2.13: Identify and Apply Number Properties in Integer Operations
Did you know that you can simplify an expression by using a property? Take a look at this dilemma.
Travis is working on his homework. He was doing great until he got to this problem. Take a look.
Travis isn't sure how to simplify this expression since he hasn't been given a value for . If Travis applies the distributive property, then he can simplify this expression. Take a look at this Concept and then you can work on this dilemma at the end of it.
Guidance
There are many properties, the associative property, the commutative property, the additive identity property, and the additive inverse property.
You can use these properties to simplify integer expressions.
Simplify the following expression. Justify each step by identifying the property used.
First, think about this problem. It has losses and gains in it.
Use properties to reorganize the expression.
The commutative property states that numbers can be added in any order and this does not change the sum. When working with negative numbers, keep the negative sign with the number it belongs to. Adding parentheses around the negative number can help you keep things organized.
Apply the commutative property:
The associative property states that the grouping of numbers does not change the sum.
Apply the associative property:
The additive inverse property states that any number added to its opposite equals zero.
Apply the additive inverse property:
The additive identity property states that the sum of any number and zero is that number.
Apply the additive identity property:
This is our answer.
Exactly, and what you will find is that it is easier to perform operations with positive and negative integers when they are organized together. If you add two negative integers, then you have a negative answer. If you add two positives then you have a positive answer. Just remember to think in terms of losses and gains and it will help you keep your signs straight.
You have already learned several properties of multiplication: the associative property, the commutative property, the distributive property, the multiplicative identity, and the zero property. You can use these properties to simplify integer expressions.
Simplify the following expression. Justify each step by writing the property used.
I know that it may seem confusing, but start by looking at what is actually in the problem. You can see that there is a set of parentheses with values in it. There is also a small integer addition problem outside of the parentheses.
Example A
Simplify
Solution:
Example B
Solution:
Example C
Solution:
Now let's go back to the dilemma from the beginning of the Concept.
First, Travis has to distribute the negative four by multiplying it with both of the terms inside the parentheses.
This is the answer.
Vocabulary
- Associative Property
- The associative property says the operations can be grouped in a different order and give the same result. Addition has the associative property, because ( x + y ) + z is the same as x + ( y + z )
- Commutative Property
- The commutative property says that the order of the operands does not matter. Addition has the commutative property, because x + y is always the same as y + x . Subtraction does not have the commutative property, because x - y is different from y - x
- Distributive Property
- The product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, .
- Additive Identity Property
- The sum of any number and zero is the number itself.
- Additive Inverse Property
- Any number added to its opposite or additive inverse is equal to 0.
- Multiplicative Identity
- The multiplicative identity for multiplication of real numbers is one.
- Zero Property
- The zero property of multiplication says that the product of any number and zero is zero.
- Multiplicative Inverse
- The multiplicative inverse of a number is the reciprocal of the number. The product of a number and its multiplicative inverse will always be equal to 1.
Guided Practice
Here is one for you to try on your own.
Name the property illustrated here.
Solution
Notice that there aren't any parentheses here. This expression also does not distribute any terms. The same terms are on both sides of the equals sign. The only difference is the order of the terms.
The Commutative Property is our answer.
Video Review
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Directions: Identify each property illustrated.
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Directions: Simplify each expression.
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Image Attributions
Description
Learning Objectives
Here you'll identify and apply number properties in integer operations.