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NRF 10 - 3.1 Factors and Multiples of Whole Numbers Where does our number system come from? The simple answer is from the Hindu-Arabic system (a base 10 place holder system) which was influenced by the fact that we have 10 fingers! Our Number System has become more clearly defined with history. We now categorize numbers as follows: Note: The definition of a Rational Number is: Q ab a, b I , b 0 Translation: If it can be made into a fraction it is Rational! Classify each of the following numbers: 1. ‘5’ Natural, Whole, Integer, Rational, Irrational, Real 2. ‘1.6’ Natural, Whole, Integer, Rational, Irrational, Real 3. ‘ 8 ’ Natural, Whole, Integer, Rational, Irrational, Real 4. ‘ 2.3 Natural, Whole, Integer, Rational, Irrational, Real 5. ‘-4’ Natural, Whole, Integer, Rational, Irrational, Real 6. ‘-1.25’ Natural, Whole, Integer, Rational, Irrational, Real 7. ‘ ’ 8. ‘0.121212 . . .’ 9. ‘ ’ 10. ‘ ’ Natural, Natural, Natural, Natural, Irrational, Irrational, Irrational, Irrational, Whole, Whole, Whole, Whole, Integer, Integer, Integer, Integer, Rational, Rational, Rational, Rational, Real Real Real Real Prime Numbers When a number has exactly two factors, 1 and itself, the number is a prime number. Examples: 2, 3, 5, ___, ___, ___, ___, …. Composite Numbers have more than two factors Note: - The number 1 is a special number – neither prime or composite - The number 0 is composite – infinite number of factors Prime Factorization Every Natural number can be re-written as the product of prime numbers. This is the prime factorization of a number. Factor Trees can be used to break a number down to its prime factorization Examples: 12 75 1750 22302 Divisibility Tests/Rules are great at helping break up larger Natural numbers into their prime factors: Recall: Divisibility Rule for 2: ____________________________________ Divisibility Rule for 3: ____________________________________ Divisibility Rule for 4: ____________________________________ Divisibility Rule for 5: ____________________________________ Divisibility Rule for 6: ____________________________________ Divisibility Rule for 8: ____________________________________ Divisibility Rule for 9: ___________________________________ Divisibility Rule for 10: __________________________________ The Greatest Common Factor (GCF) The GCF of two or more numbers is the greatest factor the numbers have in common. Example 1: What is the GCF of 76 and 148? Solution: Method 1: Finding the GCF by listing the factors of each number Step1 : List the factors of each number 76 – 1, 2, 4, 19, 38, 76 148 - 1, 2, 4, 37, 74, 148 Step 2: Identify the greatest of the common factors GCF(76,148) = 4 4 is the “largest number that divides both numbers” Method 2: Finding the GCF using the prime factorizations of each numbers Step 1 : Break each number into the product of its primes (prime factorization) 76 = 2 x 38 = 2 x 2 x 19 148 = 2 x 74 = 2 x 2 x 37 Step 2 : Multiply the common factors of each prime factorization list Since both have two 2’s in common, the GCF(76,148) = 2 x 2 = 4 Example 2: What is the GCF of 70 and 84? Example 3: What is the GCF of 420, 555, and 615? Least Common Multiple (LCM) The LCM of two or more numbers is the least (smallest) number that is divisible by each number. Example: What is the LCM of 32 and 50? Method 1 – Using the list of multiples of each number Multiples of 32 – 32, 64, 96, 128, 160, 192, 224, 256, 288, 320, 352, 384, 416, 448, 480, 512, 544, 576, 608, 640, 672, 704, 736, 768, 800, 832, 864,… Multiples of 50 – 50, 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600, 650, 700, 750, 800, 850, 900,… The LCM(32,50) is 800. A lot of work….is there a more efficient way of finding the answer? Yes, there is! Consider the next method: Method 2 – Finding the LCM using the prime factorizations of each number 32 = 2 x 2 x 2 x 2 x 2 = 25 50 = 2 x 5 x 5 = 2 x 52 LCM(32,50) = 25 x 52 = 32 x 25 = 800 (The LCM is the product of the highest power of all unique primes) Example 2: Find the least common multiple of 20, 36, 38. 20 = 2 x 2 x 5 36 = 2 x 2 x 3 x 3 114 = 2 x 3 x 19 = 22 x 5 = 22 x 32 = 2 x 3 x 19 LCM = 22 x 32 x 5 x 19 = 3420 Example 3: Find the least common multiple of 28, 40, 44. Problems: 1. A builder wants to cover a wall measuring 9 ft. by 15 ft. with square pieces of plywood. a) What is the side length of the largest square that could be used to cover the wall? Assume the squares cannot be cut. b) How many square pieces of plywood would be needed? 2. Bill and Betty do chores at home. Bill mows the lawn every 8 days, and Betty bathes the dog every 14 days. Suppose Bill and Betty do their chores today. How many days will pass before they both do their chores on the same day again?