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GOOD MORNING Shania QQ:1246640685 MSN: [email protected] www.themegallery.com LOGO www.themegallery.com Contents Math words quiz Factors-prime factors Multiples-LCM Patterns and sequences SETS!!! LOGO www.themegallery.com Math words quiz 10 minutes!!! LOGO www.themegallery.com Factors Factors of a number are the whole numbers that multiply together to give the original number E.g. The factors of 12 are? 12 is the the original number So which numbers can multiply together to give 12? 1×12, 2×6, 3×4 That is, 1,2,3,4,6,12 are factors of 12. We use F(12) as a short way of writing factors of 12. F(12)={1,2,3,4,6,12} Factor pairs of 12 are (1,12), (2,6), (3,4) Among these 6 factors 2 and 3 are prime factors. [Prime factors of a number are factors of the number that are also prime number.] LOGO www.themegallery.com Writing numbers as the product of prime factors Prime factors 2,3,5,7,11,13… 12=4×3,but 4 is not prime number, we break 4 down further 12=2×2×3 that we have written 12 as the product of prime factors LOGO www.themegallery.com Steps •Step 1 Try to divide the given number by the first prime number -- 2 • Step 2 Continue until 2 will no longer divide into it •Step 3 Try the next prime number, 3, then 5, 7 and so on, until final answer is 1 LOGO www.themegallery.com Examples Write 60 as a product of prime factors. Write 3465 as a product of prime factors. So 3465=3×3×5×7×11 LOGO www.themegallery.com Several rules The number is ended by 0,2,4,6,8 can be divided by 2. The sum of all digits of the number can be divided by 3, that is, the number can be divided by 3. 3465 3+4+6+5=18 18/3=6 so 3465 can be divided by 3 So does 5, 7, 11 and 13 LOGO www.themegallery.com Multiples Definition: the multiples of number are the products of that numbers and 1,2,3,4,5…(Natural number) E.g. The multiples of 3 are??? 3, 6, 9, 12, 15… The first five multiples of 3: M(3)={3,6,9,12,15} LOGO www.themegallery.com LCM-Lowest common multiple 最小公倍数 The smallest number that is a multiple of two or more numbers 12, 24, 36 are multiples of 3 and 4. BUT, 12 is the smallest one, that is, 12 is the LCM of 3 and 4. LOGO www.themegallery.com Two ways to find LCM ONE: ①List the multiples of each numbers of each numbers ②and then pick out the lowest number that appears in every one of the lists. (applicable for small numbers) ANOTHER: Expressing each number as a product of prime factors LOGO www.themegallery.com Sets Any collection of objects – have sth in common, some connection with each other. { } braces , comma The object in a set we called element of the set ∈ 5 ways to express sets LOGO www.themegallery.com 5 ways 1. Listed set {1,2,3,4.5} 2. Described set {first five natural numbers} 3. Set builder notation to describe sets mathematically {x:x≦10 and x is an even number} 4. Represented by a name or a letter {red, blue, yellow} {Thomas, Joise} 5. Venn diagram LOGO www.themegallery.com Special sets Finite sets and infinite sets Universal set rectangular Venn diagram It can change from problem to problem { } empty set LOGO www.themegallery.com Relationships between sets Equal sets: same cardinality and same elements “=“ Equivalent sets: same number of elements Subsets: A is the subset of set B if all of the elements of A are elements of B A⊂ B(子集) B⊃ A (superset扩散集) In our book, different from Chinese book How many subsets? Include {} and equal set Use permutation and combination to prove. LOGO www.themegallery.com Continue Complement set A’ contains all of the elements of the universal set not in A. set A and its complement A’ are disjoint- A∩A’=empty set Power set: All subsets of a given set A If a set has n elements it will have 2^n subsets LOGO www.themegallery.com Intersection and union of sets A∪B : the union of sets A and B. A∩B : the intersection of sets A and B. The elements common to set A and B. LOGO www.themegallery.com Laws 1. A ∩ A = A 2. A ∩ B = B ∩ A (commutative law) 3. A ∩ B ∩ C = A ∩ (B ∩ C) (associative law) 4. A ∩ φ = φ ∩ A = φ 5. A ∪ (A ∩ B) = A 6. A ∩ (A ∪ B) = A 7. A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (distributive law) 8. A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (distributive law) LOGO Homework www.themegallery.com LOGO