Algebra Properties
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Created by:
amctaggart25 on January 16, 2012
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44 terms
Terms | Definitions |
|---|---|
 Intersection of Sets | The intersection of sets A and B, written A (upside down U) B, is the set of elements that are in both A and B. |
| Union of Sets | The union of sets A and B, written A U B, is the set of elements in either A or B (or in both). |
| Order of Operation in Evaluating Expressions | 1. Do operations within parentheses or other grouping symbols. 2. Within grouping symbols, or if there are no grouping symbols: a. Do all powers from left to right b. Do all multiplications and divisions from left to right. c. Do all additions and subtractions from left to right. |
| Square of the Square Root Property | For any nonnegative number n, √n • √n = n. |
| Pythagorean Theorem | In a right triangle with legs of lengths a and b and hypotenuse of length c, a2 + b2 = c2. |
| Area Model for Multiplication | The area A of a rectangle with length L and width w is Ew. |
| Commutative Property of Multiplication | For any real numbers a and b, ab = ba. |
| Area Model for Multiplication (discrete form) | The number of elements in a rectangular array with r rows and c columns is rc. |
| Associative Property of Multiplication | For any real numbers a, b, and c, (ab)c = a(bc). |
| Multiplicative Identity Property of 1 | For any real number a, a • 1 = 1 • a = a. |
| Property of Reciprocals | Suppose a ≠ 0. The reciprocal of a is 1/a. That is, a • 1/a = 1/a • a = 1. |
| Reciprocal of a Fraction Property | Suppose a ≠ 0 and b ≠ 0. The reciprocal of a/b is b/a. |
| Multiplication Property of Zero | For any real number a, a • 0 = 0 • a = 0. |
| Multiplying Fractions Property | For all real numbers a, b, c, and d, with b and d not zero, a/b • c/d = ac/bd. |
| Equal Fractions Property | If b ≠ 0 and k ≠ 0, then ak/bk = a/b. |
| Rate FActor Model for Multiplication | When a rate r is multiplied by another quantity x, the product is rx. So the unit of rx is the product of the units for r and x. |
| Multiplication Property of -1 | For any real number a, a • -1 = -1 • a = -a. |
| Rules for Multiplying Positive and Negative Numbers | If two numbers have the same sign, their product is positive. If two numbers have different signs, their product is negative. |
| Properties of Multiplication of Positive and Negative Numbers | 1. The product of an odd number of negative numbers is negative. 2. The product of an even number of negative numbers is positive. |
| Multiplication Property of Equality | For all real numbers a, b, and c, if a = b, then ca = cb. |
| Multiplication Property of Inequality (Part 1) | If x < y and a is positive, then ax < ay. |
| Multiplication Property of Inequality (Part 2) | If x < y and a is negative, then ax > ay. |
| Multiplication Counting Principle | If one choice can be made in m ways and a second choice can be made in n ways, then there are mn ways of making the first choice followed by the second choice. |
| Permutation Theorem | There are n! possible permutations of n different items when each item is used exactly once. |
| Putting-Together Model for Addition | If a quantity x is put together with a quantity y with the same units, and there is no overlap, then the result is the quantity x + y. |
| Slide Model for Addition | If a slide x is followed by a slide y, the result is the slide x + y. |
| Commutative Property of Addition | For any real numbers a and b, a + b = b + a. |
| Associative Property of Addition | For any real number a, b, and c, (a + b) + c = a + (b + c). |
| Additive Identity Property | For any real number a, a + 0 = 0 + a = a. |
| Property of Opposites | For any real number a, a + -a = -a + a = 0. |
| Opposite of Opposites (Op-op) Property | For any real number a, -(-a) = a. |
| Addition Property of Equality | For all real numbers a, b, and c, if a = b, then a + c = b + c. |
| Distributive Property: Adding or Subtracting Like Terms | For any real numbers a, b, and c, ac + bc = (a + b)c and ac - bc = (a - b)c. |
| The Distributive Property: Removing Parentheses | For all real numbers a, b, and c, c(a + b) = ca + cb and c(a - b) = ca - cb. |
| Distributive Property: Adding Fractions | For all real numbers a, b, and c, with c ≠ 0, a/c + b/c = a + b/c. |
| Addition Property of Inequality | For all real numbers a, b, and c, if a < b, then a + c < b + c. |
| Algebraic Definition of Subtraction | For all real numbers a and b, a - b = a + -b. |
| Take-Away Model for Subtraction | If a quantity y is taken away from an original quantity x, the quantity left is x - y. |
| Comparison Model for Subtraction | The quantity x - y tells how much the quantity x differs from the quantity y. |
| Opposite of a Sum Property | For all real numbers a and b, -(a + b) = -a + -b = -a - -b. |
| Opposite of a Difference Property | For all real number a and b, -(a - b) = -a + b. |
| Triangle Sum Theorem | If any triangle with angle measures a, b, and c in degrees, a + b + c = 180. |
| Triangle Inequality | Part 1: If A, B, and C are any three points, then AB + BC ≥ AC. Part 2: If A, B, and C are vertices of a triangle, then AB + BC > AC. |
| Third Side Property | If x and y are the lengths of two sides of a triangle, and x > y, then the length z of the third side must satisfy the inequality x - y < z < x + y. |
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