real number system homework 2

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.

The Real Number System

The real number system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count, like how many sheep a farmer owns. These are called the natural numbers , or sometimes the counting numbers .

Natural Numbers

or “Counting Numbers”

1, 2, 3, 4, 5, . . .

• The use of three dots at the end of the list is a common mathematical notation to indicate that the list keeps going forever.

At some point, the idea of “zero” came to be considered as a number. If the farmer does not have any sheep, then the number of sheep that the farmer owns is zero. We call the set of natural numbers plus the number zero the whole numbers .

Whole Numbers

Natural Numbers together with “zero”

0, 1, 2, 3, 4, 5, . . .

Even more abstract than zero is the idea of negative numbers. If, in addition to not having any sheep, the farmer owes someone 3 sheep, you could say that the number of sheep that the farmer owns is negative 3. It took longer for the idea of negative numbers to be accepted, but eventually they came to be seen as something we could call “numbers.” The expanded set of numbers that we get by including negative versions of the counting numbers is called the integers .

Whole numbers plus negatives

. . . –4, –3, –2, –1, 0, 1, 2, 3, 4, . . .

The next generalization that we can make is to include the idea of fractions. While it is unlikely that a farmer owns a fractional number of sheep, many other things in real life are measured in fractions, like a half-cup of sugar. If we add fractions to the set of integers, we get the set of rational numbers .

Rational Numbers

Rational numbers include what we usually call fractions

• Notice that the word “rational” contains the word “ratio,” which should remind you of fractions.

The bottom of the fraction is called the denominator . Think of it as the denomination —it tells you what size fraction we are talking about: fourths, fifths, etc.

The top of the fraction is called the numerator . It tells you how many fourths, fifths, or whatever.

• RESTRICTION : The denominator cannot be zero! (But the numerator can)

If the numerator is zero, then the whole fraction is just equal to zero. If I have zero thirds or zero fourths, than I don’t have anything. However, it makes no sense at all to talk about a fraction measured in “zeroths.”

• Fractions can be numbers smaller than 1, like 1/2 or 3/4 (called proper fractions ) , or they can be numbers bigger than 1 (called improper fractions ) , like two-and-a-half, which we could also write as 5/2

All integers can also be thought of as rational numbers, with a denominator of 1:

This means that all the previous sets of numbers (natural numbers, whole numbers, and integers) are subsets of the rational numbers.

Now it might seem as though the set of rational numbers would cover every possible case, but that is not so. There are numbers that cannot be expressed as a fraction, and these numbers are called irrational because they are not rational.

Irrational Numbers

• Cannot be expressed as a ratio of integers.
• As decimals they never repeat or terminate (rationals always do one or the other)

Congratulations! You have just drawn a length that cannot be measured by any rational number. According to the Pythagorean Theorem, the length of this diagonal is the square root of 2; that is, the number which when multiplied by itself gives 2.

According to my calculator,

But my calculator only stops at eleven decimal places because it can hold no more. This number actually goes on forever past the decimal point, without the pattern ever terminating or repeating.

This is because if the pattern ever stopped or repeated, you could write the number as a fraction—and it can be proven that the square root of 2 can never be written as

for any choice of integers for a and b . The proof of this was considered quite shocking when it was first demonstrated by the followers of Pythagoras 26 centuries ago.

The Real Numbers

• Rationals + Irrationals
• All points on the number line
• Or all possible distances on the number line

When we put the irrational numbers together with the rational numbers, we finally have the complete set of real numbers. Any number that represents an amount of something, such as a weight, a volume, or the distance between two points, will always be a real number. The following diagram illustrates the relationships of the sets that make up the real numbers.

An Ordered Set

The real numbers have the property that they are ordered , which means that given any two different numbers we can always say that one is greater or less than the other. A more formal way of saying this is:

For any two real numbers a and b , one and only one of the following three statements is true:

1.      a is less than b , (expressed as a < b )

2.      a is equal to b , (expressed as a  =  b )

3.      a is greater than b , (expressed as a > b )

The Number Line

The ordered nature of the real numbers lets us arrange them along a line (imagine that the line is made up of an infinite number of points all packed so closely together that they form a solid line). The points are ordered so that points to the right are greater than points to the left:

• Every real number corresponds to a distance on the number line, starting at the center (zero).
• Negative numbers represent distances to the left of zero, and positive numbers are distances to the right.
• The arrows on the end indicate that it keeps going forever in both directions.

Absolute Value

When we want to talk about how “large” a number is without regard as to whether it is positive or negative, we use the absolute value function. The absolute value of a number is the distance from that number to the origin (zero) on the number line. That distance is always given as a non-negative number.

WARNING : If there is arithmetic to do inside the absolute value sign, you must do it before taking the absolute value—the absolute value function acts on the result of whatever is inside it. For example, a common error is

The correct result is

1.8 The Real Numbers

Learning objectives.

By the end of this section, you will be able to:

• Simplify expressions with square roots
• Identify integers, rational numbers, irrational numbers, and real numbers
• Locate fractions on the number line
• Locate decimals on the number line

Be Prepared 1.8

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapters, Decimals and Properties of Real Numbers .

Simplify Expressions with Square Roots

Remember that when a number n is multiplied by itself, we write n 2 n 2 and read it “n squared.” The result is called the square of n . For example,

Similarly, 121 is the square of 11, because 11 2 11 2 is 121.

Square of a Number

If n 2 = m , n 2 = m , then m is the square of n .

Manipulative Mathematics

Complete the following table to show the squares of the counting numbers 1 through 15.

The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.

The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.

Did you notice that these squares are the same as the squares of the positive numbers?

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 10 2 = 100 , 10 2 = 100 , we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is m m is called a square root of m .

Square Root of a Number

If n 2 = m , n 2 = m , then n is a square root of m .

Notice ( −10 ) 2 = 100 ( −10 ) 2 = 100 also, so −10 −10 is also a square root of 100. Therefore, both 10 and −10 −10 are square roots of 100.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign , m , m , denotes the positive square root. The positive square root is called the principal square root . When we use the radical sign that always means we want the principal square root.

We also use the radical sign for the square root of zero. Because 0 2 = 0 , 0 2 = 0 , 0 = 0 . 0 = 0 . Notice that zero has only one square root.

Square Root Notation

m m is read “the square root of m ”

If m = n 2 , m = n 2 , then m = n , m = n , for n ≥ 0 . n ≥ 0 .

The square root of m , m , m , is the positive number whose square is m .

Since 10 is the principal square root of 100, we write 100 = 10 . 100 = 10 . You may want to complete the following table to help you recognize square roots.

Example 1.108

Simplify: ⓐ 25 25 ⓑ 121 . 121 .

Try It 1.215

Simplify: ⓐ 36 36 ⓑ 169 . 169 .

Try It 1.216

Simplify: ⓐ 16 16 ⓑ 196 . 196 .

We know that every positive number has two square roots and the radical sign indicates the positive one. We write 100 = 10 . 100 = 10 . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, − 100 = −10 . − 100 = −10 . We read − 100 − 100 as “the opposite of the square root of 10.”

Example 1.109

Simplify: ⓐ − 9 − 9 ⓑ − 144 . − 144 .

Try It 1.217

Simplify: ⓐ − 4 − 4 ⓑ − 225 . − 225 .

Try It 1.218

Simplify: ⓐ − 81 − 81 ⓑ − 100 . − 100 .

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

We have already described numbers as counting number s , whole number s , and integers . What is the difference between these types of numbers?

What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.

Rational Number

A rational number is a number of the form p q , p q , where p and q are integers and q ≠ 0 . q ≠ 0 .

A rational number can be written as the ratio of two integers.

All signed fractions, such as 4 5 , − 7 8 , 13 4 , − 20 3 4 5 , − 7 8 , 13 4 , − 20 3 are rational numbers. Each numerator and each denominator is an integer.

Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to 3 1 , 6 2 , 9 3 , 12 4 , 15 5 … 3 1 , 6 2 , 9 3 , 12 4 , 15 5 …

An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.

Since any integer can be written as the ratio of two integers, all integers are rational numbers ! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.

What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers.

We’ve already seen that integers are rational numbers. The integer −8 −8 could be written as the decimal −8.0 . −8.0 . So, clearly, some decimals are rational.

Think about the decimal 7.3. Can we write it as a ratio of two integers? Because 7.3 means 7 3 10 , 7 3 10 , we can write it as an improper fraction, 73 10 . 73 10 . So 7.3 is the ratio of the integers 73 and 10. It is a rational number.

In general, any decimal that ends after a number of digits (such as 7.3 or −1.2684 ) −1.2684 ) is a rational number. We can use the reciprocal (or multiplicative inverse) of the place value of the last digit as the denominator when writing the decimal as a fraction.

Example 1.110

Write as the ratio of two integers: ⓐ −27 −27 ⓑ 7.31.

So we see that −27 −27 and 7.31 are both rational numbers, since they can be written as the ratio of two integers.

Try It 1.219

Write as the ratio of two integers: ⓐ −24 −24 ⓑ 3.57.

Try It 1.220

Write as the ratio of two integers: ⓐ −19 −19 ⓑ 8.41.

Let’s look at the decimal form of the numbers we know are rational.

We have seen that every integer is a rational number , since a = a 1 a = a 1 for any integer, a . We can also change any integer to a decimal by adding a decimal point and a zero.

We have also seen that every fraction is a rational number . Look at the decimal form of the fractions we considered above.

What do these examples tell us?

Every rational number can be written both as a ratio of integers , ( p q , ( p q , where p and q are integers and q ≠ 0 ) , q ≠ 0 ) , and as a decimal that either stops or repeats.

Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:

Its decimal form stops or repeats.

Are there any decimals that do not stop or repeat? Yes!

The number π π (the Greek letter pi , pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.

We can even create a decimal pattern that does not stop or repeat, such as

Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational.

Irrational Number

An irrational number is a number that cannot be written as the ratio of two integers.

Its decimal form does not stop and does not repeat.

Let’s summarize a method we can use to determine whether a number is rational or irrational.

Rational or Irrational?

If the decimal form of a number

• repeats or stops , the number is rational .
• does not repeat and does not stop , the number is irrational .

Example 1.111

Given the numbers 0.58 3 – , 0.47 , 3.605551275 . . . 0.58 3 – , 0.47 , 3.605551275 . . . list the ⓐ rational numbers ⓑ irrational numbers.

Try It 1.221

For the given numbers list the ⓐ rational numbers ⓑ irrational numbers: 0.29 , 0.81 6 – , 2.515115111 … . 0.29 , 0.81 6 – , 2.515115111 … .

Try It 1.222

For the given numbers list the ⓐ rational numbers ⓑ irrational numbers: 2.6 3 – , 0.125 , 0.418302 … 2.6 3 – , 0.125 , 0.418302 …

Example 1.112

For each number given, identify whether it is rational or irrational: ⓐ 36 36 ⓑ 44 . 44 .

• ⓐ Recognize that 36 is a perfect square, since 6 2 = 36 . 6 2 = 36 . So 36 = 6 , 36 = 6 , therefore 36 36 is rational.
• ⓑ Remember that 6 2 = 36 6 2 = 36 and 7 2 = 49 , 7 2 = 49 , so 44 is not a perfect square. Therefore, the decimal form of 44 44 will never repeat and never stop, so 44 44 is irrational.

Try It 1.223

For each number given, identify whether it is rational or irrational: ⓐ 81 81 ⓑ 17 . 17 .

Try It 1.224

For each number given, identify whether it is rational or irrational: ⓐ 116 116 ⓑ 121 . 121 .

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real number s .

Real Number

A real number is a number that is either rational or irrational.

All the numbers we use in elementary algebra are real numbers. Figure 1.15 illustrates how the number sets we’ve discussed in this section fit together.

Can we simplify −25 ? −25 ? Is there a number whose square is −25 ? −25 ?

None of the numbers that we have dealt with so far has a square that is −25 . −25 . Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to −25 . −25 .

The square root of a negative number is not a real number.

Example 1.113

For each number given, identify whether it is a real number or not a real number: ⓐ −169 −169 ⓑ − 64 . − 64 .

• ⓐ There is no real number whose square is −169 . −169 . Therefore, −169 −169 is not a real number.
• ⓑ Since the negative is in front of the radical, − 64 − 64 is −8 , −8 , Since −8 −8 is a real number, − 64 − 64 is a real number.

Try It 1.225

For each number given, identify whether it is a real number or not a real number: ⓐ −196 −196 ⓑ − 81 . − 81 .

Try It 1.226

For each number given, identify whether it is a real number or not a real number: ⓐ − 49 − 49 ⓑ −121 . −121 .

Example 1.114

Given the numbers −7 , 14 5 , 8 , 5 , 5.9 , − 64 , −7 , 14 5 , 8 , 5 , 5.9 , − 64 , list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers.

• ⓐ Remember, the whole numbers are 0, 1, 2, 3, … and 8 is the only whole number given.
• ⓑ The integers are the whole numbers, their opposites, and 0. So the whole number 8 is an integer, and −7 −7 is the opposite of a whole number so it is an integer, too. Also, notice that 64 is the square of 8 so − 64 = −8 . − 64 = −8 . So the integers are −7 , 8 , − 64 . −7 , 8 , − 64 .
• ⓒ Since all integers are rational, then −7 , 8 , − 64 −7 , 8 , − 64 are rational. Rational numbers also include fractions and decimals that repeat or stop, so 14 5 and 5.9 14 5 and 5.9 are rational. So the list of rational numbers is −7 , 14 5 , 8 , 5.9 , − 64 . −7 , 14 5 , 8 , 5.9 , − 64 .
• ⓓ Remember that 5 is not a perfect square, so 5 5 is irrational.
• ⓔ All the numbers listed are real numbers.

Try It 1.227

For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: −3 , − 2 , 0. 3 – , 9 5 , 4 , 49 . −3 , − 2 , 0. 3 – , 9 5 , 4 , 49 .

Try It 1.228

For the given numbers, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers: − 25 , − 3 8 , −1 , 6 , 121 , 2.041975 … − 25 , − 3 8 , −1 , 6 , 121 , 2.041975 …

Locate Fractions on the Number Line

The last time we looked at the number line , it only had positive and negative integers on it. We now want to include fraction s and decimals on it.

Let’s start with fractions and locate 1 5 , − 4 5 , 3 , 7 4 , − 9 2 , −5 , and 8 3 1 5 , − 4 5 , 3 , 7 4 , − 9 2 , −5 , and 8 3 on the number line.

We’ll start with the whole numbers 3 3 and −5 . −5 . because they are the easiest to plot. See Figure 1.16 .

The proper fractions listed are 1 5 and − 4 5 . 1 5 and − 4 5 . We know the proper fraction 1 5 1 5 has value less than one and so would be located between 0 and 1. 0 and 1. The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts 1 5 , 2 5 , 3 5 , 4 5 . 1 5 , 2 5 , 3 5 , 4 5 . We plot 1 5 . 1 5 . See Figure 1.16 .

Similarly, − 4 5 − 4 5 is between 0 and −1 . −1 . After dividing the unit into 5 equal parts we plot − 4 5 . − 4 5 . See Figure 1.16 .

Finally, look at the improper fractions 7 4 , − 9 2 , 8 3 . 7 4 , − 9 2 , 8 3 . These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See Figure 1.16 .

Figure 1.16 shows the number line with all the points plotted.

Example 1.115

Locate and label the following on a number line: 4 , 3 4 , − 1 4 , −3 , 6 5 , − 5 2 , and 7 3 . 4 , 3 4 , − 1 4 , −3 , 6 5 , − 5 2 , and 7 3 .

Locate and plot the integers, 4 , −3 . 4 , −3 .

Locate the proper fraction 3 4 3 4 first. The fraction 3 4 3 4 is between 0 and 1. Divide the distance between 0 and 1 into four equal parts then, we plot 3 4 . 3 4 . Similarly plot − 1 4 . − 1 4 .

Now locate the improper fractions 6 5 , − 5 2 , 7 3 . 6 5 , − 5 2 , 7 3 . It is easier to plot them if we convert them to mixed numbers and then plot them as described above: 6 5 = 1 1 5 , − 5 2 = −2 1 2 , 7 3 = 2 1 3 . 6 5 = 1 1 5 , − 5 2 = −2 1 2 , 7 3 = 2 1 3 .

Try It 1.229

Locate and label the following on a number line: −1 , 1 3 , 6 5 , − 7 4 , 9 2 , 5 , − 8 3 . −1 , 1 3 , 6 5 , − 7 4 , 9 2 , 5 , − 8 3 .

Try It 1.230

Locate and label the following on a number line: −2 , 2 3 , 7 5 , − 7 4 , 7 2 , 3 , − 7 3 . −2 , 2 3 , 7 5 , − 7 4 , 7 2 , 3 , − 7 3 .

In Example 1.116 , we’ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.

• a < b “ a is less than b ” when a is to the left of b on the number line
• a > b “ a is greater than b ” when a is to the right of b on the number line

As we move from left to right on a number line, the values increase.

Example 1.116

Order each of the following pairs of numbers, using < or >. It may be helpful to refer Figure 1.17 .

ⓐ − 2 3 ___ −1 − 2 3 ___ −1 ⓑ −3 1 2 ___ −3 −3 1 2 ___ −3 ⓒ − 3 4 ___ − 1 4 − 3 4 ___ − 1 4 ⓓ −2 ___ − 8 3 −2 ___ − 8 3

Try It 1.231

Order each of the following pairs of numbers, using < or >:

ⓐ − 1 3 ___ −1 − 1 3 ___ −1 ⓑ −1 1 2 ___ −2 −1 1 2 ___ −2 ⓒ − 2 3 ___ − 1 3 − 2 3 ___ − 1 3 ⓓ −3 ___ − 7 3 . −3 ___ − 7 3 .

Try It 1.232

ⓐ −1 ___ − 2 3 −1 ___ − 2 3 ⓑ −2 1 4 ___ −2 −2 1 4 ___ −2 ⓒ − 3 5 ___ − 4 5 − 3 5 ___ − 4 5 ⓓ −4 ___ − 10 3 . −4 ___ − 10 3 .

Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Example 1.117

Locate 0.4 on the number line.

A proper fraction has value less than one. The decimal number 0.4 is equivalent to 4 10 , 4 10 , a proper fraction, so 0.4 is located between 0 and 1. On a number line, divide the interval between 0 and 1 into 10 equal parts. Now label the parts 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0. We write 0 as 0.0 and 1 and 1.0, so that the numbers are consistently in tenths. Finally, mark 0.4 on the number line. See Figure 1.18 .

Try It 1.233

Locate on the number line: 0.6.

Try It 1.234

Locate on the number line: 0.9.

Example 1.118

Locate −0.74 −0.74 on the number line.

The decimal −0.74 −0.74 is equivalent to − 74 100 , − 74 100 , so it is located between 0 and −1 . −1 . On a number line, mark off and label the hundredths in the interval between 0 and −1 . −1 . See Figure 1.19 .

Try It 1.235

Locate on the number line: −0.6 . −0.6 .

Try It 1.236

Locate on the number line: −0.7 . −0.7 .

Which is larger, 0.04 or 0.40? If you think of this as money, you know that $0.40 (forty cents) is greater than $0.04 (four cents). So,

0.40 > 0.04 0.40 > 0.04

Again, we can use the number line to order numbers.

Where are 0.04 and 0.40 located on the number line? See Figure 1.20 .

We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that 0.40 > 0.04.

How does 0.31 compare to 0.308? This doesn’t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.

Because 310 > 308, we know that 310 1000 > 308 1000 . 310 1000 > 308 1000 . Therefore, 0.31 > 0.308.

Notice what we did in converting 0.31 to a fraction—we started with the fraction 31 100 31 100 and ended with the equivalent fraction 310 1000 . 310 1000 . Converting 310 1000 310 1000 back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!

We say 0.31 and 0.310 are equivalent decimals .

Equivalent Decimals

Two decimals are equivalent if they convert to equivalent fractions.

We use equivalent decimals when we order decimals.

The steps we take to order decimals are summarized here.

Order Decimals.

• Step 1. Write the numbers one under the other, lining up the decimal points.
• Step 2. Check to see if both numbers have the same number of digits. If not, write zeros at the end of the one with fewer digits to make them match.
• Step 3. Compare the numbers as if they were whole numbers.
• Step 4. Order the numbers using the appropriate inequality sign.

Example 1.119

Order 0.64 ___ 0.6 0.64 ___ 0.6 using < < or > . > .

Try It 1.237

Order each of the following pairs of numbers, using < or > : 0.42 ___ 0.4 . < or > : 0.42 ___ 0.4 .

Try It 1.238

Order each of the following pairs of numbers, using < or > : 0.18 ___ 0.1 . < or > : 0.18 ___ 0.1 .

Example 1.120

Order 0.83 ___ 0.803 0.83 ___ 0.803 using < < or > . > .

Try It 1.239

Order the following pair of numbers, using < or > : 0.76 ___ 0.706 . < or > : 0.76 ___ 0.706 .

Try It 1.240

Order the following pair of numbers, using < or > : 0.305 ___ 0.35 . < or > : 0.305 ___ 0.35 .

When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because −2 −2 lies to the right of −3 −3 on the number line, we know that −2 > −3 . −2 > −3 . Similarly, smaller numbers lie to the left on the number line. For example, because −9 −9 lies to the left of −6 −6 on the number line, we know that −9 < −6 . −9 < −6 . See Figure 1.21 .

If we zoomed in on the interval between 0 and −1 , −1 , as shown in Example 1.121 , we would see in the same way that −0.2 > −0.3 and − 0.9 < −0.6 . −0.2 > −0.3 and − 0.9 < −0.6 .

Example 1.121

Use < < or > > to order −0.1 ___ −0.8 . −0.1 ___ −0.8 .

Try It 1.241

Order the following pair of numbers, using < or >: −0.3 ___ −0.5 . −0.3 ___ −0.5 .

Try It 1.242

Order the following pair of numbers, using < or >: −0.6 ___ −0.7 . −0.6 ___ −0.7 .

Section 1.8 Exercises

Practice makes perfect.

In the following exercises, simplify.

− 100 − 100

− 121 − 121

In the following exercises, write as the ratio of two integers.

ⓐ − 12 − 12 ⓑ 9.279

ⓐ − 16 − 16 ⓑ 4.399

In the following exercises, list the ⓐ rational numbers, ⓑ irrational numbers

0.75 , 0.22 3 – , 1.39174 … 0.75 , 0.22 3 – , 1.39174 …

0.36 , 0.94729 … , 2.52 8 – 0.36 , 0.94729 … , 2.52 8 –

0.4 5 – , 1.919293 … , 3.59 0.4 5 – , 1.919293 … , 3.59

0.1 3 – , 0.42982 … , 1.875 0.1 3 – , 0.42982 … , 1.875

In the following exercises, identify whether each number is rational or irrational.

ⓐ 25 25 ⓑ 30 30

ⓐ 44 44 ⓑ 49 49

ⓐ 164 164 ⓑ 169 169

ⓐ 225 225 ⓑ 216 216

In the following exercises, identify whether each number is a real number or not a real number.

ⓐ − 81 − 81 ⓑ −121 −121

ⓐ − 64 − 64 ⓑ −9 −9

ⓐ −36 −36 ⓑ − 144 − 144

ⓐ −49 −49 ⓑ − 144 − 144

In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.

−8 , 0 , 1.95286 … , 12 5 , 36 , 9 −8 , 0 , 1.95286 … , 12 5 , 36 , 9

−9 , −3 4 9 , − 9 , 0.40 9 – , 11 6 , 7 −9 , −3 4 9 , − 9 , 0.40 9 – , 11 6 , 7

− 100 , −7 , − 8 3 , −1 , 0.77 , 3 1 4 − 100 , −7 , − 8 3 , −1 , 0.77 , 3 1 4

−6 , − 5 2 , 0 , 0. 714285 ——— , 2 1 5 , 14 −6 , − 5 2 , 0 , 0. 714285 ——— , 2 1 5 , 14

In the following exercises, locate the numbers on a number line.

3 4 , 8 5 , 10 3 3 4 , 8 5 , 10 3

1 4 , 9 5 , 11 3 1 4 , 9 5 , 11 3

3 10 , 7 2 , 11 6 , 4 3 10 , 7 2 , 11 6 , 4

7 10 , 5 2 , 13 8 , 3 7 10 , 5 2 , 13 8 , 3

2 5 , − 2 5 2 5 , − 2 5

3 4 , − 3 4 3 4 , − 3 4

3 4 , − 3 4 , 1 2 3 , −1 2 3 , 5 2 , − 5 2 3 4 , − 3 4 , 1 2 3 , −1 2 3 , 5 2 , − 5 2

2 5 , − 2 5 , 1 3 4 , −1 3 4 , 8 3 , − 8 3 2 5 , − 2 5 , 1 3 4 , −1 3 4 , 8 3 , − 8 3

In the following exercises, order each of the pairs of numbers, using < or >.

−1 ___ − 1 4 −1 ___ − 1 4

−1 ___ − 1 3 −1 ___ − 1 3

−2 1 2 ___ −3 −2 1 2 ___ −3

−1 3 4 ___ −2 −1 3 4 ___ −2

− 5 12 ___ − 7 12 − 5 12 ___ − 7 12

− 9 10 ___ − 3 10 − 9 10 ___ − 3 10

−3 ___ − 13 5 −3 ___ − 13 5

−4 ___ − 23 6 −4 ___ − 23 6

Locate Decimals on the Number Line In the following exercises, locate the number on the number line.

In the following exercises, order each pair of numbers, using < or >.

0.37 ___ 0.63 0.37 ___ 0.63

0.86 ___ 0.69 0.86 ___ 0.69

0.91 ___ 0.901 0.91 ___ 0.901

0.415 ___ 0.41 0.415 ___ 0.41

−0.5 ___ −0.3 −0.5 ___ −0.3

−0.1 ___ −0.4 −0.1 ___ −0.4

−0.62 ___ −0.619 −0.62 ___ −0.619

−7.31 ___ −7.3 −7.31 ___ −7.3

Everyday Math

Field trip All the 5th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 people. Each bus holds 44 people.

ⓐ How many busses will be needed? ⓑ Why must the answer be a whole number? ⓒ Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

Child care Serena wants to open a licensed child care center. Her state requires there be no more than 12 children for each teacher. She would like her child care center to serve 40 children.

ⓐ How many teachers will be needed? ⓑ Why must the answer be a whole number? ⓒ Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?

Writing Exercises

In your own words, explain the difference between a rational number and an irrational number.

Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.

ⓑ On a scale of 1 − 10 , 1 − 10 , how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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• Book URL: https://openstax.org/books/elementary-algebra-2e/pages/1-introduction
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• Kindergarten
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• Subtraction
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The Real Number System

The Real Number System - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are Sets of numbers in the real number system, Components of the real number system, 6th number grade system, Sets of real numbers date period, Real numbers precalculus, Real numbers, Real numbers and number operations, Introduction to 1 real numbers and algebraic expressions.

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1. Sets of Numbers in the Real Number System

2. components of the real number system -, 3. 6th number grade system, 4. sets of real numbers date period, 5. p.1 real numbers precalculus, 6. real numbers, 7. 1.1 real numbers and number operations, 8. introduction to 1 real numbers and algebraic expressions.

Real Number System Unit | Fractions, Decimals, Square Roots, and Real Numbers

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What educators are saying

Description.

An 8 day CCSS-Aligned Real Number System Unit includes squares and square roots, rational vs. irrational numbers, classifying real numbers, and comparing and ordering real numbers.

Students will practice with both skill-based problems, real-world application questions, and error analysis to support higher level thinking skills. You can reach your students and teach the standards without all of the prep and stress of creating materials!

Standards: 8.NS.1, 8.NS.2, 8.EE.2; Texas Teacher? Grab the TEKS-Aligned Real Number System Unit. Please don’t purchase both as there is overlapping content.

Learning Focus:

• approximate the value of an irrational number and locate the value on a number line
• classify, compare, and order real numbers
• convert between fractions and decimals and evaluate square roots

More details on what is included:

1. Unit Overviews

• Streamline planning with unit overviews that include essential questions, big ideas, vertical alignment, vocabulary, and common misconceptions.
• A pacing guide and tips for teaching each topic are included to help you be more efficient in your planning.

2. Student Handouts

• Student-friendly guided notes are scaffolded to support student learning. 
• Available as a PDF and the student handouts/homework/study guides have been converted to Google Slides™ for your convenience.

3. Independent Practice

• Daily homework is aligned directly to the student handouts and is versatile for both in class or at home practice. 

4. Assessments

• 1-2 quizzes, a unit study guide, and a unit test allow you to easily assess and meet the needs of your students.
• The Unit Test is available as an editable PPT, so that you can modify and adjust questions as needed.

5. Answer Keys

• All answer keys are included.

***Please download a preview to see sample pages and more information.***

How to use this resource:

• Use as a whole group, guided notes setting
• Use in a small group, math workshop setting
• Chunk each student handout to incorporate whole group instruction, small group practice, and independent practice.
• Incorporate our Real Number System Activity Bundle for hands-on activities as additional and engaging practice opportunities.

Time to Complete:

• Each student handout is designed for a single class period. However, feel free to review the problems and select specific ones to meet your student needs. There are multiple problems to practice the same concepts, so you can adjust as needed.

Is this resource editable?

• The unit test is editable with Microsoft PPT. The remainder of the file is a PDF and not editable.

Looking for hands-on classroom activities?

• Check out the corresponding Real Number System Activity Bundle , which includes activities like scavenger hunts, find it and fix its, mazes and more to allow students to engage and practice the concepts. Win-win!

More 8th Grade Units:

Unit 1: Real Number System 

Unit 2: Exponents and Scientific Notation

Unit 3: Linear Equations

Unit 4: Linear Relationships

Unit 5: Functions

Unit 6: Systems of Equations

Unit 7: Transformations

Unit 8: Angle Relationships

Unit 9: Pythagorean Theorem

Unit 10: Volume Unit 11: Scatter Plots and Data

More 8th Grade Activity Bundles:

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7.E: The Properties of Real Numbers (Exercises)

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7.1 - Rational and Irrational Numbers

In the following exercises, write as the ratio of two integers.

In the following exercises, determine which of the numbers is rational.

• 0.42, 0.\(\overline{3}\), 2.56813…
• 0.75319…, 0.\(\overline{16}\), 1.95

In the following exercises, identify whether each given number is rational or irrational.

• (a) 49 (b) 55
• (a) 72 (b) 64

In the following exercises, list the (a) whole numbers, (b) integers, (c) rational numbers, (d) irrational numbers, (e) real numbers for each set of numbers.

• −9, 0, 0.361...., \(\dfrac{8}{9}, \sqrt{16}\), 9
• −5, \(− 2 \dfrac{1}{4}, − \sqrt{4}, 0.\overline{25}, \dfrac{13}{5}\), 4

7.2 - Commutative and Associative Properties

In the following exercises, use the commutative property to rewrite the given expression.

• 6 + 4 = ____
• −14 • 5 = ____
• a + 8 = ____

In the following exercises, use the associative property to rewrite the given expression.

• (13 • 5) • 2 = _____
• (22 + 7) + 3 = _____
• (4 + 9x) + x = _____
• \(\dfrac{1}{2}\)(22y) = _____

In the following exercises, evaluate each expression for the given value.

• y + 0.7 + (− y)
• y + (− y) + 0.7
• z + 5.39 + (− z)
• z + (− z) + 5.39
• \(\dfrac{4}{9} \left(\dfrac{9}{4} k\right)\)
• \(\left(\dfrac{4}{9} \cdot \dfrac{9}{4}\right) k\)
• \(− \dfrac{2}{5} \left(\dfrac{5}{2} m\right)\)
• \(\left(− \dfrac{2}{5} \cdot \dfrac{5}{2}\right) m\)

In the following exercises, simplify using the commutative and associative properties.

• 6y + 37 + (−6y)
• \(\dfrac{1}{4} + \dfrac{11}{15} + \left(− \dfrac{1}{4}\right)\)
• \(\dfrac{14}{11} \cdot \dfrac{35}{9} \cdot \dfrac{14}{11}\)
• −18 • 15 • \(\dfrac{2}{9}\)
• \(\left(\dfrac{7}{12} + \dfrac{4}{5}\right) + \dfrac{1}{5}\)
• (3.98d + 0.75d) + 1.25d
• −12(4m)
• 30\(\left(\dfrac{5}{6} q\right)\)
• 11x + 8y + 16x + 15y
• 52m + (−20n) + (−18m) + (−5n)

7.3 - Distributive Property

In the following exercises, simplify using the distributive property.

• 9(u − 4)
• −3(6m − 1)
• −8(−7a − 12)
• \(\dfrac{1}{3}\)(15n − 6)
• (y + 10) • p
• (a − 4) − (6a + 9)
• 4(x + 3) − 8(x − 7)

In the following exercises, evaluate using the distributive property.

• 3(8u + 9) and
• 3 • 8u + 3 • 9 to show that 3(8u + 9) = 3 • 8u + 3 • 9
• 8\(\left(n + \dfrac{1}{4}\right)\) and
• 8 • n + 8 • \(\dfrac{1}{4}\) to show that 8\(\left(n + \dfrac{1}{4}\right)\) = 8 • n + 8 • \(\dfrac{1}{4}\)
• −100(0.1d + 0.35) and
• −100 • (0.1d) + (−100)(0.35) to show that −100(0.1d + 0.35) = −100 • (0.1d) + (−100)(0.35)
• −(y − 18) and
• −y + 18 to show that −(y − 18) = − y + 18

7.4 - Properties of Identities, Inverses, and Zero

In the following exercises, identify whether each example is using the identity property of addition or multiplication.

• −35(1) = −35
• 29 + 0 = 29
• (6x + 0) + 4x = 6x + 4x
• 9 • 1 + (−3) = 9 + (−3)

In the following exercises, find the additive inverse.

• \(\dfrac{3}{5}\)
• \(− \dfrac{7}{15}\)

In the following exercises, find the multiplicative inverse.

• \(\dfrac{9}{2}\)
• \(\dfrac{1}{10}\)
• \(− \dfrac{4}{9}\)

In the following exercises, simplify.

• 83 • 0
• \(\dfrac{0}{9}\)
• \(\dfrac{5}{0}\)
• 0 ÷ \(\dfrac{2}{3}\)
• 43 + 39 + (−43)
• (n + 6.75) + 0.25
• \(\dfrac{5}{13} \cdot 57 \cdot \dfrac{13}{5}\)
• \(\dfrac{1}{6}\) • 17 • 12
• \(\dfrac{2}{3} \cdot 28 \cdot \dfrac{3}{7}\)
• 9(6x − 11) + 15

7.5 - Systems of Measurement

In the following exercises, convert between U.S. units. Round to the nearest tenth.

• A floral arbor is 7 feet tall. Convert the height to inches.
• A picture frame is 42 inches wide. Convert the width to feet.
• Kelly is 5 feet 4 inches tall. Convert her height to inches.
• A playground is 45 feet wide. Convert the width to yards.
• The height of Mount Shasta is 14,179 feet. Convert the height to miles.
• Shamu weighs 4.5 tons. Convert the weight to pounds.
• The play lasted \(1 \dfrac{3}{4}\) hours. Convert the time to minutes.
• How many tablespoons are in a quart?
• Naomi’s baby weighed 5 pounds 14 ounces at birth. Convert the weight to ounces.
• Trinh needs 30 cups of paint for her class art project. Convert the volume to gallons.

In the following exercises, solve, and state your answer in mixed units.

• John caught 4 lobsters. The weights of the lobsters were 1 pound 9 ounces, 1 pound 12 ounces, 4 pounds 2 ounces, and 2 pounds 15 ounces. What was the total weight of the lobsters?
• Every day last week, Pedro recorded the amount of time he spent reading. He read for 50, 25, 83, 45, 32, 60, and 135 minutes. How much time, in hours and minutes, did Pedro spend reading?
• Fouad is 6 feet 2 inches tall. If he stands on a rung of a ladder 8 feet 10 inches high, how high off the ground is the top of Fouad’s head?
• Dalila wants to make pillow covers. Each cover takes 30 inches of fabric. How many yards and inches of fabric does she need for 4 pillow covers?

In the following exercises, convert between metric units.

• Donna is 1.7 meters tall. Convert her height to centimeters.
• Mount Everest is 8,850 meters tall. Convert the height to kilometers.
• One cup of yogurt contains 488 milligrams of calcium. Convert this to grams.
• One cup of yogurt contains 13 grams of protein. Convert this to milligrams.
• Sergio weighed 2.9 kilograms at birth. Convert this to grams.
• A bottle of water contained 650 milliliters. Convert this to liters.

In the following exercises, solve.

• Minh is 2 meters tall. His daughter is 88 centimeters tall. How much taller, in meters, is Minh than his daughter?
• Selma had a 1-liter bottle of water. If she drank 145 milliliters, how much water, in milliliters, was left in the bottle?
• One serving of cranberry juice contains 30 grams of sugar. How many kilograms of sugar are in 30 servings of cranberry juice?
• One ounce of tofu provides 2 grams of protein. How many milligrams of protein are provided by 5 ounces of tofu?

In the following exercises, convert between U.S. and metric units. Round to the nearest tenth.

• Majid is 69 inches tall. Convert his height to centimeters.
• A college basketball court is 84 feet long. Convert this length to meters.
• Caroline walked 2.5 kilometers. Convert this length to miles.
• Lucas weighs 78 kilograms. Convert his weight to pounds.
• Steve’s car holds 55 liters of gas. Convert this to gallons.
• A box of books weighs 25 pounds. Convert this weight to kilograms.

In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.

In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

• −5°C
• −12°C

PRACTICE TEST

• For the numbers 0.18349…, 0.\(\overline{2}\), 1.67, list the (a) rational numbers and (b) irrational numbers.
• Is \(\sqrt{144}\) rational or irrational?
• From the numbers −4, \(− 1 \dfrac{1}{2}\), 0, \(\dfrac{5}{8}\), \(\sqrt{2}\), 7, which are (a) integers (b) rational (c) irrational (d) real numbers?
• Rewrite using the commutative property: x • 14 = _________
• Rewrite the expression using the associative property: (y + 6) + 3 = _______________
• Rewrite the expression using the associative property: (8 · 2) · 5 = ___________
• Evaluate \(\dfrac{3}{16} \left(\dfrac{16}{3} n\right)\) when n = 42.
• For the number \(\dfrac{2}{5}\) find the (a) additive inverse (b) multiplicative inverse.

In the following exercises, simplify the given expression.

• \(\dfrac{3}{4}\)(−29)\(\left(\dfrac{4}{3}\right)\)
• −3 + 15y + 3
• (1.27q + 0.25q) + 0.75q
• \(\left(\dfrac{8}{15} + \dfrac{2}{9}\right) + \dfrac{7}{9}\)
• −18\(\left(\dfrac{3}{2} n\right)\)
• 14y + (−6z) + 16y + 2z
• 6(5x − 4)
• −10(0.4n + 0.7)
• \(\dfrac{1}{4}\)(8a + 12)
• 8(6p − 1) + 2(9p + 3)
• (12a + 4) − (9a + 6)
• \(\dfrac{0}{8}\)
• \(\dfrac{4.5}{0}\)
• 0 ÷ \(\left(\dfrac{2}{3}\right)\)

In the following exercises, solve using the appropriate unit conversions.

• Azize walked \(4 \dfrac{1}{2}\) miles. Convert this distance to feet. (1 mile = 5,280 feet).
• One cup of milk contains 276 milligrams of calcium. Convert this to grams. (1 milligram = 0.001 gram)
• Larry had 5 phone customer phone calls yesterday. The calls lasted 28, 44, 9, 75, and 55 minutes. How much time, in hours and minutes, did Larry spend on the phone? (1 hour = 60 minutes)
• Janice ran 15 kilometers. Convert this distance to miles. Round to the nearest hundredth of a mile. (1 mile = 1.61 kilometers)
• Yolie is 63 inches tall. Convert her height to centimeters. Round to the nearest centimeter. (1 inch = 2.54 centimeters)
• Use the formula F = \(\dfrac{9}{5}\)C + 32 to convert 35°C to degrees F.

Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/[email protected] ."

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Real Number System Homework 1

Displaying top 8 worksheets found for - Real Number System Homework 1 .

Some of the worksheets for this concept are Sets of numbers in the real number system, Algebra 1 name homework 8 the real number system date, Lesson 1 classification and real numbers, Real numbers and number operations, Unit 1 real number system homework, Teksstaar based lessons, Lesson format resources, Science 7th grade number system crossword 1 name.

Found worksheet you are looking for? To download/print, click on pop-out icon or print icon to worksheet to print or download. Worksheet will open in a new window. You can & download or print using the browser document reader options.

1. Sets of Numbers in the Real Number System

2. algebra 1 name: homework #8: the real number system date ..., 3. lesson 1 (classification and real numbers), 4. 1.1 real numbers and number operations, 5. unit 1 real number system homework, 6. teks/staar-based lessons, 7. lesson format: resources, 8. science: 7th grade number system crossword 1 name.

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