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Math is filled with theorems, axioms, and properties. Though the following properties may seem simplistic, the ISEE will couple these properties with difficult concepts, such as algebraic functions. It is important to understand the following properties thoroughly. Reflexive Property A value is equal to itself x = x. This is useful because even when you do not know the individual values (such as in an algebraic equation), you know a value is equal to itself. Other examples of the reflexive property at work: a + b2 = a + b2 x2 + 7x + 10 = x2 + mx + 10 We can see in the second example that mx must be 7x, based on the reflexive property. Symmetric Property Equal values may be placed on either side of the equals sign. x = y y = x This is important for manipulating equations. For example, say you’re given the following system of equations. You can rewrite the equations to see how to easily use elimination. 8x + 2y = 14 is the same as 8x + 2y = 14 16 = 4x - 2y is the same as 4x - 2y = 16 Now you can see that the 2y and -2y can easily cancel. Transitive Property If one value equals a second value and the second value equals a third value, then the third value equals the first value. x = y y = z x = z For example, if you know: x + y = z × w z × w = a - b Then you also know that: x + y = a - b Commutative Property When you are only adding values, you may change the order without changing the result. The same is true for multiplication. x + y = y + x x × y = y × x We can use the commutative property to solve the following problem: By rearranging the denominators using the commutative property, we can see how to reduce each fraction easily. Associative Property When you are only adding or multiplying values, you may change the groupings of values without changing the result. The groups are separated by parentheses. x + (y + z) = (x + y) + z x × (y × z) = (x × y) × z The associative property can help solve a quadratic equation: You can rearrange parentheses with the associative property to create standard quadratic notation, and then solve by factoring. Addition Property You must add the same value to both sides of the equals sign to maintain the equality of the values. x + a = y + a Multiplication Property You must multiply the same value to both sides of the equals sign to maintain the equality of the values. a(x) = a(y) Distributive Property You may multiply a value to a sum by first multiplying the value to the sum’s addends and then adding those products together. x(y + z) = xy + xz Definition of Division Dividing values is the same as multiplying the value in the numerator by the reciprocal of the value in the denominator. x/y = x × 1/y x/(1/y) = x × y/1 Other examples of division: Multiplication Property of Zero The product of any value and zero will be zero. x × 0 = 0 Zero Product Property If the product of two values is zero, one of the values is zero. xy = 0 Either x = 0 or y = 0 This property allows us to use factoring to solve quadratic equations. Returning to the previous example: Using the zero product property, we know that (x + 5) or (x + 2) must be 0, thus x must either be -5 or -2. Definition of Zero as a Dividend The quotient of zero and any value (except zero) is zero. 0 ÷ x = 0 In other words, 0 divided by any number will always be 0. That is, unless you are dividing 0 by 0, in which case there is another definition you need to understand—the definition of zero as a divisor. Definition of Zero as a Divisor The quotient of any value and zero is considered undefined. x ÷ 0 = undefined This is because any attempt at a definition leads to a contradiction. We can understand why a fraction with 0 as a divisor is undefined by considering the following expression: b = a ÷ 0 According to this equation, it would also follow that the dividend, a, is equal to the product of b and 0: a = b × 0 However, in ordinary arithmetic, b × 0 will only equal a if a = 0. Otherwise, there is no solution to the equation. More colloquially, imagine you have 6 apples. If you want to evenly separate them into 2 groups, you will create 2 groups of 3 apples—in other words, this is 6 ÷ 2 = 3. If you take a grouping of 6 apples and separate them into 1 group, you will have a single group with all 6 apples—in other words, 6 ÷ 1 = 6. Now try to separate the apples into 0 groups—in other words 6 ÷ 0 = ?—and you will see that this is not possible. Later on in your math education, you may alternate ways of approaching the problem, but for now, any fraction with 0 as a divisor is undefined. Practice Problems Answers Question 1 Distributive Property Multiplication Property Multiplication Property of 0 Definition of Division Commutative Property D A A D
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