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Learning Objectives
 Identify rational numbers and irrational numbers
 Classify different types of real numbers
be prepared!
Before you get started, take this readiness quiz.
 Write 3.19 as an improper fraction. If you missed this problem, review Example 5.1.4.
 Write \(\dfrac{5}{11}\) as a decimal. If you missed this problem, review Example 5.5.3.
 Simplify: \(\sqrt{144}\). If you missed this problem, review Example 5.12.1.
Identify Rational Numbers and Irrational Numbers
Congratulations! You have completed the first six chapters of this book! It's time to take stock of what you have done so far in this course and think about what is ahead. You have learned how to add, subtract, multiply, and divide whole numbers, fractions, integers, and decimals. You have become familiar with the language and symbols of algebra, and have simplified and evaluated algebraic expressions. You have solved many different types of applications. You have established a good solid foundation that you need so you can be successful in algebra.
In this chapter, we'll make sure your skills are firmly set. We'll take another look at the kinds of numbers we have worked with in all previous chapters. We'll work with properties of numbers that will help you improve your number sense. And we'll practice using them in ways that we'll use when we solve equations and complete other procedures in algebra.
We have already described numbers as counting numbers, whole numbers, and integers. Do you remember what the difference is among these types of numbers?
counting numbers  1, 2, 3, 4… 
whole numbers  0, 1, 2, 3, 4… 
integers  …−3, −2, −1, 0, 1, 2, 3, 4… 
Rational Numbers
What type of numbers would you get if you started with all the integers and then included all the fractions? The numbers you would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.
Definition: Rational Numbers
A rational number is a number that can be written in the form \(\dfrac{p}{q}\), where p and q are integers and q ≠ 0.
All fractions, both positive and negative, are rational numbers. A few examples are
\[\dfrac{4}{5},  \dfrac{7}{8}, \dfrac{13}{4},\; and\;  \dfrac{20}{3}\]
Each numerator and each denominator is an integer.
We need to look at all the numbers we have used so far and verify that they are rational. The definition of rational numbers tells us that all fractions are rational. We will now look at the counting numbers, whole numbers, integers, and decimals to make sure they are rational.
Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. An easy way to do this is to write it as a fraction with denominator one.
\[3 = \dfrac{3}{1} \quad 8 = \dfrac{8}{1} \quad 0 = \dfrac{0}{1}\]
Since any integer can be written as the ratio of two integers, all integers are rational numbers. Remember that all the counting numbers and all 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 could be written as the decimal −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 \(7 \dfrac{3}{10}\), we can write it as an improper fraction, \(7 \dfrac{3}{10}\). 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) 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 \(\PageIndex{1}\):
Write each as the ratio of two integers: (a) −15 (b) 6.81 (c) \(−3 \dfrac{6}{7}\).
Solution
(a) −15
Write the integer as a fraction with denominator 1.  $$\dfrac{15}{1}$$ 
(b) 6.81
Write the decimal as a mixed number.  $$6 \dfrac{81}{100}$$ 
Then convert it to an improper fraction.  $$\dfrac{681}{100}$$ 
(c) \(−3 \dfrac{6}{7}\)
Convert the mixed number to an improper fraction.  $$ \dfrac{27}{7}$$ 
Exercise \(\PageIndex{1}\):
Write each as the ratio of two integers: (a) −24 (b) 3.57.
 Answer a

\(\frac{24}{1}\)
 Answer b

\(\frac{357}{100}\)
Exercise \(\PageIndex{2}\):
Write each as the ratio of two integers: (a) −19 (b) 8.41.
 Answer a

\(\frac{19}{1}\)
 Answer b

\(\frac{841}{100}\)
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 = \(\dfrac{a}{1}\) for any integer, a. We can also change any integer to a decimal by adding a decimal point and a zero.
\[\begin{split} Integer \qquad &2,\quad 1,\quad 0,\quad 1,\; \; 2,\; 3 \\ Decimal \qquad &2.0, 1.0, 0.0, 1.0, 2.0, 3.0 \end{split}\]
These decimal numbers stop.
We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we just considered.
\[\begin{split} Ratio\; of\; Integers \qquad \dfrac{4}{5},\quad \dfrac{7}{8},\quad \dfrac{13}{4},\;& \dfrac{20}{3} \\ Decimal\; forms \qquad 0.8, 0.875, 3.25, &6.666 \ldots \\ &6.\overline{66} \end{split}\]
These decimals either stop or repeat.
What do these examples tell you? Every rational number can be written both as a ratio of integers and as a decimal that either stops or repeats. The table below shows the numbers we looked at expressed as a ratio of integers and as a decimal.
Rational Numbers  

Fractions  Integers  
Number  $$\dfrac{4}{5},  \dfrac{7}{8}, \dfrac{13}{4}, \dfrac{20}{3}$$  $$2, 1, 0, 1, 2, 3$$ 
Ratio of Integer  $$\dfrac{4}{5}, \dfrac{7}{8}, \dfrac{13}{4}, \dfrac{20}{3}$$  $$\dfrac{2}{1}, \dfrac{1}{1}, \dfrac{0}{1}, \dfrac{1}{1}, \dfrac{2}{1}, \dfrac{3}{1}$$ 
Decimal number  $$0.8, 0.875, 3.25, 6.\overline{6}$$  $$2.0, 1.0, 0.0, 1.0, 2.0, 3.0$$ 
Irrational Numbers
Are there any decimals that do not stop or repeat? Yes. The number \(\pi\) (the Greek letter pi, pronounced ‘pie’), which is very important in describing circles, has a decimal form that does not stop or repeat.
\[\pi = 3.141592654 \ldots \ldots\]
Similarly, the decimal representations of square roots of whole numbers that are not perfect squares never stop and never repeat. For example,
\[\sqrt{5} = 2.236067978 \ldots \ldots\]
A decimal that does not stop and does not repeat cannot be written as the ratio of integers. We call this kind of number an irrational number.
Definition: 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.
If the decimal form of a number
 stops or repeats, the number is rational.
 does not stop and does not repeat, the number is irrational.
Example \(\PageIndex{2}\):
Identify each of the following as rational or irrational: (a) 0.58\(\overline{3}\) (b) 0.475 (c) 3.605551275…
Solution
(a) 0.58\(\overline{3}\)
The bar above the 3 indicates that it repeats. Therefore, 0.583 – is a repeating decimal, and is therefore a rational number.
(b) 0.475
This decimal stops after the 5, so it is a rational number.
(c) 3.605551275…
The ellipsis (…) means that this number does not stop. There is no repeating pattern of digits. Since the number doesn't stop and doesn't repeat, it is irrational.
Exercise \(\PageIndex{3}\):
Identify each of the following as rational or irrational: (a) 0.29 (b) 0.81\(\overline{6}\) (c) 2.515115111…
 Answer a

rational
 Answer b

rational
 Answer c

irrational
Exercise \(\PageIndex{4}\):
Identify each of the following as rational or irrational: (a) 0.2\(\overline{3}\) (b) 0.125 (c) 0.418302…
 Answer a

rational
 Answer b

rational
 Answer c

irrational
Let's think about square roots now. Square roots of perfect squares are always whole numbers, so they are rational. But the decimal forms of square roots of numbers that are not perfect squares never stop and never repeat, so these square roots are irrational.
Example \(\PageIndex{3}\):
Identify each of the following as rational or irrational: (a) 36 (b) 44
Solution
(a) The number 36 is a perfect square, since 6^{2} = 36. So \(\sqrt{36}\) = 6. Therefore \(\sqrt{36}\) is rational.
(b)Remember that 6^{2} = 36 and 7^{2} = 49, so 44 is not a perfect square. This means \(\sqrt{44}\) is irrational.
Exercise \(\PageIndex{5}\):
Identify each of the following as rational or irrational: (a) \(\sqrt{81}\) (b) \(\sqrt{17}\)
 Answer a

rational
 Answer b

irrational
Exercise \(\PageIndex{6}\):
Identify each of the following as rational or irrational: (a) \(\sqrt{116}\) (b) \(\sqrt{121}\)
 Answer a

irrational
 Answer b

rational
Classify Real Numbers
We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers, we get the set of real numbers. Figure \(\PageIndex{1}\) illustrates how the number sets are related.
Figure \(\PageIndex{1}\)  This diagram illustrates the relationships between the different types of real numbers.
Definition: Real Numbers
Real numbers are numbers that are either rational or irrational.
Does the term “real numbers” seem strange to you? Are there any numbers that are not “real”, and, if so, what could they be? For centuries, the only numbers people knew about were what we now call the real numbers. Then mathematicians discovered the set of imaginary numbers. You won't encounter imaginary numbers in this course, but you will later on in your studies of algebra.
Example \(\PageIndex{4}\):
Determine whether each of the numbers in the following list is a (a) whole number, (b) integer, (c) rational number, (d) irrational number, and (e) real number.
\[−7, \dfrac{14}{5}, 8, \sqrt{5}, 5.9, − \sqrt{64}\]
Solution
 The whole numbers are 0, 1, 2, 3,… The number 8 is the only whole number given.
 The integers are the whole numbers, their opposites, and 0. From the given numbers, −7 and 8 are integers. Also, notice that 64 is the square of 8 so \(− \sqrt{64}\) = −8. So the integers are −7, 8, \(− \sqrt{64}\).
 Since all integers are rational, the numbers −7, 8, and \(− \sqrt{64}\) are also rational. Rational numbers also include fractions and decimals that terminate or repeat, so \(\dfrac{14}{5}\) and 5.9 are rational.
 The number 5 is not a perfect square, so \(\sqrt{5}\) is irrational.
 All of the numbers listed are real.
We'll summarize the results in a table.
Number  Whole  Integer  Rational  Irrational  Real 

7  \(\checkmark\)  \(\checkmark\)  \(\checkmark\)  
\(\dfrac{14}{5}\)  \(\checkmark\)  \(\checkmark\)  
8  \(\checkmark\)  \(\checkmark\)  \(\checkmark\)  \(\checkmark\)  
\(\sqrt{5}\)  \(\checkmark\)  \(\checkmark\)  
5.9  \(\checkmark\)  \(\checkmark\)  
\( \sqrt{64}\)  \(\checkmark\)  \(\checkmark\)  \(\checkmark\) 
Exercise \(\PageIndex{7}\):
Determine whether each number is a (a) whole number,(b) integer,(c) rational number,(d) irrational number, and (e) real number: −3, \(− \sqrt{2}, 0.\overline{3}, \dfrac{9}{5}\), 4, \(\sqrt{49}\).
 Answer

Number Whole Integer Rational Irrational Real 3 \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\sqrt{2}\) \(\checkmark\) \(\checkmark\) \(0.\overline{3}\) \(\checkmark\) \(\checkmark\) \(\dfrac{9}{5}\) \(\checkmark\) \(\checkmark\) \(4\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\sqrt{49}\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\)
Exercise \(\PageIndex{8}\):
Determine whether each number is a (a) whole number,(b) integer,(c) rational number,(d) irrational number, and (e) real number: \(− \sqrt{25}, − \dfrac{3}{8}\), −1, 6, \(\sqrt{121}\), 2.041975…
 Answer

Number Whole Integer Rational Irrational Real \(− \sqrt{25}\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\dfrac{3}{8}\) \(\checkmark\) \(\checkmark\) \(1\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(6\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\sqrt{121}\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(\checkmark\) \(2.041975…\) \(\checkmark\) \(\checkmark\)
ACCESS ADDITIONAL ONLINE RESOURCES
Sets of Real Numbers
Real Numbers
Practice Makes Perfect
Rational Numbers
In the following exercises, write as the ratio of two integers.
 (a) 5 (b) 3.19
 (a) 8 (b) −1.61
 (a) −12 (b) 9.279
 (a) −16 (b) 4.399
In the following exercises, determine which of the given numbers are rational and which are irrational.
 0.75, 0.22\(\overline{3}\), 1.39174…
 0.36, 0.94729…, 2.52\(\overline{8}\)
 0.\(\overline{45}\), 1.919293…, 3.59
 0.1\(\overline{3}\), 0.42982…, 1.875
In the following exercises, identify whether each number is rational or irrational.
 (a) 25 (b) 30
 (a) 44 (b) 49
 (a) 164 (b) 169
 (a) 225 (b) 216
Classifying Real Numbers
In the following exercises, determine whether each number is whole, integer, rational, irrational, and real.
 −8, 0, 1.95286...., \(\dfrac{12}{5}, \sqrt{36}\), 9
 −9 , \(−3 \dfrac{4}{9}, − \sqrt{9}, 0.4\overline{09}, \dfrac{11}{6}\), 7
 \(− \sqrt{100}\), −7, \(− \dfrac{8}{3}\), −1, 0.77, \(3 \dfrac{1}{4}\)
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 buses will be needed?
 Why must the answer be a whole number?
 Why shouldn't you round the answer the usual way?
 Child care Serena wants to open a licensed child care center. Her state requires that 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?
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.
Self Check
(a) After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
(b) If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.
Contributors and Attributions
Lynn Marecek(Santa Ana College) and MaryAnne AnthonySmith (Formerly of Santa Ana College). This content islicensed underCreative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1fa2...49835c3c@5.191."
FAQs
7.1: Rational and Irrational Numbers? ›
All decimal fractions, such as those who are terminating like 7.1 can be represented on real number line and hence they are real numbers. In fact nonterminating but repeating can be expressed as a fraction (i.e. they are rational numbers) and can also represented on real number line and hence are real numbers.
Is 7.1 rational or irrational? ›All decimal fractions, such as those who are terminating like 7.1 can be represented on real number line and hence they are real numbers. In fact nonterminating but repeating can be expressed as a fraction (i.e. they are rational numbers) and can also represented on real number line and hence are real numbers.
Is 7.2 rational or irrational? ›The number 7.2 is a rational number. A rational number is defined as a number that results when one integer is divided by another integer. Since you can obtain the number 7.2 by dividing the integer 72 by the integer 10, as in 72/10, 7.2 is considered a rational number.
Is 7.0 rational or irrational? ›Therefore, the given number 7 is a rational number. Note: Based on the above solution, we can make a conclusion that all natural numbers are rational numbers as they all can be expressed as a fraction.
Is 7.12 repeating rational or irrational? ›Furthermore, the numerator and the denominator of the fraction above are integers. Therefore, we can conclude that the answer to "Is 7.12 a rational number?" is yes. Rational Number?
Is 7.1 a natural number? ›A set of natural numbers looks like this: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14...}
Is 7.1 a integer? ›Furthermore, the numerator and the denominator of the fraction above are integers. Therefore, we can conclude that the answer to "Is 7.1 a rational number?" is yes.
Is 7.25 rational or irrational? ›All integers and whole numbers are part of a bigger group called rational numbers. This group also includes fractions and decimals. That means 35 and 7.25 are rational numbers.
Is 7.3 an irrational number? ›Because 7.3 means 7310, we can write it as an improper fraction, 7310. 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) is a rational number.
Is 7.5 rational or irrational? ›7.5 is a rational number. A rational number is a quantity that can be expressed as a fraction.
Is negative 7 rational or irrational? ›
Answer and Explanation: Negative 7 is a rational number. By definition, a rational number is one that results when one integer is divided by another. An integer is simply a whole number.
Is 3.14 a rational number? ›Since we represented the number '3.14' in the form of \[\dfrac{p}{q}\] where \[p,q\] are integers which are in simplest form and \[q\ne 0\], we can conclude that '3.14' is a rational number.
Is square root 7 rational irrational? ›A rational number includes any whole number, fraction, or decimal that ends or repeats. An irrational number is any number that cannot be turned into a fraction, so any number that does not fit the definition of a rational number.
Is 5.676677666777 a irrational number? ›No, because integers cannot be negative. Q. Jeremy says that 5.676677666777... is a rational number because it is a decimal that goes on forever with a pattern.
Is this number rational or irrational? ›How can we identify if a number is rational or irrational? If a number has a terminating or repeating decimal, it is rational; for example, 1/2 = 0.5. If a number has a nonterminating and nonrepeating decimal, it is irrational, for example, o. 31545673…
Is this a irrational number? ›An irrational number is a real number that cannot be expressed as a ratio of integers; for example, √2 is an irrational number. We cannot express any irrational number in the form of a ratio, such as p/q, where p and q are integers, q≠0.
Is 2.7 a rational number? ›The fraction above is the same as 2.7. Furthermore, the numerator and the denominator of the fraction above are integers. Therefore, we can conclude that the answer to "Is 2.7 a rational number?" is yes.
Is 7.0 a natural number? ›Natural Numbers The numbers that we use when we are counting or ordering {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 …} Whole Numbers The numbers that include natural numbers and zero. Not a fraction or decimal.
Which number is rational? ›A rational number is any number that can be written as a fraction, where both the numerator (the top number) and the denominator (the bottom number) are integers, and the denominator is not equal to zero. In other words, a rational number can be expressed as p/q, where p and q are both integers and q ≠ 0.
What is 7.1 as a fraction? ›Solution: 7.1 as a fraction is 71/10.
Which are irrational numbers? ›
irrational number, any real number that cannot be expressed as the quotient of two integers—that is, p/q, where p and q are both integers. For example, there is no number among integers and fractions that equals Square root of√2.
How do you identify an 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.
Is 7.15 rational or irrational? ›Furthermore, the numerator and the denominator of the fraction above are integers. Therefore, we can conclude that the answer to "Is 7.15 a rational number?" is yes.
What decimals are irrational? ›Irrational Numbers: Any real number that cannot be written in fraction form is an irrational number. These numbers include nonterminating, nonrepeating decimals, for example , 0.45445544455544445555..., or . Any square root that is not a perfect root is an irrational number.
Is 1.75 rational or irrational? ›Number  As a Fraction  Rational or Irrational? 

1.75  74  Rational 
.001  11000  Rational 
√2 (square root of 2)  ?  Irrational ! 
3.7 rational. 0.0000001  rational.
Is 6.2 A irrational number? ›Answer and Explanation: The number 6.2 is a rational number.
Is 1.7 A irrational number? ›The number 1.7 is both rational are real. Therefore it is neither irrational nor imaginary. It is not a counting number, an integer, or a whole number.
Is 3.1 rational or irrational? ›The decimal 3.1 is a rational number. It can be expressed as the mixed number 3 1/10 or the fraction 31/10. Since 31 and 10 are both integers (whole numbers), we know that 31/10 and 3.1 are rational numbers.
Is 14.28 rational? ›Furthermore, the numerator and the denominator of the fraction above are integers. Therefore, we can conclude that the answer to "Is 14.28 a rational number?" is yes.
Is 8 irrational or irrational? ›
The number 8 is a rational number because it can be written as the fraction 8/1.
Is 0.7 a rational number? ›The decimal 0.7 is a rational number. It is read as seven tenths and is equivalent to the fraction 7/10. Since it can be written as a fraction, it is a rational number.
What is 7 8 as a rational number? ›7/8 is a rational number. Both numbers in the ratio 7/8 are integers because both 7 and 8 are natural numbers. Since 7/8 is a ratio between two integers, the resulting number, 0.875, is a rational number.
Is negation of √7 irrational? ›Answer: root7 is not an irrational number : False . Negation of the statement is equal to the opposite of that statement ...... √7 is an irrational number is a correct statement.
Is 0.5 A irrational number? ›0.5 is a repeating decimal, so 0.5 is a rational number.
Is 3.141141114 a rational number? ›D) 3.141141114 is an irrational number because it has not terminating non repeating decimal expansion.
Is 1.5 A rational number? ›(C) 1.5 is a terminating decimal and therefore is a rational number.
Why is radical 7 irrational? ›Any number that does not seem to terminate after the decimal point is called an irrational number. In this case, √7 is an irrational number because its value, 2.645751311064591, doesn't terminate after the decimal, and it is a neverending number.
Why is root 7 irrational? ›All square roots except perfect squares are irrational numbers. 7 is not a perfect square. Hence, the square root of 7 is irrational.
Is 2 √ 7 7 √ 7 rational or irrational? ›Thus, 2√7 / 7√7 is a rational number.
Is 0.101100101010 an irrational number? ›
Thus, 0.101100101010……. is an irrational number.
Is 3.3333 an irrational number? ›No, it not irrational as it is a rational number. In fact any decimal number which ends after a limited number of places beyond decimal point, or in which digits repeat endlessly after decimal place, are rational number.
Is 1.414213 an irrational number? ›1.414213 is a rational number.
Is the number 3 √ 7 )( 3 √ 7 rational or irrational? ›Hence , (3+√7)(3√7) is also a rational number.
Is 2 square root irrational? ›Sal proves that the square root of 2 is an irrational number, i.e. it cannot be given as the ratio of two integers.
Is ✓ 2 a rational number? ›Hence, 2 is an irrational number.
Is 7.11 rational or irrational? ›Therefore, we can conclude that the answer to "Is 7.11 a rational number?" is yes.
Is 3.14 irrational? ›To summarize, though π is often said to have value 3.14, it is actually an irrational number and 3.14 is just π rounded to two decimal places. However, the actual number 3.14 can be written as the fraction 314/100, or 157/50, so 3.14 is a rational number.
Is 0.277277277 irrational? ›0.277277277 can be written as a fraction: $\frac{277}{999}$, so it is rational.
Is negative 7 a irrational number? ›Answer and Explanation: Negative 7 is a rational number.
Is 7.31 rational or irrational? ›
and 7.31 are both rational numbers, since they can be written as the ratio of two integers.
Is 7 √ 3 a irrational number? ›ExpertVerified Answer
But our assumption is HCF(a,b) = 1. ∴Our assumption is wrong. Hence, 7√3 is an irrational no.
No. Square roots of all positive integers are not irrational. ex: 4,9,16 etc are positive integers and their square roots are 2,9 and 4 which are rational numbers .
Can a negative fraction be irrational? ›Explanation: Negative fractions are rational numbers  they are not irrational. Any number that can be expressed in the form mn where m,n are integers and n≠0 is a rational number. That includes negative fractions.
Is a 7 irrational number? ›Hence, 7 is irrational.
Is the root of 7 irrational? ›For example, because of this proof we can quickly determine that √3, √5, √7, or √11 are irrational numbers.
Is 7.25 a rational or irrational number? ›All integers and whole numbers are part of a bigger group called rational numbers. This group also includes fractions and decimals. That means 35 and 7.25 are rational numbers.
Is 7.8 rational or irrational? ›Furthermore, the numerator and the denominator of the fraction above are integers. Therefore, we can conclude that the answer to "Is 7.8 a rational number?" is yes.
Is 9.27 rational or irrational? ›The decimal expansion of √927 is 0.5773502692....... and it is nonterminating nonrecurring. Therefore, √927 is an irrational number.
Is √ 7 3 a rational number? ›Yes, 7/3 is a rational number.
Is 7 3 ✓ 2 irrational? ›
Now ,from equation (i) ,we get that √2 is rational but we know that √2 is irrational. so actually 7 + 3√2 is irrational not rational. thus our assumption is wrong. The number is irrational.