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Categories of Numbers

Categories of Numbers

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Categories of Numbers

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  1. Categories of Numbers Natural Numbers– Counting numbers1, 2, 3, 4, 5, and so on. No Zero, no negatives, no decimals. 1, 2, 3, 4, 5, 6, ... Whole Numbers - Natural Numbers in it plus the number 0. 0, 1, 2, 3, 4, 5, 6, ... Integers - Whole Numbers and their opposites, or, Positive Numbers, Negative Numbers and Zero. ..., –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, ... Rational Numbers - Any number that can be expressed as a ratio of two integers. Note the ratio in rational. Based on how the decimals act. Decimals either do not exist, as in 5, (which is 5/1). Or the decimals terminate, as in 2.4, which is 24/10. Or the decimals repeat with a pattern, as in 2.333, which is 7/3. Summary: The behavior of the decimals is just the result when you divide an integer by another integer. Irrational Numbers - This is any number that cannot be express as an integer divided by an integer, These numbers have decimals that never terminate and never repeat with a pattern. If a number falls into a category, it automatically falls into all the categories above that category There is no number which is both an irrational number and a rational number.

  2. Order of Operations (STEPS) • Step 1: Evaluate the expressions inside grouping symbols • Step 2: Multiply and/or divide in order from left to right • Step 3: Add and/or subtract in order from left to right • Expressions versus Equations (Discuss parts of an expression) • Numerical expression: contains a combination of numbers and operations • Numerical equation: numerical expression with an equal sign • Algebraic expression: numerical expression that contains at least one variable • Algebraic equation: algebraic expression with an equal sign • Properties (RULES) – (Algebra pg 16) • Commutative 4+5 = 5+4 4x5 = 5x4 • Associative (4+5)+3 = 4+(5+3) (4x5)x3 = 4x(5x3) • Identity 3+0 = 3 3x0 = 0 • Distributive 5(3+a) = 5(3)+5(a) 5(3)+5(a) = 5(3+a) • Squares and Square Roots • The square of a number is the product of a number and itself 4 x 4 = 42 = 16 • A square root of a number is one of its two equal factors 16 = 4

  3. Integer Rules Addition Rules 1. When the signs are the same, add the numbers and keep the sign. 7 + 8 = 15 -7 - 8 = -15 2. When the signs are different, subtract the numbers and take the sign of the larger number. -7 + 8 = 1 7 – 8 = - 1 Note: Use Addition and Subtraction Rules when you “combine like terms” Subtraction RulesInverse (opposite) of Addition Rules 1. When the signs are the same, subtract the numbers and take the sign of the larger number. 7 – (+ 8) = - 1 -7 – (- 8) = 1 2. When the signs are different, add the numbers and keep the sign. 7 – (- 8) = 15 -7 – (+ 8) = -15 Multiplication & Division Rules 1. Negative x Negative = Positive Negative ÷ Negative = Positive - 6 x (-3) = 18 -6 ÷ (- 2) = 3 2. Positive x Positive = Positive Positive ÷ Positive = Positive 6 x 3 = 18 6 ÷ 2 = 3 3. Negative x Positive = Negative Negative ÷ Positive = Negative - 6 x 3 = -18 -6 ÷ 2 = -3 4. Positive x Negative = Negative Positive ÷ Negative = Negative 6 x (-3) = 18 6 ÷ (-2) = 3

  4. STEPS - Order of Operations (3 x 2)2 x 2 ÷ 9 +3 – 1 = G (6)2 x 2 ÷ 9 +3 – 1 = E 36 x 2 ÷ 9 +3 – 1 = M 72 ÷ 9 +3 – 1 = D 8 + 3 – 1 = A 11 – 1 = S 10

  5. Key Concepts (continued) • Absolute Value • The absolute value of a number is the distance the number is from zero on the number line • Comparing and Ordering Integers • When two numbers are graphed on a number line, the number to the left is always less than the number to the right ( -8 < 8) • Graphing Points • On a coordinate plane, the horizontal number line is the x-axis and the vertical number line is the y-axis. The origin is (0,0) and is the point where the number lines intersect. The x-axis and y-axis separate the plane into four quadrants. • Function: Relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Quadrant I Quadrant II 0,0 X Quadrant III Quadrant IV Y

  6. Function versus Relation Function Relation Input Output Input Output Domain Range Domain Range

  7. Changing Improper Fractions to Mixed Numbers • An improper fraction is a fraction that has a numerator larger than or equal to its denominator. • A proper fraction is a fraction with the numerator smaller than the denominator. • A mixed number consists of an integer followed by a proper fraction.

  8. Note the following pattern for repeating decimals:0.22222222... = 2/90.54545454... = 54/990.298298298... = 298/999Division by 9's causes the repeating pattern. Note the pattern if zeros proceed the repeating decimal:0.022222222... = 2/900.00054545454... = 54/990000.00298298298... = 298/99900Adding zero's to the denominator adds zero's before the repeating decimal. To convert a decimal that begins with a non-repeating part to a fraction: 0.21456456... Write it as the sum of the non-repeating part and the repeating part:  0.21 + 0.00456456456456456... Next, convert each of these decimals to fractions. The first decimal has a divisor of power ten. The second decimal (which repeats) is converted according to the pattern given above.  21/100 + 456/99900 Next, add these fraction by expressing both with a common divisor and add:  20979/99900 + 456/99900 = 21435/99900 Finally, simplify it to lowest terms and check on your calculator or with long division:  1429/6660 = 0.2145645645…

  9. Reflective Property: Any quantity equal to itself (a = a) Symmetric Property: If one quantity equals a second quantity, then the second quantity equals the first (a = b, then b = a) Transitive Property: If one quantity equals a second quantity and the second quantity equals a third quantity, then the first quantity equals the third quantity (a = b, and b = c, then a = c) Substitution Property: A quantity may be substituted for its equal value (a = b, then b can be substituted for a)

  10. Problem: 2z - 5(z + 1) = 3z + 1 Remove all parentheses using the distributive property 2z - 5z - 5 = 3z + 1 Combine like terms that occur on the same side of the equation using the commutative and associative properties -3z - 5 = 3z + 1 Two-Step Operations -3z - 5 = 3z + 1 -3z -3z -6z - 5 = 0 + 1 +5 +5 -6z - 0 = 6 Perform all addition and subtraction operations Perform all multiplication and division problems -6z = 6 ÷ -6÷ -6 z = -1 Answer: z = -1

  11. Greatest Common Factor • The greatest common factor of two or more whole numbers is the largest whole number that divides evenly into each of the numbers. There are three ways to find the greatest common factor. • First method – Factorization Tree Example: Find the Prime Factors for 36 and 54. 36 54 6 6 6 9 2 3 2 3 2 3 3 3 • The prime factorization of 36 is 2 x 2 x 3 x 3 • The prime factorization of 54 is 2 x 3 x 3 x 3 • The prime factorizations of 36 and 54 both have one2 and two3s in common. Next, multiply these common prime factors to find the greatest common factor. Like this...2 x 3 x 3 = 18

  12. Second method – Division by a prime factor, then list the common factors and multiply them. • Example: Find the GCF of 36 and 54. • 2 36 54 • 3 18 27 • 3 6 9 • 2 3 • The prime factorizations of 36 and 54 both have one2 and two3s in common. Next, multiply these common prime factors to find the greatest common factor. • Like this...2 x 3 x 3 = 18 • Third method - list all of the factors of each number, then list the common factors and choose the largest one. • Example: Find the GCF of 36 and 54. • The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. • The factors of 54 are 1, 2, 3, 6, 9, 18, 27, and 54. • The common factors of 36 and 54 are 1, 2, 3, 6, 9, 18 • Although the numbers in bold are all common factors of both 36 and 54, 18 is the greatest common factor.

  13. Graphing Absolute Value Functions

  14. Metric Measures and Conversions Some things to Remember when converting any type of measures: To convert from a larger to smaller metric unit you always multiply To convert from a smaller to larger unit you always divide The latin prefixes used in the metric system literally mean the number they represent. Example: 1 kilogram = 1000 grams. A kilo is 1000 of something just like a dozen is 12 of something. This is the metric conversion stair chart. You basically take a place value chart turn it sideways and expand it so it looks like stairs. The Latin prefixes literally mean the number indicated. Meter, liter or gram can be used interchangeably. You use this chart to convert metric measurements like this: • If you are measuring length use meter. • If you are measuring dry weight use grams. • If you are measuring liquid capacity use liter For every step upward on the chart you are dividing by 10 or moving the decimal one place to the left. Example: To convert 1000 milligrams to grams you are moving upward on the stairs. Pretend you are standing on the milli-gram stair tread and to get to the 1-gram stair tread you move up 3 steps dividing by 10 each time. 1000/10 = 100 100/10 = 10 10/10 = 1 or 1000/1000 = 1 or use the shortcut and just move the decimal place one place to the left with each step 1000 milligrams = 1 gram. When you move down the stairs you are multiplying by 10 for each step. SO you are adding a zero to your original number and moving the decimal one place to the right with each step. Example: To convert 2 kilometers to meters you move 3 steps down on the chart so you add 3 zeros to the 2. 2 kilometers = 2000 meters Problems: 1.) 3 meters = _300_ centimeters (multiply by 100 or just add 2 zeros) 2.) 40 liters = _4_ dekaliters (dividing by 10, or move your decimal place one place to the left) 3.) 600 milligrams = _0.6_ grams (dividing by 1000, or move your decimal 3 places to the left) 4.) 5 kilometers = _50_ hectometers 5.) 70 centimeters = _0.7_ meters 6.) 900 deciliters= _9_ dekaliters 7.) John's pet python measured 600 centimeters long. How many meters long was the snake? 6m 8.) Faith weighed 5 kilograms at birth. How many grams did she weigh? 500 grams 9.) Jessica drank 4 liters of tea today. How many deciliters did she drink? 40 deciliters

  15. COMMON MEASUREMENTS