InterMath Dictionary
Properties
Field Axioms
What happens when you add two numbers that are opposites, like 5 and -5? In a sense, they "cancel" each other, resulting in zero. In other words, the sum of a number and its additive inverse is zero.
Examples of Additive Inverse Property
2 + (-2) = 0
x + (-x) = 0
-0.5 + 0.5 = 0
Non-Examples of Additive Inverse Property
2 + (-1) = 1
5 + (-6) = -1
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External Links to Extend Basic Understanding
Glossary of Properties: Learn more about the basic properties. Associative Property for Addition: Definition with example. Associative Property for Multiplication: Definition with example. Commutative Property for Addition: Definition with example. Commutative Property for Multiplication: Definition with example. Distributive Property: Definition with example. Where Will I See These Properties Again?: An answer, and a nice discussion of how the distributive property works.
Related Investigations on InterMath to Challenge Teachers
Formulaic Properties: Find values of x, y, and z, so that a given expression equals 18.
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Examples of Inverse Operations
Addition and subtraction are inverse operations.
Multiplication and division are inverse operations.
Order of Operations
When you are simplifying expressions, operations should be performed in the following order to ensure accuracy. Also any expression in parenthesis should be simplified first.
1. Multiplication 2. Division 3. Addition 4. Subtraction
This order is an arbitrary order that was established so that we would all simplify expressions in the same way.
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Order of Operations Definition: Clarify your understanding of order of operations. Practice with Order of Operations: A lesson on order of operations.
Formulaic Properties: Find values of x, y, and z, so that a given expression equals 18. Multiple Solutions: Find the set of solutions of a system of two equations with three variables.
Examples of Addition Property of Equality Suppose we had the equation a = b. When we add the same number, say 3, to both sides of the equation, we still have an equivalent equation:
a + 3 = b + 3.
To look at it in a different way, consider the following balance scale.
The scale is balanced with 1 box on each side. If we place 3 balls on both sides of the scale, the scale remains balanced.
We could write this mathematically, using x to represent the weight of one box as:
x = x x + 3 = x + 3.
Non-Examples Adding different numbers will not produce an equivalent equation.
Here we added 5 balls to one side of the scale and only 3 balls to the other side. As a result, the scale becomes unbalanced.
Simplifying with Addition and Subtraction: General tips for using these properties to solve equations.
Counterfeit Coin: How can you determine which coin is counterfeit?
"Canceling" is usually indicated by using slashes through the numbers, as the following examples show.
Examples of Canceling
Example 1: First, the numerator and denominator are written as a product of their factors. Then common factors are "cancelled."
Example 2: When common factors exists on opposite sides of an equation, you can also cancel as the following shows:
Example 3: We can also "cancel" using addition and subtraction.
In the above example, since there is a 3 on both sides of the equation, we can "cancel" both 3's.
Non-Examples of Canceling In order to "cancel" a number, the operations being performed on the numbers must be the same. Consider the following expression.
What do you think can be canceled?
Click here for a hint. hint (2x/x) + (1/2)
Can you determine why?
Click here for a hint.
Put these Properties to Use: See an online lesson on solving equations that use these properties!
Reciprocal Sums: Given the sum of two positive integers, determine the smallest possible sum of their reciprocals. Reciprocal Functions: Describe any patterns you find when comparing a function to its reciprocal function.
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