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Real Numbers

The real number system.

All the numbers mentioned in this lesson belong to the set of Real numbers. The set of real numbers is denoted by the symbol [latex]\mathbb{R}[/latex]. There are five subsets  within the set of real numbers. Let’s go over each one of them.

Five (5) Subsets of Real Numbers

1) The Set of Natural or Counting Numbers 

 The set of the natural numbers (also known as counting numbers) contains the elements

The ellipsis “…” signifies that the numbers go on forever in that pattern.

2) The Set of Whole Numbers

 The set of whole numbers includes all the elements of the natural numbers plus the number zero ( 0 ).

The slight addition of the element zero to the set of natural numbers generates the new set of whole numbers. Simple as that!

3) The Set of Integers

The set of integers includes all the elements of the set of whole numbers and the opposites or “negatives” of all the elements of the set of counting numbers.

4) The Set of Rational Numbers

 The set of rational numbers includes all numbers that can be written as a fraction or as a ratio of integers. However, the denominator cannot be equal to zero.

A rational number may also appear in the form of a decimal. If a decimal number is repeating or terminating, it can be written as a fraction, therefore, it must be a rational number.

Examples of terminating decimals :

Examples of repeating decimals :

5) The Set of Irrational Numbers 

The set of irrational numbers can be described in many ways. These are the common ones.

  • Irrational numbers are numbers that cannot be written as a ratio of two integers. This description is exactly the opposite of that of rational numbers.
  • Irrational numbers are the leftover numbers after all rational numbers are removed from the set of the real numbers. You may think of it as,

irrational numbers = real numbers “minus” rational numbers

  • Irrational numbers if written in decimal forms don’t terminate and don’t repeat.

There’s really no standard symbol to represent the set of irrational numbers. But you may encounter the one below.

b) Euler’s number

c) The square root of 2

Here’s a quick diagram that can help you classify real numbers.

Practice Problems on How to Classify Real Numbers

Example 1 : Tell if the statement is true or false.  Every whole number is a natural number.

Solution: The set of whole numbers includes all natural or counting numbers and the number zero (0). Since zero is a whole number that is NOT a natural number, therefore the statement is FALSE.

Example 2 : Tell if the statement is true or false.  All integers are whole numbers.

Solution: The number -1 is an integer that is NOT a whole number. This makes the statement FALSE.

Example 3 : Tell if the statement is true or false. The number zero (0) is a rational number.

Solution: The number zero can be written as a ratio of two integers, thus it is indeed a rational number. This statement is TRUE.

Example 4 : Name the set or sets of numbers to which each real number belongs.

1) [latex]7[/latex]

It belongs to the sets of natural numbers, {1, 2, 3, 4, 5, …}. It is a whole number because the set of whole numbers includes the natural numbers plus zero. It is an integer since it is both a natural and a whole number. Finally, since 7 can be written as a fraction with a denominator of 1, 7/1, then it is also a rational number.

2) [latex]0[/latex]

This is not a natural number because it cannot be found in the set {1, 2, 3, 4, 5, …}. This is definitely a whole number, an integer, and a rational number. It is rational since 0 can be expressed as fractions such as 0/3, 0/16, and 0/45.

3) [latex]0.3\overline {18}[/latex]

This number obviously doesn’t belong to the set of natural numbers, set of whole numbers, and set of integers. Observe that 18 is repeating, and so this is a rational number. In fact, we can write it as a ratio of two integers.

4) [latex]\sqrt 5 [/latex]

This is not a rational number because it is not possible to write it as a fraction. If we evaluate it, the square root of 5 will have a decimal value that is non-terminating and non-repeating. This makes it an irrational number.

Real Numbers

Introduction.

Real numbers are the set of all rational and irrational numbers . They can be represented on the number line and include positive and negative numbers , as well as zero.

Types of Real Numbers

Real numbers can be categorized into different types:

  • Natural Numbers (N): These are the counting numbers (1, 2, 3, ...).
  • Whole Numbers (W): These are the natural numbers along with zero (0, 1, 2, 3, ...).
  • Integers (Z): These include all the whole numbers and their negatives, along with zero (... -3, -2, -1, 0, 1, 2, 3, ...).
  • Rational Numbers (Q): These are numbers that can be expressed as a ratio of two integers , where the denominator is not zero (e.g., 1/2, -3/4, 5).
  • Irrational Numbers : These are numbers that cannot be expressed as a ratio of two integers and have non-repeating, non-terminating decimal expansions (e.g., √2, π).

Operations on Real Numbers

Real numbers can be operated on using the following operations :

  • Addition (+)
  • Subtraction (-)
  • Multiplication (x or *)
  • Division (÷ or /)

Properties of Real Numbers

Real numbers follow certain properties under the basic operations :

  • Commutative Property : a + b = b + a; a x b = b x a
  • Associative Property : (a + b) + c = a + (b + c); (a x b) x c = a x (b x c)
  • Distributive Property : a x (b + c) = a x b + a x c
  • Identity Property : a + 0 = a; a x 1 = a
  • Inverse Property : a + (-a) = 0; a x (1/a) = 1 (for a ≠ 0)

Study Guide

When studying real numbers , it's important to understand the different types of real numbers and their properties. Practice representing real numbers on a number line and performing operations with them. Make sure to review the properties of real numbers and how they apply to addition , subtraction , multiplication , and division . Additionally, familiarize yourself with rational and irrational numbers , and how they differ from each other.

It can also be helpful to practice solving problems involving real numbers , including simplifying expressions and solving equations . Work on identifying patterns and relationships between different types of real numbers , and how they interact with each other in mathematical operations .

Lastly, don't forget to review the properties of real numbers and how they can be applied to solve problems and simplify mathematical expressions .

Real numbers form the basis of much of mathematics, and having a strong understanding of their properties and operations is crucial for success in various mathematical topics. By mastering the concepts and properties of real numbers , you'll be better equipped to tackle more advanced mathematical concepts and problem-solving tasks.

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1.1 Real Numbers: Algebra Essentials

Learning objectives.

In this section, you will:

  • Classify a real number as a natural, whole, integer, rational, or irrational number.
  • Perform calculations using order of operations.
  • Use the following properties of real numbers: commutative, associative, distributive, inverse, and identity.
  • Evaluate algebraic expressions.
  • Simplify algebraic expressions.

It is often said that mathematics is the language of science. If this is true, then an essential part of the language of mathematics is numbers. The earliest use of numbers occurred 100 centuries ago in the Middle East to count, or enumerate items. Farmers, cattle herders, and traders used tokens, stones, or markers to signify a single quantity—a sheaf of grain, a head of livestock, or a fixed length of cloth, for example. Doing so made commerce possible, leading to improved communications and the spread of civilization.

Three to four thousand years ago, Egyptians introduced fractions. They first used them to show reciprocals. Later, they used them to represent the amount when a quantity was divided into equal parts.

But what if there were no cattle to trade or an entire crop of grain was lost in a flood? How could someone indicate the existence of nothing? From earliest times, people had thought of a “base state” while counting and used various symbols to represent this null condition. However, it was not until about the fifth century CE in India that zero was added to the number system and used as a numeral in calculations.

Clearly, there was also a need for numbers to represent loss or debt. In India, in the seventh century CE, negative numbers were used as solutions to mathematical equations and commercial debts. The opposites of the counting numbers expanded the number system even further.

Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. In this section, we will explore sets of numbers, calculations with different kinds of numbers, and the use of numbers in expressions.

Classifying a Real Number

The numbers we use for counting, or enumerating items, are the natural numbers : 1, 2, 3, 4, 5, and so on. We describe them in set notation as { 1 , 2 , 3 , ... } { 1 , 2 , 3 , ... } where the ellipsis (…) indicates that the numbers continue to infinity. The natural numbers are, of course, also called the counting numbers . Any time we enumerate the members of a team, count the coins in a collection, or tally the trees in a grove, we are using the set of natural numbers. The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the opposites of the natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . It is useful to note that the set of integers is made up of three distinct subsets: negative integers, zero, and positive integers. In this sense, the positive integers are just the natural numbers. Another way to think about it is that the natural numbers are a subset of the integers.

The set of rational numbers is written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } . Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed as a terminating or repeating decimal. Any rational number can be represented as either:

  • ⓐ a terminating decimal: 15 8 = 1.875 , 15 8 = 1.875 , or
  • ⓑ a repeating decimal: 4 11 = 0.36363636 … = 0. 36 ¯ 4 11 = 0.36363636 … = 0. 36 ¯

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Writing Integers as Rational Numbers

Write each of the following as a rational number.

Write a fraction with the integer in the numerator and 1 in the denominator.

  • ⓐ 7 = 7 1 7 = 7 1
  • ⓑ 0 = 0 1 0 = 0 1
  • ⓒ −8 = − 8 1 −8 = − 8 1

Identifying Rational Numbers

Write each of the following rational numbers as either a terminating or repeating decimal.

  • ⓐ − 5 7 − 5 7
  • ⓑ 15 5 15 5
  • ⓒ 13 25 13 25

Write each fraction as a decimal by dividing the numerator by the denominator.

  • ⓐ − 5 7 = −0. 714285 ——— , − 5 7 = −0. 714285 ——— , a repeating decimal
  • ⓑ 15 5 = 3 15 5 = 3 (or 3.0), a terminating decimal
  • ⓒ 13 25 = 0.52 , 13 25 = 0.52 , a terminating decimal
  • ⓐ 68 17 68 17
  • ⓑ 8 13 8 13
  • ⓒ − 17 20 − 17 20

Irrational Numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even 3 2 , 3 2 , but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers . Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.

Differentiating Rational and Irrational Numbers

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

  • ⓑ 33 9 33 9
  • ⓓ 17 34 17 34
  • ⓔ 0.3033033303333 … 0.3033033303333 …
  • ⓐ 25 : 25 : This can be simplified as 25 = 5. 25 = 5. Therefore, 25 25 is rational.

So, 33 9 33 9 is rational and a repeating decimal.

  • ⓒ 11 : 11 11 : 11 is irrational because 11 is not a perfect square and 11 11 cannot be expressed as a fraction.

So, 17 34 17 34 is rational and a terminating decimal.

  • ⓔ 0.3033033303333 … 0.3033033303333 … is not a terminating decimal. Also note that there is no repeating pattern because the group of 3s increases each time. Therefore it is neither a terminating nor a repeating decimal and, hence, not a rational number. It is an irrational number.
  • ⓐ 7 77 7 77
  • ⓒ 4.27027002700027 … 4.27027002700027 …
  • ⓓ 91 13 91 13

Real Numbers

Given any number n , we know that n is either rational or irrational. It cannot be both. The sets of rational and irrational numbers together make up the set of real numbers . As we saw with integers, the real numbers can be divided into three subsets: negative real numbers, zero, and positive real numbers. Each subset includes fractions, decimals, and irrational numbers according to their algebraic sign (+ or –). Zero is considered neither positive nor negative.

The real numbers can be visualized on a horizontal number line with an arbitrary point chosen as 0, with negative numbers to the left of 0 and positive numbers to the right of 0. A fixed unit distance is then used to mark off each integer (or other basic value) on either side of 0. Any real number corresponds to a unique position on the number line.The converse is also true: Each location on the number line corresponds to exactly one real number. This is known as a one-to-one correspondence. We refer to this as the real number line as shown in Figure 1 .

Classifying Real Numbers

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?

  • ⓐ − 10 3 − 10 3
  • ⓒ − 289 − 289
  • ⓓ −6 π −6 π
  • ⓔ 0.615384615384 … 0.615384615384 …
  • ⓐ − 10 3 − 10 3 is negative and rational. It lies to the left of 0 on the number line.
  • ⓑ 5 5 is positive and irrational. It lies to the right of 0.
  • ⓒ − 289 = − 17 2 = −17 − 289 = − 17 2 = −17 is negative and rational. It lies to the left of 0.
  • ⓓ −6 π −6 π is negative and irrational. It lies to the left of 0.
  • ⓔ 0.615384615384 … 0.615384615384 … is a repeating decimal so it is rational and positive. It lies to the right of 0.
  • ⓑ −11.411411411 … −11.411411411 …
  • ⓒ 47 19 47 19
  • ⓓ − 5 2 − 5 2
  • ⓔ 6.210735 6.210735

Sets of Numbers as Subsets

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, such as Figure 2 .

Sets of Numbers

The set of natural numbers includes the numbers used for counting: { 1 , 2 , 3 , ... } . { 1 , 2 , 3 , ... } .

The set of whole numbers is the set of natural numbers plus zero: { 0 , 1 , 2 , 3 , ... } . { 0 , 1 , 2 , 3 , ... } .

The set of integers adds the negative natural numbers to the set of whole numbers: { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } . { ... , −3 , −2 , −1 , 0 , 1 , 2 , 3 , ... } .

The set of rational numbers includes fractions written as { m n | m and  n are integers and  n ≠ 0 } . { m n | m and  n are integers and  n ≠ 0 } .

The set of irrational numbers is the set of numbers that are not rational, are nonrepeating, and are nonterminating: { h | h is not a rational number } . { h | h is not a rational number } .

Differentiating the Sets of Numbers

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

  • ⓔ 3.2121121112 … 3.2121121112 …
a. X X X X
b. X
c. X
d. –6 X X
e. 3.2121121112... X
  • ⓐ − 35 7 − 35 7
  • ⓔ 4.763763763 … 4.763763763 …

Performing Calculations Using the Order of Operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example, 4 2 = 4 ⋅ 4 = 16. 4 2 = 4 ⋅ 4 = 16. We can raise any number to any power. In general, the exponential notation a n a n means that the number or variable a a is used as a factor n n times.

In this notation, a n a n is read as the n th power of a , a , or a a to the n n where a a is called the base and n n is called the exponent . A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, 24 + 6 ⋅ 2 3 − 4 2 24 + 6 ⋅ 2 3 − 4 2 is a mathematical expression.

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.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. 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.

Let’s take a look at the expression provided.

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify 4 2 4 2 as 16.

Next, perform multiplication or division, left to right.

Lastly, perform addition or subtraction, left to right.

Therefore, 24 + 6 ⋅ 2 3 − 4 2 = 12. 24 + 6 ⋅ 2 3 − 4 2 = 12.

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.

Order of Operations

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :

P (arentheses) E (xponents) M (ultiplication) and D (ivision) A (ddition) and S (ubtraction)

Given a mathematical expression, simplify it using the order of operations.

  • Step 1. Simplify any expressions within grouping symbols.
  • Step 2. Simplify any expressions containing exponents or radicals.
  • Step 3. Perform any multiplication and division in order, from left to right.
  • Step 4. Perform any addition and subtraction in order, from left to right.

Using the Order of Operations

Use the order of operations to evaluate each of the following expressions.

  • ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 )
  • ⓑ 5 2 − 4 7 − 11 − 2 5 2 − 4 7 − 11 − 2
  • ⓒ 6 − | 5 − 8 | + 3 ( 4 − 1 ) 6 − | 5 − 8 | + 3 ( 4 − 1 )
  • ⓓ 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2 14 − 3 ⋅ 2 2 ⋅ 5 − 3 2
  • ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1
  • ⓐ ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction. ( 3 ⋅ 2 ) 2 − 4 ( 6 + 2 ) = ( 6 ) 2 − 4 ( 8 ) Simplify parentheses. = 36 − 4 ( 8 ) Simplify exponent. = 36 − 32 Simplify multiplication. = 4 Simplify subtraction.

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.

  • ⓒ 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition. 6 − | 5 − 8 | + 3 | 4 − 1 | = 6 − | −3 | + 3 ( 3 ) Simplify inside grouping symbols. = 6 - ( 3 ) + 3 ( 3 ) Simplify absolute value. = 6 - 3 + 9 Simplify multiplication. = 12 Simplify addition.

In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

  • ⓔ 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add. 7 ( 5 ⋅ 3 ) − 2 [ ( 6 − 3 ) − 4 2 ] + 1 = 7 ( 15 ) − 2 [ ( 3 ) − 4 2 ] + 1 Simplify inside parentheses. = 7 ( 15 ) − 2 ( 3 − 16 ) + 1 Simplify exponent. = 7 ( 15 ) − 2 ( −13 ) + 1 Subtract. = 105 + 26 + 1 Multiply. = 132 Add.
  • ⓐ 5 2 − 4 2 + 7 ( 5 − 4 ) 2 5 2 − 4 2 + 7 ( 5 − 4 ) 2
  • ⓑ 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6 1 + 7 ⋅ 5 − 8 ⋅ 4 9 − 6
  • ⓒ | 1.8 − 4.3 | + 0.4 15 + 10 | 1.8 − 4.3 | + 0.4 15 + 10
  • ⓓ 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2 1 2 [ 5 ⋅ 3 2 − 7 2 ] + 1 3 ⋅ 9 2
  • ⓔ [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 ) [ ( 3 − 8 ) 2 − 4 ] − ( 3 − 8 )

Using Properties of Real Numbers

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.

Commutative Properties

The commutative property of addition states that numbers may be added in any order without affecting the sum.

We can better see this relationship when using real numbers.

Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product.

Again, consider an example with real numbers.

It is important to note that neither subtraction nor division is commutative. For example, 17 − 5 17 − 5 is not the same as 5 − 17. 5 − 17. Similarly, 20 ÷ 5 ≠ 5 ÷ 20. 20 ÷ 5 ≠ 5 ÷ 20.

Associative Properties

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.

Consider this example.

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.

Are subtraction and division associative? Review these examples.

As we can see, neither subtraction nor division is associative.

Distributive Property

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.

A special case of the distributive property occurs when a sum of terms is subtracted.

For example, consider the difference 12 − ( 5 + 3 ) . 12 − ( 5 + 3 ) . We can rewrite the difference of the two terms 12 and ( 5 + 3 ) ( 5 + 3 ) by turning the subtraction expression into addition of the opposite. So instead of subtracting ( 5 + 3 ) , ( 5 + 3 ) , we add the opposite.

Now, distribute −1 −1 and simplify the result.

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.

Identity Properties

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.

For example, we have ( −6 ) + 0 = −6 ( −6 ) + 0 = −6 and 23 ⋅ 1 = 23. 23 ⋅ 1 = 23. There are no exceptions for these properties; they work for every real number, including 0 and 1.

Inverse Properties

The inverse property of addition states that, for every real number a , there is a unique number, called the additive inverse (or opposite), denoted by (− a ), that, when added to the original number, results in the additive identity, 0.

For example, if a = −8 , a = −8 , the additive inverse is 8, since ( −8 ) + 8 = 0. ( −8 ) + 8 = 0.

The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a , there is a unique number, called the multiplicative inverse (or reciprocal), denoted 1 a , 1 a , that, when multiplied by the original number, results in the multiplicative identity, 1.

For example, if a = − 2 3 , a = − 2 3 , the reciprocal, denoted 1 a , 1 a , is − 3 2 − 3 2 because

Properties of Real Numbers

The following properties hold for real numbers a , b , and c .

Addition Multiplication
There exists a unique real number called the additive identity, 0, such that, for any real number There exists a unique real number called the multiplicative identity, 1, such that, for any real number
Every real number a has an additive inverse, or opposite, denoted , such that Every nonzero real number has a multiplicative inverse, or reciprocal, denoted such that

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

  • ⓐ 3 ⋅ 6 + 3 ⋅ 4 3 ⋅ 6 + 3 ⋅ 4
  • ⓑ ( 5 + 8 ) + ( −8 ) ( 5 + 8 ) + ( −8 )
  • ⓒ 6 − ( 15 + 9 ) 6 − ( 15 + 9 )
  • ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) 4 7 ⋅ ( 2 3 ⋅ 7 4 )
  • ⓔ 100 ⋅ [ 0.75 + ( −2.38 ) ] 100 ⋅ [ 0.75 + ( −2.38 ) ]
  • ⓐ 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify. 3 ⋅ 6 + 3 ⋅ 4 = 3 ⋅ ( 6 + 4 ) Distributive property. = 3 ⋅ 10 Simplify. = 30 Simplify.
  • ⓑ ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition. ( 5 + 8 ) + ( −8 ) = 5 + [ 8 + ( −8 ) ] Associative property of addition. = 5 + 0 Inverse property of addition. = 5 Identity property of addition.
  • ⓒ 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify. 6 − ( 15 + 9 ) = 6 + [ ( −15 ) + ( −9 ) ] Distributive property. = 6 + ( −24 ) Simplify. = −18 Simplify.
  • ⓓ 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication. 4 7 ⋅ ( 2 3 ⋅ 7 4 ) = 4 7 ⋅ ( 7 4 ⋅ 2 3 ) Commutative property of multiplication. = ( 4 7 ⋅ 7 4 ) ⋅ 2 3 Associative property of multiplication. = 1 ⋅ 2 3 Inverse property of multiplication. = 2 3 Identity property of multiplication.
  • ⓔ 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify. 100 ⋅ [ 0.75 + ( − 2.38 ) ] = 100 ⋅ 0.75 + 100 ⋅ ( −2.38 ) Distributive property. = 75 + ( −238 ) Simplify. = −163 Simplify.
  • ⓐ ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ] ( − 23 5 ) ⋅ [ 11 ⋅ ( − 5 23 ) ]
  • ⓑ 5 ⋅ ( 6.2 + 0.4 ) 5 ⋅ ( 6.2 + 0.4 )
  • ⓒ 18 − ( 7 −15 ) 18 − ( 7 −15 )
  • ⓓ 17 18 + [ 4 9 + ( − 17 18 ) ] 17 18 + [ 4 9 + ( − 17 18 ) ]
  • ⓔ 6 ⋅ ( −3 ) + 6 ⋅ 3 6 ⋅ ( −3 ) + 6 ⋅ 3

Evaluating Algebraic Expressions

So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see expressions such as x + 5 , 4 3 π r 3 , x + 5 , 4 3 π r 3 , or 2 m 3 n 2 . 2 m 3 n 2 . In the expression x + 5 , x + 5 , 5 is called a constant because it does not vary and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division.

We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way.

In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables.

Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before.

Describing Algebraic Expressions

List the constants and variables for each algebraic expression.

  • ⓑ 4 3 π r 3 4 3 π r 3
  • ⓒ 2 m 3 n 2 2 m 3 n 2
Constants Variables
a. + 5 5
b.
c. 2
  • ⓐ 2 π r ( r + h ) 2 π r ( r + h )
  • ⓑ 2( L + W )
  • ⓒ 4 y 3 + y 4 y 3 + y

Evaluating an Algebraic Expression at Different Values

Evaluate the expression 2 x − 7 2 x − 7 for each value for x.

  • ⓐ x = 0 x = 0
  • ⓑ x = 1 x = 1
  • ⓒ x = 1 2 x = 1 2
  • ⓓ x = −4 x = −4
  • ⓐ Substitute 0 for x . x . 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7 2 x − 7 = 2 ( 0 ) − 7 = 0 − 7 = −7
  • ⓑ Substitute 1 for x . x . 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5 2 x − 7 = 2 ( 1 ) − 7 = 2 − 7 = −5
  • ⓒ Substitute 1 2 1 2 for x . x . 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6 2 x − 7 = 2 ( 1 2 ) − 7 = 1 − 7 = −6
  • ⓓ Substitute −4 −4 for x . x . 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15 2 x − 7 = 2 ( − 4 ) − 7 = − 8 − 7 = −15

Evaluate the expression 11 − 3 y 11 − 3 y for each value for y.

  • ⓐ y = 2 y = 2
  • ⓑ y = 0 y = 0
  • ⓒ y = 2 3 y = 2 3
  • ⓓ y = −5 y = −5

Evaluate each expression for the given values.

  • ⓐ x + 5 x + 5 for x = −5 x = −5
  • ⓑ t 2 t −1 t 2 t −1 for t = 10 t = 10
  • ⓒ 4 3 π r 3 4 3 π r 3 for r = 5 r = 5
  • ⓓ a + a b + b a + a b + b for a = 11 , b = −8 a = 11 , b = −8
  • ⓔ 2 m 3 n 2 2 m 3 n 2 for m = 2 , n = 3 m = 2 , n = 3
  • ⓐ Substitute −5 −5 for x . x . x + 5 = ( −5 ) + 5 = 0 x + 5 = ( −5 ) + 5 = 0
  • ⓑ Substitute 10 for t . t . t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19 t 2 t − 1 = ( 10 ) 2 ( 10 ) − 1 = 10 20 − 1 = 10 19
  • ⓒ Substitute 5 for r . r . 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π 4 3 π r 3 = 4 3 π ( 5 ) 3 = 4 3 π ( 125 ) = 500 3 π
  • ⓓ Substitute 11 for a a and –8 for b . b . a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85 a + a b + b = ( 11 ) + ( 11 ) ( −8 ) + ( −8 ) = 11 − 88 − 8 = −85
  • ⓔ Substitute 2 for m m and 3 for n . n . 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12 2 m 3 n 2 = 2 ( 2 ) 3 ( 3 ) 2 = 2 ( 8 ) ( 9 ) = 144 = 12
  • ⓐ y + 3 y − 3 y + 3 y − 3 for y = 5 y = 5
  • ⓑ 7 − 2 t 7 − 2 t for t = −2 t = −2
  • ⓒ 1 3 π r 2 1 3 π r 2 for r = 11 r = 11
  • ⓓ ( p 2 q ) 3 ( p 2 q ) 3 for p = −2 , q = 3 p = −2 , q = 3
  • ⓔ 4 ( m − n ) − 5 ( n − m ) 4 ( m − n ) − 5 ( n − m ) for m = 2 3 , n = 1 3 m = 2 3 , n = 1 3

An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation 2 x + 1 = 7 2 x + 1 = 7 has the solution of 3 because when we substitute 3 for x x in the equation, we obtain the true statement 2 ( 3 ) + 1 = 7. 2 ( 3 ) + 1 = 7.

A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area A A of a circle in terms of the radius r r of the circle: A = π r 2 . A = π r 2 . For any value of r , r , the area A A can be found by evaluating the expression π r 2 . π r 2 .

Using a Formula

A right circular cylinder with radius r r and height h h has the surface area S S (in square units) given by the formula S = 2 π r ( r + h ) . S = 2 π r ( r + h ) . See Figure 3 . Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of π . π .

Evaluate the expression 2 π r ( r + h ) 2 π r ( r + h ) for r = 6 r = 6 and h = 9. h = 9.

The surface area is 180 π 180 π square inches.

A photograph with length L and width W is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm 2 ) is found to be A = ( L + 16 ) ( W + 16 ) − L ⋅ W . A = ( L + 16 ) ( W + 16 ) − L ⋅ W . See Figure 4 . Find the area of a mat for a photograph with length 32 cm and width 24 cm.

Simplifying Algebraic Expressions

Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions.

Simplify each algebraic expression.

  • ⓐ 3 x − 2 y + x − 3 y − 7 3 x − 2 y + x − 3 y − 7
  • ⓑ 2 r − 5 ( 3 − r ) + 4 2 r − 5 ( 3 − r ) + 4
  • ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) ( 4 t − 5 4 s ) − ( 2 3 t + 2 s )
  • ⓓ 2 m n − 5 m + 3 m n + n 2 m n − 5 m + 3 m n + n
  • ⓐ 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify. 3 x − 2 y + x − 3 y − 7 = 3 x + x − 2 y − 3 y − 7 Commutative property of addition. = 4 x − 5 y − 7 Simplify.
  • ⓑ 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify. 2 r − 5 ( 3 − r ) + 4 = 2 r − 15 + 5 r + 4 Distributive property. = 2 r + 5 r − 15 + 4 Commutative property of addition. = 7 r − 11 Simplify.
  • ⓒ ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify. ( 4 t − 5 4 s ) − ( 2 3 t + 2 s ) = 4 t − 5 4 s − 2 3 t − 2 s Distributive property. = 4 t − 2 3 t − 5 4 s − 2 s Commutative property of addition. = 10 3 t − 13 4 s Simplify.
  • ⓓ 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify. 2 m n − 5 m + 3 m n + n = 2 m n + 3 m n − 5 m + n Commutative property of addition. = 5 m n − 5 m + n Simplify.
  • ⓐ 2 3 y − 2 ( 4 3 y + z ) 2 3 y − 2 ( 4 3 y + z )
  • ⓑ 5 t − 2 − 3 t + 1 5 t − 2 − 3 t + 1
  • ⓒ 4 p ( q − 1 ) + q ( 1 − p ) 4 p ( q − 1 ) + q ( 1 − p )
  • ⓓ 9 r − ( s + 2 r ) + ( 6 − s ) 9 r − ( s + 2 r ) + ( 6 − s )

Simplifying a Formula

A rectangle with length L L and width W W has a perimeter P P given by P = L + W + L + W . P = L + W + L + W . Simplify this expression.

If the amount P P is deposited into an account paying simple interest r r for time t , t , the total value of the deposit A A is given by A = P + P r t . A = P + P r t . Simplify the expression. (This formula will be explored in more detail later in the course.)

Access these online resources for additional instruction and practice with real numbers.

  • Simplify an Expression.
  • Evaluate an Expression 1.
  • Evaluate an Expression 2.

1.1 Section Exercises

Is 2 2 an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?

What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

For the following exercises, simplify the given expression.

10 + 2 × ( 5 − 3 ) 10 + 2 × ( 5 − 3 )

6 ÷ 2 − ( 81 ÷ 3 2 ) 6 ÷ 2 − ( 81 ÷ 3 2 )

18 + ( 6 − 8 ) 3 18 + ( 6 − 8 ) 3

−2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2 −2 × [ 16 ÷ ( 8 − 4 ) 2 ] 2

4 − 6 + 2 × 7 4 − 6 + 2 × 7

3 ( 5 − 8 ) 3 ( 5 − 8 )

4 + 6 − 10 ÷ 2 4 + 6 − 10 ÷ 2

12 ÷ ( 36 ÷ 9 ) + 6 12 ÷ ( 36 ÷ 9 ) + 6

( 4 + 5 ) 2 ÷ 3 ( 4 + 5 ) 2 ÷ 3

3 − 12 × 2 + 19 3 − 12 × 2 + 19

2 + 8 × 7 ÷ 4 2 + 8 × 7 ÷ 4

5 + ( 6 + 4 ) − 11 5 + ( 6 + 4 ) − 11

9 − 18 ÷ 3 2 9 − 18 ÷ 3 2

14 × 3 ÷ 7 − 6 14 × 3 ÷ 7 − 6

9 − ( 3 + 11 ) × 2 9 − ( 3 + 11 ) × 2

6 + 2 × 2 − 1 6 + 2 × 2 − 1

64 ÷ ( 8 + 4 × 2 ) 64 ÷ ( 8 + 4 × 2 )

9 + 4 ( 2 2 ) 9 + 4 ( 2 2 )

( 12 ÷ 3 × 3 ) 2 ( 12 ÷ 3 × 3 ) 2

25 ÷ 5 2 − 7 25 ÷ 5 2 − 7

( 15 − 7 ) × ( 3 − 7 ) ( 15 − 7 ) × ( 3 − 7 )

2 × 4 − 9 ( −1 ) 2 × 4 − 9 ( −1 )

4 2 − 25 × 1 5 4 2 − 25 × 1 5

12 ( 3 − 1 ) ÷ 6 12 ( 3 − 1 ) ÷ 6

For the following exercises, evaluate the expression using the given value of the variable.

8 ( x + 3 ) – 64 8 ( x + 3 ) – 64 for x = 2 x = 2

4 y + 8 – 2 y 4 y + 8 – 2 y for y = 3 y = 3

( 11 a + 3 ) − 18 a + 4 ( 11 a + 3 ) − 18 a + 4 for a = –2 a = –2

4 z − 2 z ( 1 + 4 ) – 36 4 z − 2 z ( 1 + 4 ) – 36 for z = 5 z = 5

4 y ( 7 − 2 ) 2 + 200 4 y ( 7 − 2 ) 2 + 200 for y = –2 y = –2

− ( 2 x ) 2 + 1 + 3 − ( 2 x ) 2 + 1 + 3 for x = 2 x = 2

For the 8 ( 2 + 4 ) − 15 b + b 8 ( 2 + 4 ) − 15 b + b for b = –3 b = –3

2 ( 11 c − 4 ) – 36 2 ( 11 c − 4 ) – 36 for c = 0 c = 0

4 ( 3 − 1 ) x – 4 4 ( 3 − 1 ) x – 4 for x = 10 x = 10

1 4 ( 8 w − 4 2 ) 1 4 ( 8 w − 4 2 ) for w = 1 w = 1

For the following exercises, simplify the expression.

4 x + x ( 13 − 7 ) 4 x + x ( 13 − 7 )

2 y − ( 4 ) 2 y − 11 2 y − ( 4 ) 2 y − 11

a 2 3 ( 64 ) − 12 a ÷ 6 a 2 3 ( 64 ) − 12 a ÷ 6

8 b − 4 b ( 3 ) + 1 8 b − 4 b ( 3 ) + 1

5 l ÷ 3 l × ( 9 − 6 ) 5 l ÷ 3 l × ( 9 − 6 )

7 z − 3 + z × 6 2 7 z − 3 + z × 6 2

4 × 3 + 18 x ÷ 9 − 12 4 × 3 + 18 x ÷ 9 − 12

9 ( y + 8 ) − 27 9 ( y + 8 ) − 27

( 9 6 t − 4 ) 2 ( 9 6 t − 4 ) 2

6 + 12 b − 3 × 6 b 6 + 12 b − 3 × 6 b

18 y − 2 ( 1 + 7 y ) 18 y − 2 ( 1 + 7 y )

( 4 9 ) 2 × 27 x ( 4 9 ) 2 × 27 x

8 ( 3 − m ) + 1 ( − 8 ) 8 ( 3 − m ) + 1 ( − 8 )

9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x ) 9 x + 4 x ( 2 + 3 ) − 4 ( 2 x + 3 x )

5 2 − 4 ( 3 x ) 5 2 − 4 ( 3 x )

Real-World Applications

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.

Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

How much money does Fred keep?

For the following exercises, solve the given problem.

According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by π . π . Is the circumference of a quarter a whole number, a rational number, or an irrational number?

Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact?

For the following exercises, consider this scenario: There is a mound of g g pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

Write the equation that describes the situation.

Solve for g .

For the following exercise, solve the given problem.

Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, then the department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. They must spend the budget such that 2,500,000 − x = 0. 2,500,000 − x = 0. What property of addition tells us what the value of x must be?

For the following exercises, use a graphing calculator to solve for x . Round the answers to the nearest hundredth.

0.5 ( 12.3 ) 2 − 48 x = 3 5 0.5 ( 12.3 ) 2 − 48 x = 3 5

( 0.25 − 0.75 ) 2 x − 7.2 = 9.9 ( 0.25 − 0.75 ) 2 x − 7.2 = 9.9

If a whole number is not a natural number, what must the number be?

Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

Determine whether the statement is true or false: The product of a rational and irrational number is always irrational.

Determine whether the simplified expression is rational or irrational: −18 − 4 ( 5 ) ( −1 ) . −18 − 4 ( 5 ) ( −1 ) .

Determine whether the simplified expression is rational or irrational: −16 + 4 ( 5 ) + 5 . −16 + 4 ( 5 ) + 5 .

The division of two natural numbers will always result in what type of number?

What property of real numbers would simplify the following expression: 4 + 7 ( x − 1 ) ? 4 + 7 ( x − 1 ) ?

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This Real Numbers Unit Bundle includes guided notes, homework assignments, three quizzes, a study guide, and a unit test that cover the following topics:

• Integers and Integer Operations

• Absolute Value

• Simplifying Fractions

• Converting Fractions, Decimals, and Percents

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• Exponents

• Zero Exponent and Negative Exponents

• Perfect Squares and Perfect Cubes

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• The Real Number System

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This 9-day CCSS-Aligned Number System Unit for 6th-grade includes introducing the number line, comparing and ordering rational numbers, and absolute value. | maneuveringthemiddle.com

Number System Unit 6th Grade CCSS

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This Number System Unit 6th Grade CCSS includes: introducing the number line, comparing and ordering rational numbers, and absolute value.

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  1. PDF Scanned with CamScanner

    Scanned with CamScanner. Answer Key Name Date Unit: Real Number System Homework 6 COMPARING & ORDERING REAL NUMBERS correct tn each blank. 10 In questions 1- 5.6% < q-ã7. 15. 2Tt < 8. The time that it took four students in Mrs. Alvarez's class to solve a rubiks cube is listed 7. Bennett planted four tomato plants in his garden, and he recorded ...

  2. PDF Unit 1 Real Number System Homework

    M8 Unit 1: Real Number System HOMEWORK Page 17 . M8 1-2: Scientific Notation HW . Answer the following questions on your lined paper. 1. A computer can perform 4.66 x 108 calculations per second. What is this number in standard form? 2. The size of the Indian Ocean is 2.7 x 1010 square miles. The Arctic Ocean is 4.5 X 105 square miles.

  3. PDF Grade 8

    One method to get an estimate is to divide 3 (the distance between 25 and 28) by 11 (the distance between the perfect squares of 25 and 36) to get 0.27. The estimate of √28 would be 5.27 (the actual is 5.29). Questions for 8.NS.2. 1. Use the algebra unit tiles to represent √15 as a mixed number. 2.

  4. PDF Intro to Algebra

    rational numbers and explain why the numbers are rational. Answers will vary. An example is 7.8 because this number is a terminating decimal. Another example ξis 9 because it is a perfect square. 3.173095 Give two examples of rationals below: irrational numbers and explain why the numbers are irrational. Answers will vary.

  5. PDF Sets of Numbers in the Real Number System

    23. Sets of Numbers in the Real Number System. Reals. A real number is either a rational number or an irrational number. 2 4, 7,0, , 11 3 −. Rationals. A rational number is any number that can be put in the form. p q. where p. and qare integers and 0q≠ . 12 5 1 8 3, , ,4 , 62713 −.

  6. The Real Number System

    Solution: The number -1 is an integer that is NOT a whole number. This makes the statement FALSE. Example 3: Tell if the statement is true or false. The number zero (0) is a rational number. Solution: The number zero can be written as a ratio of two integers, thus it is indeed a rational number. This statement is TRUE.

  7. PDF 8 Algebra CC Answer Key Unit 1 Review: The Real Number System

    um of. rational numbers is always an irration. The sum can be rational or irrational. . 2 2 irrational sum2 2 0 rational sum6. The pro. ct of. rrational numbers is always an irrational number. False The product can be. 1. irrational product 2 2 4 rational product7.

  8. PDF 4-7 The Real Number System

    The square root of 95 is between the integers 9 and 10. Since 95 is closer to 100 than to 81, LV closer to 10 than to 9. $16:(5 10 62/87,21 The first perfect square less than 54 is 49. The first perfect square greater than 54 is 64. The negative square root of 54 is between the integers ±7 and ±8.

  9. PDF 1.1 The Real Number System

    line there corresponds exactly one real number, and this number is called the coordinate of that point. If a real number x is less than a real number y , we write x < y . On the number line, x is to the left of y. Example 4: For each pair of real numbers, place one of the symbols < , =, or > in the blank. a) 2 ____ 2 b) -5 _____ - 6 c) 4

  10. PDF Algebra Unit 1: The Number System Name: Answer Key

    ant:A constant is a value that doesn't change. All real numbers are constants. 2 This includes rational numbers like 4, -. , 3.25, and as well as irrational numbers like 3. Difference: The result of a subtraction problem. The dif. rence between 4 and 8 is -4 because 4 - 8 = -4. The difference betwee.

  11. Real numbers. 8th Grade Math Worksheets, Study Guides and Answer key

    Real numbers. 8th Grade Math Worksheets and Answer key, Study Guides. Covers the following skills: Compare real numbers; locate real numbers on a number line. Identify the square root of a perfect square to 400 or, if it is not a perfect square root, locate it as an irrational number between two consecutive positive integers. Homework. U.S. National Standards.

  12. 1.1 Real Numbers: Algebra Essentials

    Answer Key. Chapter 1; Chapter 2; Chapter 3; Chapter 4; Chapter 5; Chapter 6; Chapter 7; Chapter 8; Chapter 9; ... Because of the evolution of the number system, we can now perform complex calculations using these and other categories of real numbers. ... The real numbers can be visualized on a horizontal number line with an arbitrary point ...

  13. Unit 1

    Unit 1 - The Real Number System. real numbers. Click the card to flip 👆. the set of rational and irrational numbers. Click the card to flip 👆. 1 / 16.

  14. 1.1: The Real Number System

    The real number system is by no means the only field. The {} (which are the real numbers that can be written as r = p / q, where p and q are integers and q ≠ 0) also form a field under addition and multiplication. The simplest possible field consists of two elements, which we denote by 0 and 1, with addition defined by 0 + 0 = 1 + 1 = 0, 1 ...

  15. 1.1: Real numbers and the Number Line

    A point on the real number line that is associated with a coordinate is called its graph. To construct a number line, draw a horizontal line with arrows on both ends to indicate that it continues without bound. Next, choose any point to represent the number zero; this point is called the origin. Figure 1.1.2.

  16. Algebra 1A- Unit 1-Post Test: The Real Number System

    Post Test: The Real Number System Learn with flashcards, games, and more — for free. ... algebra 2a - unit 2: key features and graphing quadratics lesson 5-8. 76 terms. Cyb3rkrw. Preview. Math Vocab. 24 terms. Dacoda_Fields. Preview. ... See an expert-written answer! We have an expert-written solution to this problem!

  17. Math Unit 1- The Real Number System Flashcards

    this property shows that any number multiplied by 1 is always equal to itself. inverse property. this property shows that each real number (except zero) there exists an opposite number and a reciprocal. distributive property. this property shows that multiplication can be distributed over addition and subtraction.

  18. PDF Grade 6 Answer Key

    The equation that Joyce can use to determine the number of miles, m, after h hours is 300 ÷ 4 = 75. 300 miles in 4 hours = 75 miles per hour. = 75h. The equation that Ashley can use to determine the number of miles, m, after h hours is 350 ÷ 5 = 70. 350 miles in 5 hours = 70 miles per hour. = 70h.

  19. The Real Numbers (Pre-Algebra Curriculum

    Description. This Real Numbers Unit Bundle includes guided notes, homework assignments, three quizzes, a study guide, and a unit test that cover the following topics: • Integers and Integer Operations. • Absolute Value. • Simplifying Fractions. • Converting Fractions, Decimals, and Percents.

  20. PDF Scanned with CamScanner

    If you square 6, the result is 36. This would be written as 62 = 36. Numbers that result from squaring an integer are called a perfect square. The numbers that are perfect squares and less than 200 are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196. The square root of a number is the number that, when multiplied by itself, is equal ...

  21. PDF Scanned with CamScanner

    Scanned with CamScanner. Unit: Real Number System Homework 3 Answer Key Name Date RATIONAL VS. IRRATIONAL NUMBERS Fill out the table below: EXPLANATION Can be written as a fraction Non-perfect square Non-terminating, non-repeating decimal Can be written as a fraction Repeating decimal; can be written as a fraction 7.

  22. Number System Unit 6th Grade CCSS

    Standards: 6.NS.5, 6.NS.7, 6.NS.7a, 6.NS.7b, 6.NS.7c, 6.NS.7d; Texas Teacher? Grab the TEKS-Aligned Numerical Representations Unit. Please don't purchase both as there is overlapping content. Learning Focus: compare and order integers on a number line; interpret statements of inequality; understand ordering and absolute value of rational numbers

  23. Real Number System answer key.docx

    View Real Number System answer key.docx from DHHD GHGFHGDHD at HCT Fujairah Womens College. Pre-Algebra Unit 2 Real Number System Name_ Homework Day 1: Real Number System Block_ Date_ Name all sets. AI Homework Help. ... Please refer to the attachment to answer this question. This question was created from MIS 107 Midterm 2- Set 1.docx.