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Unit 1 Overview: Number Theory ARRAYS arrays must have the same number of objects in each row arrays must have the same number of objects in each column this is an array for 10: x x these are NOT arrays for 10: x x x x x x x x x x x x xx xx row x column = product factor x factor = product (or multiple) x x or x x x x this is an x x outline this is too uneven x x x x SQUARE NUMBERS Sometimes you can make a square array. You can only make square arrays with certain numbers. The number of rows and the number in each row must be the same. 1 4 9 16 x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x The shorthand for those multiplication problems uses an exponent. 2 4 = 2 x 2 = 2 read it as ‘2 squared’ or ‘2 to the second power’ 2 9 = 3 x 3 = 3 read it as ‘3 squared’ or ‘3 to the second power’ 2 16 = 4 x 4 = 4 read it as ‘4 squared’ or ‘4 to the second power’ Since you can square a number, you can also ‘unsquare’ it. When you do that, you find the square root of the number. It is the number in a row or column of the square array. Or, the number you multiply by itself to get the root. To find the square root of 9, think OR make/picture a square array of 9 x’s What number times itself equals 9? How many in a row or column? 3x3=9 x x x x x x x x x either way you think about it, the square root of 9 is 3 ****************************************************************************** FACTORS 3 x 2 =6 3 and 2 are factors of 6 6 is the product of 3 and 2 1 x 6 = 6 1 and 6 are factors of 6 6 is the product of 1 and 6 to list all the factors of a number: list the factor pairs in order, or use Factor Rainbows the factors of 24 are the factors of 23 are (1, 2, 3, 4, 6, 8, 12, 24) (1, 23) DIVISIBILITY RULES or (1, 24, 2, 12, 3, 8, 4, 6) the factors of 2 are (1, 2) any even number is ÷ 2 a number ending in 0 or 5 is ÷ 5 a number ending in 0 is ÷ 10 for ÷ 3, add the digits -- if the sum is ÷ 3, so is the number for ÷ 9, add the digits -- if the sum is ÷ 9, so is the number any even number that is also ÷3 is ÷6 (2 & 3 are factors of 6) PRIME OR COMPOSITE? EVEN OR ODD? 24 and 27 both have more than 2 factors, so they are composite numbers 23 and 11 have 2 and only 2 factors, one and the number itself, they are prime numbers odd numbers end in 1, 3, 5, 7, or 9 even numbers end in 0, 2, 4, 6, or 8 23 and 27 are odd numbers 24 and 2 are even numbers FACTOR STRINGS AND PRIME FACTORIZATION a factor string is a name for a number written as a multiplication number sentence (no 1’s) factor strings can be short or long some factor strings for 36 are 2 x 18 2x3x6 4x3x3 some factor strings for 28 are 2 x 14 4x7 2x2x7 the longest string (which has only prime numbers) is the prime factorization of that number the prime factorization of 36 is 3x3x4 the prime factorization of 28 is 2x2x7 finding the prime factorization is easy *think of any fact for the number *circle any prime numbers, write a new fact for the composite numbers for 50 for 48 5 x 10 6x8 5x2x5 2x3x2x4 2x3x2x2x2 *once all the numbers are prime, you’re done 5 x 2 x 5 2x3x2x2x2 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ PLACE VALUE Know large places to one million million, hundred ten thousands, hundreds tens ones thoudsands thousands 4, 8 5 9, 2 7 3 four million, eight hundred fifty nine thousand, two hundred seventy-three