This is known as a one-to-one correspondence. We refer to this as the real number line. Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line? Beginning with the natural numbers, we have expanded each set to form a larger set, meaning that there is a subset relationship between the sets of numbers we have encountered so far. These relationships become more obvious when seen as a diagram.
Sets of numbers. Watch this video for an overview of the sets of numbers, and how to identify which set a number belongs to. When we multiply a number by itself, we square it or raise it to a power of 2. We can raise any number to any power. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order.
We use the order of operations. This is a sequence of rules for evaluating such expressions. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols. The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right. There are no grouping symbols, so we move on to exponents or radicals.
For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated.
Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result. Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :. Note that in the first step, the radical is treated as a grouping symbol, like parentheses.
Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped. Watch the following video for more examples of using the order of operations to simplify an expression. For some activities we perform, the order of certain operations does not matter, but the order of other operations does.
For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics. The commutative property of addition states that numbers may be added in any order without affecting the sum. Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.
It is important to note that neither subtraction nor division is commutative. The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same. The associative property of addition tells us that numbers may be grouped differently without affecting the sum.
This property can be especially helpful when dealing with negative integers. Consider this example. The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum. This property combines both addition and multiplication and is the only property to do so. Let us consider an example. Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by —7, and adding the products.
To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example. Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.
This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example. The identity property of addition states that there is a unique number, called the additive identity 0 that, when added to a number, results in the original number.
The identity property of multiplication states that there is a unique number, called the multiplicative identity 1 that, when multiplied by a number, results in the original number. There are no exceptions for these properties; they work for every real number, including 0 and 1. Exponents are a way to represent repeated multiplication; the order of operations places it before any other multiplication, division, subtraction, and addition is performed.
In the video that follows, an expression with exponents on its terms is simplified using the order of operations. The rules of the order of operations require computation within grouping symbols to be completed first, even if you are adding or subtracting within the grouping symbols and you have multiplication outside the grouping symbols. After computing within the grouping symbols, divide or multiply from left to right and then subtract or add from left to right. When there are grouping symbols within grouping symbols, calculate from the inside to the outside.
That is, begin simplifying within the innermost grouping symbols first. Remember that parentheses can also be used to show multiplication. In the example that follows, both uses of parentheses—as a way to represent a group, as well as a way to express multiplication—are shown. There are brackets and parentheses in this problem. Compute inside the innermost grouping symbols first. In the following video, you are shown how to use the order of operations to simplify an expression with grouping symbols, exponents, multiplication, and addition.
These problems are very similar to the examples given above. How are they different and what tools do you need to simplify them? This problem has parentheses, exponents, multiplication, subtraction, and addition in it, as well as decimals instead of integers. Use the box below to write down a few thoughts about how you would simplify this expression with decimals and grouping symbols. Show Solution. Use the box below to write down a few thoughts about how you would simplify this expression with fractions and grouping symbols.
According to the order of operations, simplify the terms with the exponents first, then multiply, then add. In this section, we will use the skills from the last section to simplify mathematical expressions that contain many grouping symbols and many operations.
We are using the term compound to describe expressions that have many operations and many grouping symbols. More care is needed with these expressions when you apply the order of operations. Additionally, you will see how to handle absolute value terms when you simplify expressions. Grouping symbols are handled first. Start with the innermost set of parentheses that are a grouping symbol. Begin working out from there. The fraction line acts as a type of grouping symbol, too; you simplify the numerator and denominator independently, and then divide the numerator by the denominator at the end.
The top of the fraction is all set, but the bottom denominator has remained untouched. Apply the order of operations to that as well. The video that follows contains an example similar to the written one above. Note how the numerator and denominator of the fraction are simplified separately. Parentheses are used to group or combine expressions and terms in mathematics.
You may see them used when you are working with formulas, and when you are translating a real situation into a mathematical problem so you can find a quantitative solution. For example, you are on your way to hang out with your friends, and call them to ask if they want something from your favorite drive-through. Three people want the same combo meal of 2 tacos and one drink. You can use the distributive property to find out how many total tacos and how many total drinks you should take to them.
The distributive property allows us to explicitly describe a total that is a result of a group of groups. In the case of the combo meals, we have three groups of two tacos plus one drink. The following definition describes how to use the distributive property in general terms. What this means is that when a number multiplies an expression inside parentheses, you can distribute the multiplication to each term of the expression individually.
To multiply three by the sum of three and y, you use the distributive property —. The next example shows how to use the distributive property when one of the terms involved is negative. This expression has two sets of parentheses with variables locked up in them. We will use the distributive property to remove the parentheses. Note how we placed the negative sign that was on b in front of the 2 when we applied the distributive property. It is important to be careful with negative signs when you are using the distributive property.
Absolute value expressions are one final method of grouping that you may see. Recall that the absolute value of a quantity is always positive or 0. When you see an absolute value expression included within a larger expression, treat the absolute value like a grouping symbol and evaluate the expression within the absolute value sign first. Then take the absolute value of that expression. The example below shows how this is done. Grouping symbols, including absolute value, are handled first.
Simplify the numerator, then the denominator. The following video uses the order of operations to simplify an expression in fraction form that contains absolute value terms. Note how the absolute values are treated like parentheses and brackets when using the order of operations. C There are an infinite number of solutions.
D There are no solutions. Try substituting any value in for y in this equation and think about what you find. The correct answer is: There are an infinite number of solutions to the equation.
Try substituting any two values in for y in this equation and think about what you find. When dealing with sets of parentheses, make sure to evaluate the inner parentheses first, and then move to the outer set. When you evaluate the expressions on either side of the equals sign, you get. Since you have a true statement, the equation is true for all values of y.
Application Problems. The power of algebra is how it can help you model real situations in order to answer questions about them. This requires you to be able to translate real-world problems into the language of algebra, and then be able to interpret the results correctly. The sum of their ages is Use an algebraic equation to find the ages of Amanda and her dad.
For example, if Amanda is 20, then her father would be 40 because he is twice as old as she is, but then their combined age is 60, not What if she is 12?
As you can see, picking random numbers is a very inefficient strategy! You can represent this situation algebraically, which provides another way to find the answer. Find the ages of Amanda and her dad. What is the problem asking? Assign a variable to the unknown. Solve the equation for the variable. Do the answers make sense? Amanda is 22 years old, and her father is 44 years old. Consider that the rental fee for a landscaping machine includes a one-time fee plus an hourly fee. You could use algebra to create an expression that helps you determine the total cost for a variety of rental situations.
An equation containing this expression would be useful for trying to stay within a fixed expense budget. A landscaper wants to rent a tree stump grinder to prepare an area for a garden. Write an expression for the rental cost for any number of hours.
The problem asks for an algebraic expression for the rental cost of the stump grinder for any number of hours. An expression will have terms, one of which will contain a variable, but it will not contain an equal sign. Look at the values in the problem:. Think about what this means, and try to identify a pattern.
Since multiplication is repeated addition, you could also represent it like this:. What information is important to finding an answer? What is the variable? What expression models this situation? Using the information provided in the problem, you were able to create a general expression for this relationship. This means that you can find the rental cost of the machine for any number of hours! The machine cannot be rented for part of an hour. The landscaper can rent the machine for 5 hours.
It is often helpful to follow a list of steps to organize and solve application problems. Solving Application Problems. Follow these steps to translate problem situations into algebraic equations you can solve. Read and understand the problem. Determine the constants and variables in the problem. Write an equation to represent the problem.
Write a sentence that answers the question in the application problem. Gina has found a great price on paper towels. She wants to stock up on these for her cleaning business.
Write an equation that Gina could use to solve this problem and show the solution. The problem asks for how many packages of paper towels Gina can purchase. What is the problem asking you? What are the constants? What equation represents this situation? Solve for p. Check your solution.
Substitute 48 in for p in your equation. Gina can purchase 48 packages of paper towels.
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