Operations on Real Numbers: Definition & Examples

We are familiar with the system of natural numbers, whole numbers, integers, rational numbers and irrational numbers and their representation on the number line. We also know the 4 fundamental operations, which are addition, subtraction, multiplication, and division. We are familiar to perform arithmetic operations on integers. Let’s learn about operations on real numbers.

Collecting rational numbers together with all irrational numbers forms the collection of real numbers. Every real number is either a rational number or an irrational number. In this article, we will learn and study to perform operations on rational numbers and irrational numbers.

What are Real Numbers?

The combination of rational numbers and irrational numbers are known as real numbers. Real numbers can be both positive and negative, denoted as \(R\). Natural numbers, fractions, decimals all come under this category. Let us look at the below-given figure to understand the various number systems.

Decimal Expansion of Real Numbers

As we are familiar with the real numbers, we shall find the expansion of real numbers and see if we can use these expansions to distinguish between rational and irrational numbers. We will pay attention to the remainders and see if they follow any pattern. For example, let us find the decimal expansions of the following numbers.
a) \(\frac {25}{16}\)

b) \(\frac {1}{7}\)

We observed that each remainder is smaller than the divisor from the above division, which must be true for all the divisors. Also, the remainder is either zero after a certain stage or starts repeating. Thus, in the cases where the decimal expansions terminate after a finite number of steps, such decimal expansions are called terminating.

In the cases where the remainder never becomes zero, such decimal expansions are known as non-terminating recurring decimals. The numbers that have a decimal expansion which is non-terminating and non-recurring are irrational numbers.

Rational Numbers and Irrational Numbers

Any number can be expressed in the form of \(\frac {p}{q}\), where \(p\) and \(q\) are both integers and \(q ≠ 0\) is called a rational number. For example, \(\frac {2}{3},\;\frac{-5}{12},\;\frac{-3}{111},\;\frac {205}{361}\), etc., are in the standard form, whereas a number that cannot be expressed in the form of \(\frac {p}{q}\), where \(p\) and \(q\) are both integers and \(q ≠ 0\), \(p\) and \(q\) have no common factors (except \(1\)), which is called an irrational number. For example, \(\sqrt 2,\;\sqrt[3]{6},\;\sqrt {20},\;\frac{1}{{\sqrt 5 }}\) etc.,

Properties of Real Numbers

The following results hold for the system of real numbers.

1. If \(a,\,b\) are any two real numbers, then \(a + b\) is also a real number.
2. If \(a,\,b\) are any two real numbers, then \(a – b\) is also a real number.
3. If \(a,\,b\) are any two real numbers, then \(a \times b\) is also a real number.
4. If \(a,\,b \ne 0\)  are any two real numbers, then \(\frac {a}{b}\) is also a real number.

Thus, the system of real numbers is closed under all the four fundamental operations of arithmetic. Only the exception case is of division by \(0\).

1. The set of real numbers is ordered, i.e., if \(a,\;b\) are any two real numbers, then either \(a > b\) or \(a < b\) or \(a = b.\) This is called the trichotomy law.
2. If \(a,\;b\) are two real numbers, then \(\frac {a + b}{2}\) is a real number and it lies in between them, i.e., if, \(a < b\) then \(a < \frac{{a + b}}{2} < b.\) Continuing the process, we find that there are infinitely many real numbers between two different real numbers. For example, \(\frac{{\sqrt 2 + \sqrt 3 }}{2}\) is a real number that lies between \({\sqrt 2 }\) and \({\sqrt 3 }\).

Representation of Real Numbers on a Number Line

Real numbers can be represented on a number line. Let us look at the representation of rational numbers and irrational numbers separately on a number line.

Representation of Rational Number on a Number Line

We have learnt that both positive and negative integers can be represented on the number line.

On a horizontal number line:

1. All the positive numbers are to the right of zero.
2. All the negative numbers are to the left of zero.
3. Numbers are arranged in ascending order from left to right.
4. Every number is smaller than any number on its right and greater than any number on its left.

All rational numbers can be represented on the number line.

As you can see from the number line, to show the rational number \(\frac {1}{4}\), we divide the line segment between \(0\) and \(1\) into four equal divisions and then mark the first division as \(\frac {1}{4}\). Similarly, we can mark \(\frac {5}{4},\) \(\frac {-5}{4},\) etc.

Representation of Irrational Number on a Number Line

Let us understand the representation of irrational numbers on a number line with the help of an example.
Locate \(\sqrt {13}\) on the number line.
We can write \(13\) as the sum of squares of two natural numbers.
\(13 = 9 + 4 = {3^2} + {2^2}\)

Let \(l\) be the number line. If point \(O\) represents number \(0\) and point \(A\) represents number \(3\), then the line segment \(OA = 3\) units.
At \(A\), draw \(AC ⊥ OA\). From \(AC\), cut off \(AB = 2\) units.
We observe that \(OAB\) is a right-angled triangle at \(A\).

By Pythagoras theorem, we get,
\(OB^2 = OA^2 + AB^2 = 3^2 + 2^2 = 13\)
\(OB = 13\)
With \(O\) as centre and radius \(= OB\), we draw an arc of a circle to meet the number line \(l\) at point \(P\). As \(OP = OB = \sqrt {13}\) units, the point \(P\) will represent the number \(\sqrt {13}\) on the number line.

Operations on Real Numbers

The major operations on real numbers can be classified as below:

1. Operations on two rational numbers
2. Operations on two irrational numbers
3. Operations on a rational and an irrational number

We have learnt the operations of rational numbers like addition, subtraction, multiplication and division.
Now, let us see how irrational numbers are operated. We are already aware of the four basic fundamental operations in arithmetic. Let us look at some examples where we will apply the arithmetic operations on real numbers.

Example 1: Simplify the following:
a) \(\left( {5 + \sqrt 7 } \right)\left( {5 – \sqrt 7 } \right)\)
Solution: \(\left( {5 + \sqrt 7 } \right)\left( {5 – \sqrt 7 } \right) = {5^2} – {\left( {\sqrt 7 } \right)^2}\)
\(25 – 7\)
\(= 18\)
Hence, the required answer is \(18\).

Example 2: Add \(\left( {2\sqrt 5 – 4\sqrt 2 } \right)\) and \(\left( {3\sqrt 5 + 5\sqrt 2 } \right)\)
Answer: We have,
\(\left( {2\sqrt 5 – 4\sqrt 2 } \right) + \left( {3\sqrt 5 + 5\sqrt 2 } \right)\)
\( = 2\sqrt 5 – 4\sqrt 2 + 3\sqrt 5 + 5\sqrt 2 \)
\( = 2\sqrt 5 + 3\sqrt 5 + 5\sqrt 2 – 4\sqrt 2 \)
\( = 5\sqrt 5 + \sqrt 2 \)
Hence, the required answer is \( = 5\sqrt 5 + \sqrt 2 \)

Example 3: Write the expansion of \({\left( {2 – 4\sqrt 3 } \right)^2}\)
Answer: \({\left( {2 – 4\sqrt 3 } \right)^2}\)
We know that, \({\left( {a – b} \right)^2} = {a^2} + {b^2} – 2ab\)
So, \({\left( {2 – 4\sqrt 3 } \right)^2} = {2^2} + {\left( {4\sqrt 3 } \right)^2} – 2 \times 2 \times 4\sqrt 3 \)
\( = 4 + \left( {16 \times 3} \right) – 16\sqrt 3 \)
\( = 4 + 48 – 16\sqrt 3 \)
\( = 52 – 16\sqrt 3 \)
Therefore, \({\left( {2 – 4\sqrt 3 } \right)^2} = 52 – 16\sqrt 3 \)

Example 4: Express \(1.3\bar 2 + 0.\overline {35} \) in the form \(\frac {p}{q}\) Where \(p\) and \(q\) are integers, \(p,\;q \ne 0.\)
Solution: Let \(x = 1.3\bar 2 = 1.32222\) …….(i)
Multiplying both sides of (i) by \(10\), we get,
\(10x = 13.2222\) ……..(ii)
Multiplying both sides of (ii) by \(10\), we get,
\(100x = 132.2222\) ……….(iii)
Subtracting (ii) from (iii), we get,
\(100x – 10x = 132.222 – 13.222\)
\(90x = 119\)
\(x = \frac{{119}}{{90}}\) ……..(iv)
Let, \(y = 0.\overline {35} = 0.353535\) ……..(v)
Multiplying both sides of (v) by \(100\), we get,
\(100y = 35.3535\) ………(vi)
Subtracting (v) by (vi), we get,
\(100y – y = 35.3535 – 0.3535\)
\(99y = 35\)
\(y = \frac{{35}}{{99}}\) ………(vii)
Therefore, \(1.3\bar 2 + 0.\overline {35} = \frac{{119}}{{90}} + \frac{{35}}{{99}}\)
\( = \frac{{1309 + 350}}{{990}} = \frac{{1659}}{{990}}.\)

Example 5: Prove the following:
a) The sum of a rational and an irrational number is irrational.
b) The difference between two irrational numbers may not be irrational.
Answer: a) Let \(x\) be a rational number and \(y\) be an irrational number.
We have to prove that \(x + y\) is irrational.
Let us consider that \(x + y\) is rational.
As \(x + y\) and \(x\) are rational, it follows that \((x + y) – x\) is rational as the difference between two rational numbers is rational.
\(x + y – x = y\) should be rational, which is wrong as \(y\) is an irrational number.
Hence, the sum of a rational and irrational number is irrational.
b) We have to prove that the difference between two irrational numbers may not be irrational.
Consider two irrational numbers, \(a = 5 + \sqrt 5 \) and \(b = 7 + \sqrt 5 \), then \(a\) and \(b\) are both irrational.
Thus, their difference \( = a – b = 5 + \sqrt 5 \; – 7 – \sqrt 5 = \,- 2\), which is rational.
Hence, the difference between two irrational numbers may not be irrational.

Solved Examples – Operations on Real Numbers

Q.1. If \(\sqrt 3 = 1.732\), then find the value of:
\(\sqrt {27} – 3\sqrt {75} + 5\sqrt {48} + 2\sqrt {108} \)
Ans: \(\sqrt {27} – 3\sqrt {75} + 5\sqrt {48} + 2\sqrt {108} = \sqrt {9 \times 3} – 3\sqrt {25 \times 3} + 5\sqrt {16 \times 3} + 2\sqrt {36 \times 3} \)
\( = 3\sqrt 3 – 3 \times 5\sqrt 3 + 5 \times 4\sqrt 3 + 2 \times 6\sqrt 3 \)
\( = 3\sqrt 3 – 15\sqrt 3 + 20\sqrt 3 + 12\sqrt 3 \)
\( = \left( {3 – 15 + 20 + 12} \right)\sqrt 3 \)
\( = 20 \times 1.732\)
\( = 34.64\)
Hence, the required answer is \( = 34.64\)

Q.2. Simplify the following:
\({\left( {\sqrt 5 – \sqrt 3 } \right)^2}\)
Ans: Given to simplify: \({\left( {\sqrt 5 – \sqrt 3 } \right)^2}\)
Using the algebraic identity, \({\left( {a – b} \right)^2} = {a^2} + {b^2} – 2ab\), we get
\({\left( {\sqrt 5 – \sqrt 3 } \right)^2} = {\left( {\sqrt 5 } \right)^2} – 2 \times \sqrt 5 \times \sqrt 3 + {\left( {\sqrt 3 } \right)^2}\)
\( = 5 – 2\sqrt {15} + 3\)
\( = 8 – 2\sqrt {15} \)
Hence, the required answer is \( 8 – 2\sqrt {15} .\)

Q.3. Simplify: \(\frac{7}{3} – \frac{5}{6} + \frac{{11}}{2}\)
Ans: \(\frac{7}{3} – \frac{5}{6} + \frac{{11}}{2}\)
LCM of \(3,\;6\) and \(2\) is \(6\).
\( = \frac{{14 – 5 + 33}}{6}\)
\( = \frac{{9 + 33}}{6}\)
\( = \frac{{42}}{6} = 7\)
Hence, the required answer is \(7\).

Q.4. Evaluate \(\frac{5}{2} \times \frac{3}{4} \div \frac{5}{8}\)
Ans: \(\frac{5}{2} \times \frac{3}{4} \div \frac{5}{8} = \frac{5}{2} \times \frac{3}{4} \times \frac{8}{5}\)
\( = \frac{{120}}{{40}}\)
\( = 3\)
Hence, the required answer is \(3.\)

Q.5. Simplify: \(\sqrt {5 + 2\sqrt 6 } + \sqrt {8 – 2\sqrt {15} } \)
Ans: \(\sqrt {5 + 2\sqrt 6 } + \sqrt {8 – 2\sqrt {15} } \)
\(\sqrt {5 + 2\sqrt 6 } = \sqrt {3 + 2 + 2\sqrt 6 } \)
\( = \sqrt {{{\left( {\sqrt 3 + \sqrt 2 } \right)}^2}} = \sqrt 3 + \sqrt 2 \)
And, \(\sqrt {8 – 2\sqrt {15} } = \sqrt {5 + 3 – 2\sqrt {15} } \)
\( = \sqrt {{{\left( {\sqrt 5 – \sqrt 3 } \right)}^2}} = \sqrt 5 – \sqrt 3 \)
Therefore, \(\sqrt {5 + 2\sqrt 6 } + \sqrt {8 – 2\sqrt {15} } = \sqrt 3 + \sqrt 2 + \sqrt 5 – \sqrt 3 \)
\( = \sqrt 2 + \sqrt 5 \)
Hence, the simplified form of \(\sqrt {5 + 2\sqrt 6 } + \sqrt {8 – 2\sqrt {15} } \) is \( \sqrt 2 + \sqrt 5.\)

Summary

In this article, we learnt about real numbers. We learnt that real numbers comprise rational and irrational numbers, whole numbers, integers. We also learnt the properties and rules of real numbers. In addition to this, we learnt to represent the real numbers on the number line, and then we learnt to perform the operations on real numbers and mastered the concept by solving some examples.

FAQs

Q.1. What are the rational and irrational numbers?
Ans: Any number can be expressed in the form of \(\frac {p}{q}\), where \(p\) and \(q\) are both integers and \(q ≠ 0\) is called a rational number. For example, \(\frac{2}{3}\), \(\frac {-5}{12},\;\frac {-3}{111},\;\frac {205}{361},\) etc., are in the standard form, whereas a number that cannot be expressed in the form of \(\frac {p}{q}\), where \(p\) and \(q\) are both integers and \(q ≠ 0\), \(p\) and \(q\) have no common factors (except \(1\)), which is called an irrational number. For example, \(\sqrt 2 ,\;\sqrt[3]{6},\;\sqrt {20} ,\;\frac{1}{{\sqrt 5 }}\) etc.

Q.2. What are real numbers? Explain with examples.
Ans: Real numbers include positive and negative integers, rational numbers, and irrational numbers.

Q.3. What are examples of non-real numbers?
Ans: The numbers which are not real are imaginary or non-real numbers. Non-real numbers cannot be represented on the number line. For example, \(\sqrt[6]{{ – 54}},\;\sqrt { – 3} \), etc.,

Q.4. How do you identify real numbers?
Ans: Any number which can be plotted on the number line is known as a real number.

Q.5. What are the properties of real numbers?
Ans:
1. If \(a,\;b\) are any two real numbers, then \(a + b\) is also a real number.
2. If \(a,\;b\) are any two real numbers, then \(a – b\) is also a real number.
3. If \(a,\;b\) are any two real numbers, then \(a \times b\) is also a real number.
4. If \(a,\;b \ne 0\) are any two real numbers, then \(\frac {a}{b}\) is also a real number.
5. The set of real numbers is ordered, i.e., if \(a,\;b\) are any two real numbers, then either \(a > b\) or \(a < b\) or \(a = b.\) This is called the trichotomy law.
6. If \(a,\;b\) are any two real numbers, then \(\frac {a + b}{2}\) is a real number and it lies in between them, i.e., if \(a < b\), then \(a < \frac{{a + b}}{2} < b.\) Continuing the process, we find that there are infinitely many real numbers between two different real numbers. For example, \(\frac{{\sqrt 2 + \sqrt 3 }}{2}\) is a real number that lies between \(\sqrt {2}\) and \(\sqrt {3}.\)

We hope this detailed article on the operations on real numbers helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!