| A | B |
|---|
| addition property of equality | If a, b, and c are any real numbers, and a = b, then a + c = b + c and c + a = c + b |
| addtion property of order | For all real numbers a, b, and c: 1. If a < b, then a + c < b + c; 2. If a > b, then a + c > b + c |
| associative properties | For all real numbers a, b, and c: (a + b) + c = a + (b + c); (ab)c = a(bc) |
| closure properties | For all real numbers a and b: a + b is a unique real number; ab is a unique real number |
| commutative properties | For all real numbers a and b: a + b = b + a; ab = ba |
| density property for rational numbers | Between every pair of different rational numbers there is another rational number |
| distributive property (of multiplication with respect to addtion) | For all real numbers a, b, and c, a(b + c) = ab + ac and (b + c)a = ba + ca |
| distributive property (of multiplication with respect to subtraction) | For all real numbers a, b, and c, a(b - c) = ab - ac and (b - c)a = ba - ca |
| division property of equality | If a and b are any real numbers, c is any nonzero real number, and a = b, then a/c = b/c |
| identity property of addition | There is a unique real number 0 such that for every real number a, a + 0 = a and 0 + a = a |
| identity property of multiplication | There is a unique real number 1 such that for every real number a, a x 1 = a and 1 x a = a |
| multiplication property of equality | If a, b, and c are any real numbers, and a = b, then ca = cb and ac = bc |
| multiplication property of order | For all real numbers a, b, and c such that c > 0: 1. If a < b, then ac < bc; 2. If a > b, then ac > bc. For all real number a, b, and c such that c < 0: 1. If a < b, then ac > bc; 2. If a > b, then ac < bc |
| multiplicative propery of -1 | For every real number a, a(-1) = -a and (-1)a = -a |
| multiplicative property of zero | For every real number a, a x 0 = 0 and 0 x a = 0 |
| product property of square roots | For any nonnegative real numbers a and b, square root of ab = square root of a x square root of b |
| property of comparison | For all real numbers a and b, one and only one of the following statements is true: a < b, a = b, a > b |
| property of completeness | Every decimal represents a real number, and every real number can be represented by a decimal |
| property of the opposite of a sum | For all real numbers a and b: -(a + b) = (-a) + (-b) |
| property of opposites | For every real number a, there is a unique real number -a such that a + (-a) = 0 and (-a) + a = 0 |
| property of opposites in products | For all real numbers a and b, (-a)(b) = -ab, a(-b) = -ab, and (-a)(-b) = ab |
| property of quotients | If a, b, c, and d are real numbers with b not equal to 0 and d not equal to 0, then ac/bd = a/b x c/d |
| property of the reciprocal of the opposite of a number | For every nonzero number a, 1/-a = - 1/a |
| property of the reciprocal of a product | For all nonzero numbers a and b, 1/ab = 1/a x 1/b |
| property of reciprocals | For every nonzero real number a, there is a unique real number 1/a such that a x 1/a = 1 and 1/a x a = 1 |
| property of square roots of equal numbers | For any real numbers r and s: r squared = s squared if and only if r = s or r = -s |
| quotient property of square roots | For any nonnegative real number a and any positive real number b, square root of a/b = square root of a / square root of b |
| reflexive property of equality | If a is a real number, then a = a |
| substitution principle | An expression may be replaced by another expression that has the same value |
| subtraction property of equality | If a, b, and c are any real numbers, and a = b, then a - c = b - c |
| symmetric property of equality | If a and b are real numbers, and a = b, then b = a |
| transitive property of equality | For all real numbers, a, b, and c, if a = b and b = c, then a = c |
| transitive property of order | For all real numbers a, b, and c, 1. IF a < b and b < c, then a < c; 2. If c > b and b > a, then c > a |
| zero-product property | For all real numbers a and b, ab = 0 if and only if a = 0 or b = 0 |
