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What are real numbers?
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Both rational and irrational numbers.
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These arethe numbers that you encounter each day and are classified into various sets and subsets. |
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Rational Numbers
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Numbers that can be expressed as a fraction in the form a/b, where a and b are integers and b does not = 0. Rational numbers can be classified into subsets.
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These are the numbers we actually use daily, and always have a known ending. |
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Natural Numbers
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Also known as the counting numbers. {1,2,3,4,5,6,7...} Positive, whole, without 0. These are rational numbers.
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These are what little kids would think of as numbers. |
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Whole Numbers
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Natural numbers including 0. {0,1,2,3,4,5,6,7...} These are rational numbers. All positive except 0, which is neither positive or negative.
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Whole = 0. |
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Integers
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Whole numbers and their opposits. {...-3,-2,-1,0,1,2,3...} These are rational numbers. Negative, positive, include 0 (neither negative or positive)
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Add negatives in. |
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Rules of integers
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All numbers >0 are positive. All numbers <0 are negative.
Zero is neither positive or negative. Even intergers are divisible by 2 and includes 0. Odd integers are not divisible by 2.
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Everything that we generally think of as numbers. |
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Irrational numbers
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Numbers that cannot be expressed as fractions such as sqrt 2, sqrt 3, sqrt 5, and pi. When expressed as decimals, they are non-terminating and non-repeating.
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Numbers that we can't predict the end of. |
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How do you define an even integer?
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Divisible by 2
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count even numbers by what? |
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How do you define an odd integer?
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Not divisible by 2
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everything that isn't even |
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Is Zero positive or negative?
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Neither
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Imagine a numberline with 0 in it. |
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What is the order of operations used to evaluate mathematical expressions?
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PEMDAS
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Please Excuse My Dear Aunt Sally |
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What does PEMDAS stand for?
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Parentheses (or other grouping symbols)-Exponents-Multipication & Division (left to right)-Addition & Subtraction (left to right)
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Please Excuse My Dear Aunt Sally |
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What are grouping symbols?
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parenthesis, brackets, absolute value symbol, and fraction bar
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What groups things together in an equation? |
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Exponents include what 2 symbols?
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Any exponents and any radiacals.
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What is the opposite of an exponent? |
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What is the commutative property of addition?
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It states that changing the order of the addends in a sum does not change the sum.
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a+b = b+a, where a and b are any real numbers |
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What property does this equation represent?
a+b = b+a
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commutative property of addition
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does it matter who hangs out with whom in a commune? |
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What is the commutative property of multiplication
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It states that changing the order of the factors in a product does not change the product.
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a*b = b*a, where a and b are any real numbers |
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What property does this equation represent?
a*b = b*a
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Commutative Property of Multiplication
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Does it matter who multiplies whom in a commune? |
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Associative Property of Addition
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It states that changing the grouping (parenthesis or brackets) of addends in a sum does not change the resulting sum.
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a+(b+c) = (a+b)+c, where a, b, and c are any real numbers |
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What property does this equation represent?
a+(b+c) = (a+b)+c
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Associative Property of Addition
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Does it matter who adds to whom in an association? |
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Associative Property of Multipication
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It states that changing the grouping (parenthesis or brackets) of factors in a product does not change the resulting product.
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a*(b*c) = (a*b)*c, where a, b, and c are any real numbers |
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What property does this equation represent?
a*(b*c) = (a*b)*c
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Associative Property of Multiplication
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Does it matter who multiplies whom in an association? |
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The Distributive Property
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It states that multiplication distributes over addition and subtraction
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a*(b+c) = a*b+a*c, where a,b, and c are real numbers
a*(b-c) = a*b-a*c, where a, b, and c are real numbers |
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What does this equation represent?
a*(b+c) = a*b+a*c
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The distributive property of multiplication over addition
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What would distribute over what? |
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What does this equation represent?
a*(b-c) = a*b-a*c
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The distributive property of multiplication over division
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What would distribute over what? |
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What is the relationship between integers and the number line?
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Each integer has a location on a real number line, where the sign of the number determines to which side of 0 the number is located.
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Integers are {...,-3,-2,-1,0,1,2,3,...} |
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What is the absolute value of a number?
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It is the number of units a number is away from 0 on a number line.
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Consider position on number line. |
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How do you order integers?
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When comparing integers, a number farther to the left of 0 on the number line will be less in value than a number to the right of 0 on the number line.
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The value of an integer is determined by its location on the real number line, where negative numbers appear to the left of 0 and positive numbers are located to the right of 0. |
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How do you add integers?
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If the signs are the same add and keep the sign. If the signs are different subtract and take the sign of the number with the larger absolute value.
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Consider comparing the number of positives and negatives in a cloud. |
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How do you subtract integers?
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add the opposite of the number being subtracted and apply the sign of the greater number
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switch signs |
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How do you multiply negative integers?
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If there is an even number of negatives in the problem, the solution will be positive. If there is an odd number of negatives in the problem, the solution will be negative.
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evens vs odds |
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How do you divide negative integers?
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If there is an even number of negatives in the problem, the solution will be positive. If there is an odd number of negatives in the problem, the solution will be negative.
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evens vs odds |
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neg * neg = ?
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pos
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even |
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neg * pos = ?
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neg
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odd |
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neg / neg = ?
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pos
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even |
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neg / pos = ?
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neg
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odd |
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neg + neg = ?
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neg
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abs + sign |
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neg + pos = ?
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which ever sign is larger
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abs sub + sign |
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pos + pos = ?
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pos
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abs + sign |
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neg - pos = ?
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neg
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add abs + neg |
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pos - neg = ?
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larger value
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- turns pos and switches signs |
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neg - neg = ?
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larger value
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- turns pos and switches signs |
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How do you know if a number is divisible by 2?
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If the number is even.
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24, 50, and 66 are all divisible by 2 by being even |
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How do you know if a number is divisible by 3?
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If the sum of the individual digits of the number is divisible by 3.
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312 is (3+1+2) = 6 which is divisible by 3 and 9,021 is (9+0+2+1) = 12 which is divisible by 3. |
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How do you know if a number is divisible by 4?
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If the last two digits, taken as a two-digit number, are divisible by 4.
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736 is because 36 is divisible by 4. 12,716 is because 16 is divisible by 4. |
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How do you know if a number is divisible by 5?
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If the last digit of the number is a 5 or a 0.
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10, 25, 30, and 75 are divisible by 5 because they end in 5 or 0. |
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How do you know if a number is divisible by 8?
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If the last three digits, taken as a three-digit number, are divisible by 8.
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3,024 is because 024 is divisible by 8. 79,128 is because 128 is divisible by 8 (128/8 = 16). |
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How do you know if a number is divisible by 9?
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If the sum of the individual digits of the number is divisible by 9.
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9,135 is (9+1+3+5) = 18 which is divisible by 9 and 414,972 is (4+1+4+9+7+2) = 27 which is divisible by 9 |
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How do you know if a number is divisible by 6?
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If it is divisible by both 2 and 3.
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336 is divisible by both 2 and by 3 (3+3+6) = 12 |
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How do you know if a number is divisible by 12?
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If it is divisible by both 3 and 4.
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