starting at 223 (4. 14

15 Write in letters Oral number system 15 Write with digits twenty-three thousand forty-three billion, two hundred and four thousand million, twenty thousand and four. More information here: (Wikipedia) (Spanish system) Ordinal numbers Write with letters 37 o, 76 o, 85 o, 94 o, 101 o.

16 The number line An excellent tool for developing the number sense (at every level). For instance: at the end of 1st or 2nd year: Find the approximate place for numbers 87, 6, 25, At the end of elementary school, same thing for , 6005, ,

17 17 Basic arithmetic: addition and substraction Addition is an internal operation in N. Properties: conmutative, associative. Unce addition has been defined, substraction is easy: We say that a b = c if b + c = a. Comment: understanding substraction in this way right from the beginning has important consequences. For instance, makes clear the symmetric role of b and c in the expression a b = c. More en didactics. Substraction is not an internal operation in N. Using substraction, the order in N can be defined: we say that a < b if b a N.

18 Addition and substraction - Algorithms The main mistake in Spain: to introduce too soon (and too fast) standard column algorithms. Better: work with line notation. More intuitive, and related to mental calculation techniques. Works with numbers instead digits. More next year, in didactics. 18

19 Standard algorithms Traditional column algorithms are well known. Can you justify carryover? (llevadas) ( ( ( (8 National library of virtual manipulatives: 19 Numbers and operations Base blocks substraction

20 Basic arithmetic in 21st century A few years ago, computing by hand, fast and in a reliable way, was a highly valued professional skill. Which algorithms should be part of the elementary math education of the 21st century? Do you know the algorithm for square root computation? 20

21 Different approaches: Ways to calculate traditional algorithms. mental calculation techniques approximate calculation, estimation. calculator, smart phone, computer. To clarify the role of each of previous approaches in elementary mathematics education is a deep debate that has emerged in some places (but not in Spain). A radical veiw: Stop teaching calculating, start learning math. 21

22 Multiplication How should it be introduced? We have 3 dishes with 4 doughnuts in each dish. How many doughnuts are there in total? There are doughnuts. There are 3 times 4 doughnuts. 3 times 4 3 4? In (Spanish) textbooks 22

23 Multiplication What happens if we assume that: a) 3 4 means 3 times 4. b) in the times table for 2, we count by two. The traditional order for times tables does not match this proposal for the definition of multiplication. In English, both orders can be found: More in Didactics of Math. 23

24 Multiplication properties Conmutative law: a b = b a. It is not obvious that 4 times 7 is the same as 7 times times times Geometry can be a perfect tool to explain some basic facts

25 Distributive law Distributive law: a (b + c) = (a b) + (a c) (a + b) c = (a c) + (b c) Does it make sanse in elementary school? In textbooks... 7 (3 + 5) = What for?

26 Distributive law In basic mathematics (primary and secondary school) it is used in two different situations: i) algebra: 2(x + 3) = 2x + 6 ii) mental calculation (natural): 13 8 = (10 + 3) 8 = = 104 Geometry can be helpful, again. 26 (2 + 5) 4 =

27 Associative law: Associative law a (b c) = (a b) c (3 5) = (2 3)

28 Division First comment: it is important to distinguish between the meaning of division and the algorithms for division. If a 6 year old kid has 8 candies and want to share them (equally) with a friend, will he be able to to it? This idea of equal sharing is the best one to introduce division: we refer to it as se partitive division. If 20 candies are put in 4 equal bags, how may candies will contain each bag?? 28

29 Division There exists another interpretation of division: If 20 candies are put in bags and each bag contains 5 5? candies, how many bags there will be? This is the quotative division (barely studied). Related to measure: how may times does 5 fit in 20? Two observations: i) An easy way of distinguish them: think about how a 6 year old kid would solve the problem. 29 ii) In quotative division, the divisor can be a rational number: A group of frieds buys 6 pizzas, and share them equally. If each friend eats 2/3 of a pizza, how may friends ared there in the group?

30 Division A good way to understand both types of division: Make up two problems (one of each type) whose solution contain the division Another important idea, worth to spend some time with it is the division as inverse of multiplication: Because 5 4 = 20, we have 20 5 = 4 and 20 4 = 5. Why división by 0 cannot be defined? 5 0 =?? 0 = =?? 0 = 0 no solution infinite solutions 30

31 Division with remainder Given two natural numbers D (dividend) and d (divisor), there exist unique natural numbers q (quotient) and r (remainder) such that D = q d + r and 0 r b 1. Idea of every division algorithm: Aproximate from below the dividend by multiples of the divisor. Other aspect that is not properly worked: problems where the remainder is the key. The flight of an astronaut lasted 505 hours. If it started at 8 am, what time was it when he landed? 31

32 Division in N Standard algorithms for division: Extended algorithm Usual algorithm (in Spain) ( compressed )

33 Division: Meaning/Interpretation Algorithm The algorithm is not easy specially for 2-digit divisors so it requires time and practice. No standard notation to write that when 27 is dividend by 4, the quotient is 6 and the remainder is 3. An option: 27 4 = 6 R 3. I prefer just to write 27 =

34 Exercises 1. What happens whith quotient and remainder when dividend and divisor are multiplied (or divided) by the same number? 2. Knowing that 4185 = 45 93, find (without making the division) quotient and remainder when is divided by Find three 4-digit numbers that give 7 as remainder when they are divided by Find the smallest number that is bigger than 300 and has 7 as remainder when divided by

35 Introduction to algebraic language Arithmetics deals with computation with numbers. Algebra deals with computation with symbols. Algebra is very old. Greeks and Babylonians already made algebraic reasoning and computation. Middle Age mathematicians refer to the unknown (our x) as the thing. Along 16th century modern notation is introduced. 35

36 Introduction to algebraic language Makes possible to state general properties: For every a and b, it holds a + b = b + a. Makes possible to reason about unknown quantities, stablishing relationship between them: John has participated in a game. He was asked 40 questions. The prize was 150 euros for each correct answer, but there was a penalty of 60 euros for each wrong answer. It was compulsory to give an answer to all the questions. At the end of the game, John got 4530 euros. How many of his answers were correct? Algebra allows us to manipulate expressions like previous one (equations) and find their solutions. 36

37 Exercise 1. Write three generic consecutive even numbers. 2. Using the previous expression, show that the sum of three consecutive even numbers is always a multiple of Being even is not important: Write now three generic consecutive multiples of 17 and show that their sum is always a multiple of Show that, for every k N, the sum of three consecutive multiples of k is always a multiple of 3. 37

38 Look for a pattern: Algebraic language and patterns = = = = 25 Express it using usual language Express it using algebraic language 38

39 Algebraic language and patterns Figure 1 Figure 2 Figure 3 1. How many squares has Figure 8? 2. How many squares has Figure n? 39

40 Algebraic language and patterns n n n Think about several ways of counting the crossings in the figure and write the algebraic expression corresponding to each way of counting. 40 n

41 Divisibility (in N {0}) Given two integers a and c, we say that c divides a if there exist an integer q such that a = q c (i.e. if in the division a c the remainder is 0). c a means that c divides a (c is a divisor of a). If c a, we also say that a is a multiple of c. We denote by ċ the set of multiples of c. Example: 3 = {0, 3, 6, 9, 12,...}. If c does not divide a, we write c a. 41

42 Divisibility Examples: a) 2 36 (because 36 = 2 18), b) 5 19 (because 19 = ). c) For every integer a, 1 a. d) 8 0? Yes: 8 0 = 0. All natural numbers (0 N) are divisors of 0. 42

43 Divisibility Exercise: find all divisors of number 20. For every natural number n, 1 and n are allways divisors of n. The rest of divisors are called proper divisors. Def: A prime number p is a natural number bigger than 1 that does not have proper divisors (i.e., its only divisors are 1 and p). If a number is not a prime number, we say that it is a composite number. 43

44 Prime numbers: basic questions Is n a prime numbers? a) Is 97 a prime number? b) Is a prime number? Find all prime numbers smaller than n: Erathostenes sieve. (Animated) example in How many prime numbers are there? Do prime numbers have any real application? 44

45 45 Descomposition into prime factors (Factorization) Fundamental Theorem of Arithmetic: Every natural number can be written as a product of prime numbers. Furthermore, the decomposition is unique. First part (all numbers can be factorized) is easy to show: If n is prime, we are done. If n is not prime, it has a divisor a, and we have that n = a b. Now, we repeat the argument with a and b. To show that the decomposition is unique (not counting the order, of course) is not that easy, and we will not go into that. Obs: Theorem would not be true if we considered 1 as a prime number.

46 Factorizaction algorithm Therefore, 5544 = But you can also write 60 = 6 10 = Exercise: Write down all divisors of 60 and compare the factorization of 60 with the factorization of its divisors. 46

47 Prime factors and divisors Divisors of number n can be constructed from the factorization of n using the following result: c = a j b k (with a and b prime numbers) is a divisor of n if and only if a j and b k are in the factorization of n. (The same is true for more than two factors). Exercise: 1. Write down all divisors of How many of them are odd? 3. How many are multiple of 14? 47 Observe that parts 2 and 3 can be done without using part 1.

48 Counting the number of divisors A formula for the number of divisors: If then n has divisors. n = p a 1 1 pa 2 2 pa k k (a 1 + 1) (a 2 + 1) (a k + 1) Exercise: find four 3-digit numbers having 20 divisors. 48

49 More applications of factorization Find the smallest number for which 140 has to be multiplied in order to get a perfect square. Find the smallest number for which 360 has to be multiplied in order to get a perfect cube. Let us consider n = Knowing that = , find the product n = a b (a, b N, a > b) such that a b is minimum. 49

50 Back to a basic question Theorem (Euclides, 300 bc): There are infinitely many primes. Proof: Assume the number is finite. Then, we can make a list with all of them: L = {p 1, p 2,..., p n } is the list of all prime numbers. Consider q = p 1 p 2 p n q is not prime (it is not in the list) q cannot be factorized (none of the numbers in the list is a divisor of q). Impossible!

51 Some further comments Two consecutive odd prime numbers are called twin primes. Examples: 3 and 5 are twin primes, 11 and 13 are also twin primes. It is unknown whether the number of twin primes is finite. The biggest known twins: p = and p + 2. (Sept. 2013) dígitos Biggest known prime number (Sept. 2013) was p = ( digits).

52 Any real application? Security in Internet is mostly based on the fact that it is not known how to factorize big numbers. RSA: Public key criptography. Public key in the figure is a number (of about 600 digits in base 10, but it is expressed in hexadecimal form - base 16). More about this subject: 52

53 Two properties of divisibility If c a, then c (k a) (for every integer k). Namely, if c is a divisor of a, then it is also a divisor of every multiple of a. If c a and c b, then c (a + b). Combining two previous properties we have: If c a and c b, then c (k a + j b) (for every integers k and j). Ex: because 7 49 and 7 63, 7 ( ). 53

54 Greatest common divisor Greatest common divisor of numbers a and b, gcd(a, b), is the biggest natural number that divides both a and b. Why learning problems can appear here? Maybe because most of the time is not devoted to understanding the concept but to the algorithm for its computation. Exercises: a) gcd(40, 15) b) gcd(38478, 1) 54 c) gcd(384787, 0)

55 I) From the factorization. Knowing that Computation of gcd(a, b) = = find gcd(17640, 12474). The greatest common divisor of a and b is the product of the common factors in the factorizations of a and b. (With the smallest exponent). 55

56 Computation of gcd(a, b) Remark: From the factorization of a and b it is also possible to obtain all common divisors of a and b. Find all common divisors of and Using this idea, the following property can be shown: The common divisors of a and b are the divisors of gcd(a, b). 56

57 Computation of gcd(a, b) II) Euclidean algorithm. Based on the following result: Theorem: If a = q b + r, then common divisors of a and b are the same as common divisors of b and r. An example: a = 78, b = 42. For the proof, two things have to be checked: 1. If c divides both a and b, then it also divides both b and r. 2. If c divides both b and r, then it also divides both a and b. Application to the computation of the greatest common divisor: In order to compute gcd(a, b), we divide and obtain the relation a = q b + r. We know that gcd(a, b) = gcd(b, r). 57 Compute gcd(78, 42) using the Euclidean algorithm.

58 Comparing the algorithms It can be shown that Euclidean algorithm is much better than the algorithm based on factorization in the following sense: A computer can find the greatest common divisor of two big numbers (500 digits) in a few seconds. On the other hand, factorizing such a big number could require thousands of years of computation. An small example. You can solve it in both ways, using a calculator. Compute gcd(12319, 9991). 58

59 Greatest common divisor of several numbers The definition is the same: gcd(a 1, a 2,..., a k ) is the biggest common divisor of a 1, a 2,..., a k. The algorithm using factorization is exactly the same: If we know that = = = = find gcd(17640, 12474, 3591, 4998). 59

60 Greatest common divisor of several numbers Euclidean algorithm can be adapted as follows: Assume that a k a 1, a 2,..., a k 1 and assume that we compute the divisions Then, a i = q i a k + r i para i = 1,..., k 1. gcd(a 1, a 2,..., a k 1, a k ) = gcd(r 1, r 2,..., r k 1, a k ). Exercise: Find the greatest common divisor of 616, 1155, 308 and 693. Remark: For more than two numbers, it is also true that common divisors of a set of numbers are the divisors of the greatest common divisor of the set. 60

61 Mental calculation of gcd It is useful to compute at a glance examples like... a) gcd(9, 24) b) gcd(17, 284) c) gcd(8, 68) The following property can be useful: If k is a common divisor of a and de b, then gcd(a, b) = k gcd(a/k, b/k). Ex: find gcd(84, 24). 61

62 Least common multiple The least common multiple of two integers a and b, denoted lcm(a, b), is the smallest positive integer that is a multiple of both a and b. Learning difficulties are similar to the ones we already mentioned about the greatest common divisor. Exercises: a) lcm(6, 10) b) lcm(29834, 1) 62

63 Algorithm to compute lcm(a, b) We want to find the least common multiple of a = 3591 and b = knowing that 3591 = = ) m = a b = = is a common multiple. 2) Which factors of m = a b can be deleted in such a way that the resulting number is still a common multiple? 3) Therefore, lcm(a, b) = a b gcd(a, b). 63

64 Algorithm to compute lcm(a, b) We also have shown that: The least common multiple of a and b is the product of all the factors that appear in the factorizatons. Common factors are taken with the biggest exponent. Example: find lcm(3591, 14994) knowing that 3591 = = This algorithm also works to compute the least common multiple of more than two numbers. Remark: gcd(a, b, c) lcm(a, b, c) a b c 64

65 A property of the least common multiple Common multiples of two or more numbers are the multiples of their least common multiple. A lighthouse flashes every 16 minutes and a neighbour lighthouse flashes every 12 minutes. We know that both have flashed at midnight and we arrive at 5 pm. 1. how many times have they flashed simultaneously before we arrived? 2. what time do we see them to flash simultaneously for the first time? 65

66 Divisibility rules Arithmetic with remainders. A first example: even/odd. Let r(a, n) be the remainder obtained when a is divided by n. Question: can we find r(a + b, n) from r(a, n) and r(b, n)? If r(a, 3) = 2 and r(b, 3) = 2, how much is r(a + b, 3)? Using this approach, we are going to deduce divisibility rules (actually, rules to compute remainders) for 3, 4, 5, 6, 8 and 9. 66