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Definition
Numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Decimal notation for rational numbers either terminates or repeats.
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The real numbers that are not rational numbers. Decimal notation for irrational numbers neither terminates nor repeats. |
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The set of all rational numbers combined with the set of all irrational numbers. Real numbers are modeled using a number line. |
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A member belonging to a set. |
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When all the elements of one set are elements of a second set, the first set is a subset of the second set. |
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Definition
Open (a,b) {x | a < x < b} Closed [a,b] {x | a ≤ x ≤ b} Half-open [a,b) {x | a ≤ x < b} Half-open (a,b] {x | a < x ≤ b} Open (a,∞) {x | x > a} Half-open [a,∞) {x | x ≥ a} Open (-∞,b) {x | x < b} Half-open (-∞,b] {x | x ≤ b} |
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Commutative Properties of Addition and Multiplication |
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Definition
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Associative Properties of Addition and Multiplication |
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Definition
a + (b + c) = (a + b) + c a(bc) = (ab)c |
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Additive Identity Property |
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Definition
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Additive Inverse Property |
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Definition
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Multiplicative Identity Property |
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Definition
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Multiplicative Inverse Property |
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Definition
a · 1/a = 1/a · a = 1 (a ≠ 0) |
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Definition
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Definition
|a| = a if a ≥ 0 |a| = -a if a < 0 |
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Distance Between Two Points on the Number Line |
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Definition
For any real numbers a and b, the distance between a and b is |a - b| or equivalently, |b - a|. |
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Absolute Value of a Product |
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Definition
For any real numbers a and b and any nonzero number c:
|ab| = |a| · |b|. |
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Absolute Value of a Quotient |
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Definition
For any real numbers a and b and any nonzero number c:
|a/c| = |a| / |c|. |
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Absolute Value of an Exponent |
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Definition
For any real numbers a and b and any nonzero number c: |an| = an if n is an even integer.
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Absolute Value of Negative and Positive Numbers |
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Definition
For any real numbers a and b and any nonzero number c:
|-a| = |a|. |
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Definition
For any real number a, -1 · a = -a (Multiplying a number by -1 produces its opposite.) |
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Definition
For any real number a, - (-a) = a (The opposite of the opposite of a number is the number itself.) |
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Definition
- Positive numbers: Add the numbers. The result is positive.
- Negative numbers: Add the absolute values. Make the answer negative.
- A positive and a negative number: Subtract the smaller absolute value from the larger. Then:
- If the positive number is the greater absolute value, make the answer positive.
- If the negative number has the greater absolute value, make the answer negative.
- If the numbers have the same absolute value, make the answer 0.
- One number is zero: The sum is the other number.
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Definition
Those numbers used for counting {1, 2, 3, . . . } |
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The natural numbers and 0 {0, 1, 2, 3, . . . } |
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The whole numbers and their opposites {. . . -3, -2, -1, 0, 1, 2, 3, . . .}. |
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Definition
A method of denoting sets. e.g., a set containing the numbers 2, 3, 4, and 5 can
be written using the roster method: {2, 3, 4, 5}. |
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