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Mastery Test 1 Study Guide Number Systems & Number Theory Subsets of Real Numbers: Counting #’s: 1, 2, 3,…. Whole #’s: 0, 1, 2, 3,…. Integers: …., -3, -2, -1, 0, 1, 2, 3,…. Positive #’s: all numbers greater than 0 Negative #’s: all numbers less than 0 Rational #’s: all Numbers that are NOT IRRATIONAL Irrational #’s: CRAZY, most well know example is Pi (3.1416……..) #’s that go on forever without repeating or terminating Perfect Squares and Square Roots: A perfect square has a square root that works out with no remainder. List of square roots from 1-15: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. Examples of perfect squares: √16 = 4 because 4 x 4 = 16; √225 = 15 because 15 x 15 = 225 Square roots for non-perfect squares: using a “Square Root Sandwich”: 1. Find the √5: Think about the perfect squares that are one above and one below √5 2. They are the √9 and √4 3. Find the square root of the one above & below: √9 = 3; √4 = 2 4. Estimate the square root of the non-perfect square: √5 ≈ 2 because 5 is closest to four. Factors and GCF (greatest common factor): A FACTOR is a # that divides evenly into a counting # Examples: Factors of 6: 1, 2, 3, 6 (1 x 6 = 6; 2 x 3 = 6) Factors of 9: 1, 3, 9 (1 x 9 = 9; 3 x 3 = 9) Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40 Prime #’s: numbers that have ONLY 2 FACTORS: 1 and itself Ex: 2, 3, 5, 7, 11, 13, 17, 19,…. Composite #’s: numbers that have 3 OR MORE FACTORS Ex: 4, 6, 8, 9, 10, 12, 14, 15, …. The number 1 is neither prime nor composite (it only has 1 factor: 1) The GCF of a numbers is the largest factor that 2 or more #’s have in common Example: Find the GCF of 18 & 27 1. List the factors of each # 18: 1, 2, 3, 6, 9, 18 27: 1, 3, 9, 27 2. Find the greatest number they have in common: 9 Prime Factorization (factor trees): The goal of Prime Factorization is to find all of the prime factors of a given number. Example: 36 (think: what 2 #’s = 36 when I multiply them) ^ 94 (ask yourself: are these prime? NO) ^^ 3 3 2 2 (ask yourself: are these prime? YES) Next you write your answer using exponents: 36 = 22 x 32 Finding the GCF using factor trees: Make a factor tree for each number; write each answer using exponents Example: Find the GCF of 36 and 54: 36 ^ 6 6 ^ ^ 2323 54 ^ 9 6 ^ ^ 3323 36: 22 x 32 54: 2 x 33 Ask yourself: What do they have in common? GCF: 2 x 32 Now multiply to get the GCF GCF: 2 x 3 x 3 = 18 GCF: 18 Multiples and LCM (Least Common Multiple): Multiples are the numbers that you count when you say the multiplication tables of a given number. Example: Multiples of 3: 3, 6, 9, 12, …. Multiples of 5: 5, 10, 15, 20, …. Common Multiples: the multiples that 2 given numbers have in common. Example: List the common multiples of 3 & 5: 15, 30, 45, … Once you find the first number you just continue to add it each time Finding LCM using prime factorization: When given a set of numbers you can use factor trees to find their LCM Example: Find the LCM of 33, 15 & 50: 33 ^ 3 11 15 ^ 35 50 ^ 5 10 ^ 25 List the factors of each using exponents: 33: 3 x 11 15: 3 x 5 50: 2 x 52 Next make a problem with the MOST of each number LCM of 33, 15, & 50 = 2 x 3 x 52 x 11 Multiply to get the answer: 2 x 3 x 5 x 5 x 11 = 240 Changing Fractions to Decimals: Terminating decimals: decimal numbers that end Example: .23 Repeating decimals: decimal numbers that never end (they repeat) Example: .2323232323 (you would put a _ over the first .23 to write this answer. To change a fraction to a decimal, use long division: Example: change _4_ to a decimal 9 .444 .4 (you would put a _over the 4) 9 |4.000 -36 40 -36 40 -36 Placing numbers on a number line: For whole numbers, put a dot on the number, label with the letter. For fractions, divide to make a decimal, estimate where the decimal would be on the number line, label with the letter. For decimals, estimate where it would be on the number line, label with the letter. Pi is always a little past the 3 For square roots, find the square root, plot the point on the number line, label with the letter