Category: Maths Notes PPT

Mathematics involves questions of problem-solving for the students. Problems of Missing numbers are important for all the students. Logical and problem-solving questions like Missing numbers are part of a variety of entrance exams and competitive exams of Management, Architecture, Business, Administration, Civil Services exams and Engineering exams etc. 

In this article of , students can learn the definition of Missing numbers, Missing number series, Steps to find Missing numbers in the Series, quiz to find missing numbers, Missing numbers in a sequence and Solved examples to find Missing numbers in Sequence.

What are the Missing Numbers?

Missing numbers are the numbers that have been missed in the given series of a number with similar differences among them. The method of writing the missing numbers is stated as finding similar changes between those numbers and filling the missing terms in the specific series and places. In this article we will learn what are missing numbers, missing number series, how to find missing numbers in the series and sequence with examples, solved examples on missing numbers in the series and sequence, etc. 

Missing Number Series

We have seen that number series is a collection of numbers that follows a particular rule or formula. There are various types of series and missing number series is one among them. In missing number series, a series is given with one missing number and you are asked to find the missing term. To find the missing number, we first identify the rule or formula which is applied in the given missing number series. Let us learn the method to find the missing number in a series.

How to Find The Missing Number in A Series?

In the given missing number series, you can sometimes find missing numbers at the beginning or at the end of the series. The layout in the missing number series is similar to the wrong number series, you have to identify the rule and then use the rule to estimate the next number.

Here Are The Steps To Find The Missing Numbers In A Series

1. Select 2 or 3 terms to the text which rule will be applied to find the missing number. For example: If you have 5 numbers in a series then pick the first 3 terms to check the rule that is to be applied.

2. While choosing the number to check the rule, select the number that is easy to operate. These include terms that are factors of 2, 3, 5, or 10. Check the series with some common methods such as the sum of the terms, squares, cube, or other.

Let Us Understand Through An Example:

Missing Number Series Questions

1. Find the missing number 1, 2 6, 24,?

Solution: The given sequence has 4 terms. We will check which rule is applied by picking the first 3 terms. The second number in the sequence is 2 and the first number is 1 which means 1 is either added or 2 has been multiplied to obtain the second term.  The third term is 6 which we got from 2 by multiplying with 3. Hence, now we have 1 x 1, 1 x 2, 2 x 3, and 6 x 4. Thus, we have identified the rule and accordingly, the last term will be 24 x 5= 120.

Hence, the missing number is 120

2. How To Find A Missing Number In a Sequence?

Solution: Here are the steps to find missing numbers in sequence:

1. Identity, if the order of number given is ascending ( smaller to larger number)  or descending ( larger to smaller number)

2.  Calculate the differences between those that are next to each other.

3.  Estimate the difference between numbers to calculate the missing number.

Let us Understand The Above Steps Through An Example:

Find The Missing Number in the Following Sequence 30, 23? 9.

Solution: The numbers given in sequence are in decreasing order. It implies that numbers are arranged from larger to smaller.

The difference between the numbers 30-23=7

As the sequence of the numbers is in decreasing order, subtract 7 from 23. The missing number is 16 as it is 7 more than the previous number 9.

Solved Examples on Missing Numbers

1.  Find the missing number in the following sequence 1, 3, 5, 7, 11, ? 17, 19

Solution: The missing number found in the following sequence is 13.

It is because all the given numbers in the sequence 1, 3, 5, 7, 11, 17, 19 are prime numbers. The numbers given in the sequence are prime numbers as they can be divided only by 1 and itself.

Hence, the number line series will be 1, 3, 5, 7, 11, 13, 17, and 19.

2.  Find the missing number in the following sequence 1, 3, 9, 15, 25,? ,49

Solution: The missing number found in the following sequence is 35.

It is because all the numbers in the sequence are squares and (square-1) such as

12 = 2

22 = 4 and then 4 -1 = 3

32 = 9

42 = 16 and then 16-1 = 15

52 = 25 and

62 = 36 and then 36-1 = 35

72 = 49

Hence, the number line series will be 1, 3, 9, 15, 25, 35, and 49

Quiz Time

1.  Find the missing number 9, 35, 91, 189, 341, ?

      a.   559

      b.   611

      c.   521

      d.  502

2. Find the missing number.

1. 16

2. 52

3. 256

4. 112

3. Find the missing number:

1. 19

2. 24

3. 22

4. 20

Fun Facts

The number zero is first introduced by Aryabhatta.

The numbers 0-9 were introduced in India in the 6th or 7th century and are introduced in Europe through Middle Eastern Mathematician al-Khwarizmi and al-Kindi during the 12th century. These numbers are also known as Arabic Numerals.

How does help students to learn about Missing numbers and other relevant topics of math? 

1. is a free online learning platform for all students and aspirants to learn about Missing numbers and other important topics of maths.

2. In this regard, provides NCERT Texts, NCERT solutions to learn Missing numbers in both English and Hindi medium. Likewise, also provides Important questions, Revision notes and keynotes, CBSE Sample question papers and Previous Year’s question papers along with the answer key for the students to prepare for the chapter Missing Numbers.

3. Students can also solve questions on Missing Numbers from important Reference books like RD Sharma, RS Aggarwal and HC Sharma solved by the expert teachers in Maths at . 

4. Students can also learn more about Missing numbers from the Micro courses available at the website of at a pocket friendly amount of 1 rupee per course. Students can also learn about Missing numbers from the Mathlete Class 9 Youtube Channel.

5. Junior Students can also learn about Missing numbers in the personalised learning courses of Maths classes for students of age 4 to 14, foundation courses for Class 6 to 8.

6. Students can study about Missing numbers in Personalised Learning from Talented teachers through the Private Home tuitions conducted online in major cities of India and abroad.

When we talk about Multiples of 3 it is the set of all those numbers which can be expressed in the form of 3n or in other words the number which gives the remainder zero when it is divided by 3 is called a multiple of three. All the multiples of three must contain 3 as it’s one of the factors.

For example, number 18 can be expressed as 2 × 3 × 3 thus we can see that 3 is one of its factors also it is expressed in terms of 3n so this is a multiple of three. Thus, in other words, we can say that all those numbers can be divided by three or are products of 3, and any number is defined as a multiple of 3.

Multiples of 3

Here’s an article explaining various terms related to multiples. In this article, you are going to learn – what is a multiple, what are the different Multiples of 3, how to find the Multiples of 3, some important FAQs, etc. At the end of this article, you will be able to know how to find multiples of different numbers as we will be providing you with some examples with the help of which, you will easily be able to understand the topic. 

What is Multiple? 

When we multiply a number (whose multiple, we have to find) with a positive integer, the result we get is multiple. For example, if we have to find the multiple of the number 2, we can multiply by any other number, say 3, we will get 6 as an answer. Here the number 6 is the multiple of 2.

We can use a simple formula to check the multiples of a number. This formula is as follows:

Multiple of X = Xn ( ‘X’ is the number whose multiple we have to find and n is any positive integer.)

So, the multiple of a number can be defined as the number that can be written as the product of a given number and some other natural number. Multiples of the numbers can be observed in the multiplication table. Multiples of natural numbers are as given below:

E.g., multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, and so on. Hence, multiples of 2 will be even numbers and will end with 0, 2, 4, 6, or 8.

The multiples of the number 3 are 3, 6, 9, 12, 15, 18, 21,24,………….. and so on.

The multiples of the number 5 are 5, 10, 15, 20, 25,………….., and so on.

All the multiples of 5 will have the last digit as 0 or 5.

From the above-given examples we can say that multiples of 2, the number 2 can be multiplied by infinite numbers to find the “n” number of multiples.

Calculation of Multiples of 3

The multiples of the number 3 can be calculated by multiplying integers. For example, to calculate the Multiples of 3 we will use the product of 3 with the natural numbers 1, 2, 3, ………. and thus will get 3 x 1, 3 x 2, 3 x 3, 3 x 4, 3 x 5, etc., which equal 3, 6, 9, 12, 15, etc. All the Multiples of 3 that come in the table of three are Multiples of 3 i.e. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30……………….etc. Thus Multiples of 3 are expressed as 3p where p is an integer.

Example:

To find the Multiples of 3, we have to multiply them with many numbers. We will start by multiplying it with 1 and will end by multiplying it with 10. This will give us the first 10 Multiples of 3.

i.e.,

3 × 1 = 3

3 × 2 = 6

3 × 3 = 9

3 × 4 = 12

3 × 5 = 15

3 × 6 = 18

3 × 7 = 21

3 × 8 = 24 

3 × 9 = 27

3 × 10 = 30

So, the first 10 Multiples of 3 from above come out to be 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. 

Similarly, we can find more multiples of the number 3 by multiplying them with more natural numbers. 

Let’s solve some more examples which will help you to understand this topic easily. 

Example 1: Find the first 15 multiples of 7.

Solution: To find the first 15 multiples of 7, we will multiply it by the positive integers from 1 – 15.

So, the multiples are:

7 × 1 = 7

7 × 2 = 14

7 × 3 = 21

7 × 4 = 28

7 × 5 = 35

7 × 6 = 42

7 × 7 = 49

7 × 8 = 56

7 × 9 = 63

7 × 10 = 70

7 × 11 = 77

7 × 12 = 84

7 × 13 = 91

7 × 14 = 98

7 × 15 = 105

Therefore, the first 15 multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105. 

Example 2: Find the first 5 multiples of -3.

Solution: The first 5 multiples of -3 are as follows:

-3 × 1 = – 3

-3 × 2 = – 6

-3 × 3 = – 9

-3 × 4 = – 12

-3 × 5 = – 15

Hence, the first 5 multiples of -3 are -3, -6, -9, -12, -15.

Set of Multiples of 3

Multiples of 3 can be written in the form set in the roster or tabular form as well as in the inset builder.

1. In roster or tabular form, the numbers of the set are written within brackets separated by the commas so multiples of three will be written as

{0, 3, 6, 9, 12,…}

2. Inset Builder form elements of a set are written with their properties thus the Multiples of 3 will be written as

{x:x = 3n, n ∈ W, where W is the whole number}

Difference between Factors and Multiples

The exact divisors of the given number can be defined as the factor of the given number while the multiples of the number are defined as the numbers obtained when multiplied by other numbers.

The number of factors of any number is always finite while the number of multiplies of the number is infinite.

The operation which is used to find the factors of any number is a division while the operation used to find the multiples of a given number is called multiplication.

The result or the outcome of the factors in any given number will always be less than or equal to the given number while the result or the outcome of the multiples should be greater than or equal to the given number.

Now, let us assume an example:

3 × 4 = 12

Here, 3 and 4 are the factors of 12,

12 is multiple of 3 and 4

Thus, we can conclude that if X and Y are two numbers and:

X is a factor of Y if  X divides Y.

Y is a multiple of X if Y is divisible by X.

We know that the number 1 divides every integer thus it is the common factor of every integer, and also every number is divisible by 1 and every number is a multiple of 1.

The multiplicative inverse of a number is a number which when multiplied with the original number equals one. Here, the original number must never be equal to 0. The multiplicative inverse of a number X is represented as X⁻¹ or 1/X. The multiplicative inverse of a number is also referred to as its reciprocal.

In this article, we shall be learning in detail about the multiplicative inverse. The topic has been simplified by the experts at to make your learning experience engaging and fun. So, let’s get started.

Multiplicative Inverse Example

The multiplicative inverse of one is one only because 1×1=1.

The multiplicative inverse of zero does not exist. This is because 0xN=0 and 1/0 is undefined.

The multiplicative inverse of a natural number X is X⁻¹ or 1/X. For example, the multiplicative inverse of 256 is 1/256 because 256×1/256=1.

The multiplicative inverse of a natural number -Y is -Y⁻¹ or 1/-X. For example, the multiplicative inverse of -8 is 1/-8 because -8×1/-8=1.

The multiplicative inverse of a fraction x/y is y/x. In case the fraction is a unit fraction, then its multiplicative inverse will be the value present in the denominator. For example, the multiplicative inverse of 5/6 is 6/5 and the multiplicative inverse of 1/9 is 9.

[frac{4}{7}times frac{7}{4}] = 1

            

[frac{2}{3}times frac{3}{2}] = 1

Multiplicative Inverse Property 

The multiplicative inverse property states that a number P, when multiplied with its multiplicative inverse, gives the result as one.

Px1/P=1

Examples:

2×1/2=1

3/4x 4/3=1

How to Find a Multiplicative Inverse?

The easiest trick to finding the multiplicative of any rational number (except zero) is just flipping the numerator and denominator.

We can also find the multiplicative inverse by using a linear equation as follows. In the below equation y is the unknown multiplicative inverse.

8/9 * y = 1

y= 1/ (8/9)

y=1*(9/8)

y=9/8

Multiplicative Inverse Of A Complex Number 

The multiplicative inverse of any complex number x+yi is 1/(x+yi). In this multiplicative inverse, x and y are rational numbers and i is radical.

In this case, we must always remember to rationalise the multiplicative inverse. Our final answer should not contain any radicals in the denominator.

Rationalisation:

• To rationalise, multiply the numerator and denominator of 1/(x+yi) with (x-yi). This will give you (x-yi)/(x²-(yi)²).

• When we perform this operation using numbers instead of variables, we will get a constant whole number in the denominator and radicals in the numerator. At this step, our multiplicative inverse is rationalised. 

Problems: Find the Multiplicative Inverse

Problem 1: What is the reciprocal of 105/7.

Solution: The reciprocal of 105/7 is 7/105.

If we further simplify. We get,

7/105 = 1/15

So, the reciprocal of 15 is 1/15, because, 15 × 1/15 = 1.

Hence it satisfies the reciprocal property.

Problem 2: Find the reciprocal of Y²

Solution: The reciprocal of y² is 1/y² or y⁻²

Verification:  y² × y⁻² = 1

1 = 1

Did you know?

If X⁻¹ or 1/X is the multiplicative inverse of X, then X is the multiplicative inverse of X⁻¹ or 1/X. This is due to the commutative property of multiplication, which states that the result does not change if the order of numbers is changed.

One is called the multiplicative identity because when multiplied by itself, it gives itself as the result. In other words, 1 is the reciprocal of itself. This can be written as 1×1=1.

Conclusion

Hence multiplicative inverse of a number is a number which when multiplied with the original number equals one. Here, the original number must never be equal to 0. The multiplicative inverse of a number X is represented as X-1 or 1/X. The article discusses all the important and relevant information that will build strong concepts in students.

In Mathematics, the nth root of a number x is a number y which when raised to the power n, obtains x:

[y^{n} = x] 

Here, n is a positive integer, sometimes known as the degree of the root. A root of degree 2 is known as a square root, whereas the root of degree 3 is known as a cube root. Roots of higher degree are also referred to using ordinary numbers as in fourth root, fifth root, twentieth root, etc. The calculation of the nth root is a root extraction.

For example, 4 is a square root of 2, as [2^{2}] = 4, and −2 is also a square root of 4, as [(-2)^{2}] = 4.

[sqrt{x} times sqrt{x} = x] Here, the square root is used twice in multiplication to get the original value.

[sqrt[3]{x} times sqrt[3]{x} = x] Here, cube root is used thrice in multiplication to get the original value.

[sqrt[n]{x} times sqrt[n]{x} . . . sqrt[n]{x} = x] Here, the nth root is used n times in multiplication to get the original value.

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Nth Root Definition

Recall that k is a square root of y if and only if [k^{2}] = y. Similarly, k is a cube root of y if and only if [k^{3}] = y. For example, 5 is a cube root of 125 because[5^{3}] = 125. Let us understand the nth root definition with this concept.

Let n be an integer greater than 1, then y is the nth root of x if and only if yⁿ = x.

For example, -1/2 is the 5th root of -1/32 as [left ( frac{-1}{2} right )^{5}] = -1/32. There are no special names given to the nth root other than the square root (where n = 2), and the cube root (where n = 3). Other nth roots are known as the fourth root, fifth root, and so on.

Nth Root Symbol

The symbol used to represent the nth root is [sqrt[n]{x}]

It is a radical symbol used for square roots with a little n to define the nth root.

In expression [sqrt[n]{x}], n is known as the index and the x is known as the radicand.

In order to understand the definition of the nth root more precisely, the student needs to be aware of a few other topics that will play a major role in the understanding of the nth root. These topics are explained briefly below. 

Real Numbers

Real numbers are referred to as the combination of rational and irrational numbers. All the arithmetic functions are said to be performed on these numbers and they can also be represented on the number line. 

While, on the other hand, imaginary numbers are those that cannot be expressed on a number line, and are usually used to represent complex numbers. Real numbers can be both positive or negative and are usually denoted using the letter R. the natural numbers, fractions, and decimals fall under this category. 

Rational Numbers

Rational numbers fall under the heading of real numbers. These are represented as p/q, where q is not equal to 0. Any fraction that is a non-zero denominator is termed a rational number. For example,[frac{1}{3}], 1/5,[frac{3}{4}], etc. in fact the number 0 can also be called a rational number as it can be written in various forms like 0/1, 0/2, 0/3, etc. but it is to be kept in mind that 1/0, 2/0, 3/0, etc are not rational as they provide us with infinite values. 

Irrational Numbers

Irrational numbers refers to the real numbers that cannot be expressed in the form of a fraction. It cannot be denoted in the form of a ratio p/q, where the letters p and q refer to integers and q is not equal to zero. One can say that it is the opposite of the rational numbers. 

Irrational numbers are normally expressed in the form R∖Q. The backward slash refers to the ‘set minus’. It is also often expressed in the form of R-Q, which refers to the difference between a set of real numbers and a set of irrational numbers. 

Complex Numbers

Complex numbers are referred to as numbers that can be expressed in the form of a + ib. a, b are the real numbers, while i refers to the imaginary numbers. For instance, 2+3i is a complex number where 2 is a real number while 3i denotes the imaginary number.

The imaginary number is always denoted with the alphabet i or j which is equal to [sqrt{-1}], where [i^{2} = -1].

Square Roots

A square root of the number r can be referred to as x, which when squared, gives the result r. 

[r^{2} = x]

It is to be noted that every positive real number possesses two square roots, one that is positive and one that is negative. For instance, the number 25 has two square roots, one is 5 and the other is -5. The positive square root is also denoted as the principal square root.

As the square root of every number is non negative, the negative numbers do not possess a square root. But every negative real number has two imaginary square roots associated with them. For example, the square root of -25 will be 5i and -5i. The i here represents the number whose square is supposed to be -1. 

Cube Roots

The cube root of a given number x can be a number r whose cube will be x.

r3=x

How to find the Nth Root of a Number?

Ans: The nth root of a number can be calculated using the Newton method. Let us understand how to find the nth root of a number, ‘A’ using the Newton method.

Start with the initial guess x0, and then repeat using the  recurrence relation.

[x_{k+1} = frac{1}{n}(n – 1)x_{k} + frac{A}{X_{k^{n+1}}})] until the desired precision is reached.

On the basis of the application of nth root, it may be adequate to use only the first Newton approximant: [sqrt[n]{x^{n} + y} approx x + frac{y}{nx^{n-1}}].

For example, to find the fourth root of 16, note that [2^{4}] = 16 and hence x = 2, n = 4, and y = 2 in the above formula. This yields:

[sqrt[5]{34} = sqrt[5]{32 + 2} approx 2 + frac{2}{5.16} = 2.025]. The error in the approximation is only about 0,03%.

 

When does the Nth Root exist?

In a real  number system,

If n is an even whole number, the nth root of x exists whenever x is positive, and for all x.

If n is an odd whole number, the nth root of x exists for all x.

Example:

[sqrt[4]{-81}] is not a real number whereas,

[sqrt[5]{-32} = -2]

Things get more complicated in the complex number system.

Every number has a square root, cube root, fourth root, fifth root, and so on.

Example:

The fourth root of a number 81 are 3, -3, 3i, -3i, because

3⁴ = 81

-3⁴ = 81

(3i)⁴ = 3⁴ i⁴ = 81

(-3i)⁴ = (-3)⁴ i⁴ = 81

Properties of Nth Root

[sqrt[n]{a^{x}} = (a^{x})^{1/y} = a^{x/y}]

• There is exactly one positive nth root i
  n every positive real number. Hence, the rules of operation with surds including positive radicand x, and y are straightforward within a real number.

[sqrt[n]{xy} = sqrt[n]{x} sqrt[n]{y}]

[sqrt[n]{frac{x}{y}} = frac{sqrt[n]{x}}{sqrt[n]{y}}]

• Subtleties can take place while calculating the nth root of a negative or complex number. For example,

• [sqrt{-1} times sqrt{-1} = sqrt{-1 times -1} = 1]

• But instead , [sqrt{-1} times sqrt{-1} = i times i = i^{2} = -1]

• As the rule, [sqrt[n]{x} sqrt[n]{y} = sqrt[n]{xy}], strictly valid for non-negative real radicands only, its use leads to inequality in step 1 above.

Facts to Remember

• The nth root of 0 is 0 for all positive integers n, as 0n is equal to 0.

• The nth root of 1 is known as roots of unity and plays an important role in different areas of  Mathematics such as number theory, the theory of equation, etc.

Simplifying Nth Root

Ans: Let us learn to simplify the nth root through the examples below:

1. [sqrt[5]{-32}]

Solution:

The value of [sqrt[5]{-32}] is -2, because (-2)[^{5}] = -32.

2. Find [sqrt[6]{64x^{6} y^{12}}]

Solution:

Step 1: [sqrt[6]{64x^{6} y^{12}}] (Given)

Step 2: [sqrt[6]{(2)^{6} x^{6} (y^{2})^{6}}]

Step 3: [sqrt[6]{(2xy^{2})^{6}}]

Step 4: 2xy[^{2}]

A number can be defined as an arithmetic value used for representing the quantity and is used in making calculations. We count things using numbers. For example, in the image given below, this is one butterfly and these are 4 butterflies.

The collection of no objects in an element is symbolised as ‘0’ and we call it zero. Digits are used to represent numbers, which include 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For denoting a number we use a group of digits known as numerals. For example, 1234 2314 56111 are numerals. The method of expressing numbers in words is known as numeration. A number system is defined as a writing system for denoting numbers using digits or using symbols in a logical manner. 

The numeral system is used to:

• Represents a useful set of different numbers.

• Reflects the arithmetic as well as the algebraic structure of a number.

• The numeral system provides a standard representation.

One-Digit Numbers

Examples of one-digit numbers are 1, 2, 3, 4, 5,6,7,8, and 9.

Two Digits Numbers

When we add one unit to the greatest one-digit number, we get the smallest two-digit number, which is 1+9 = 10. The greatest two-digit number is 99.

Three-Digit Numbers

When we add one unit to the greatest two-digit number then we get the smallest three-digit number, which is 1+99 = 100. The greatest three-digit number is 999.

 

Four-Digit Numbers

When we add one unit to the greatest three-digit number, we get the smallest four-digit number that is, 1+999 = 1000. The smallest four-digit number is 1000 and the greatest number is 9999.

 

Five-Digit Numbers

When we add one unit to the greatest four-digit number, we get the smallest five-digit number, which is 1+ 9999 = 10000. The smallest five-digit number is 10000 and the greatest five-digit number is equal to 99999. 

Types of Numbers

The numbers can be classified into sets, which is known as the number system. The different types of numbers in maths are:

Natural numbers are also known as counting numbers that contain positive integers from 1 to infinity. The set of natural numbers is denoted as “N” and it includes N = {1, 2, 3, 4, 5, ……….} For example 0, 1, 2, 3, 4, 5, 6, 7,  8, and 9.

Prime numbers are natural numbers that are greater than 1 and have only 1 and themselves as factors. For example 2, 3, 5, 7, 11, 13… and so on.

A composite number is a natural number that is greater than 1 and has more factors including one. For example 2,4,6,8,9… and so on.

Whole numbers are the set of natural numbers in which zero is adjoined. They are known as non-negative integers and it does not include any fractional or decimal part. It is denoted as “W” and the set of whole numbers includes W = {0,1, 2, 3, 4, 5, ……….} For example 0, 1, 2, 3, 4, 5, 6, 7… and so on.

Integers are the set of all whole numbers but it includes a negative set of natural numbers also. “Z” represents integers and the set of integers are Z = { -3, -2, -1, 0, 1, 2, 3}

They are whole numbers with negative numbers adjoined. For example -9, -8, -7, -6… and so on.

All the positive and negative integers, fractional and decimal numbers without imaginary numbers are called real numbers. They are denoted by the symbol R.  Real numbers are the set of rational numbers with a set of irrational adjoins. For example- 3, 0, 1.5, √2, etc.

Any number that can be written as a ratio of one number over another number is written as a rational number. This means that any number that can be written in the form of p/q is a rational number. The symbol “Q” represents a rational number. They can also be defined as fractions with integers. For example ⅔, ⅚, etc.

The number that cannot be expressed as the ratio of one over another or that cannot be written as fractions is known as irrational number, and it is represented by the symbol ”P”.  For example √3,√5,√7, etc.

The number that can be written in the form of a+bi, where the variables “a and b” are the real numbers and variable “i” is an imaginary number, is known as complex number and is denoted by the letter “C”. For example- 2+3i, 5+2i, etc.

The imaginary numbers are known to be complex numbers that can be written in the form of the product of a real number and such numbers are denoted by the letter “i”.

What is the Indian Numeral System?

Let us consider a number, say 225. Notice that the digit 2 is used twice in this given number, however, both of them have different values. We differentiate them by stating their place value in mathematics, which is defined as the numerical value of a digit on the basis of its position in any given number. So, the place value of the leftmost 2 is Hundreds while for the 2 in the centre is Tens.

Coming back to the Indian numeral system, the place values of the various digits go in the sequence of:

1. Ones

2. Tens

3. Hundreds

4. Thousands

5. Ten Thousand

6. Lakhs

7. Ten Lakhs

8. Crores, and so on.

In the given number 10,23,45,678 the place values of each of the digits present in the number are given below:

We will begin from the right side of the number.

• 8 – Ones

• 7 – Tens

• 6 – Hundreds

• 5 – Thousands

• 4 – Ten Thousand

• 3 – Lakhs

• 2 – Ten Lakhs

• 0 – Crores

• 1 – Ten Crores

The relationship between them is given below:

• 1 hundred = 10 tens

• 1 thousand = 10 hundreds = 100 tens

• 1 lakh = 100 thousands = 1000 hundreds

• 1 crore = 100 lakhs = 10,000 thousands

International Numeral System

The place values of digits is in the sequence of Ones, Tens, Hundreds, Thousands, Ten Thousand, Hundred Thousands, Millions, Ten Million and so on, in the international numeral system. In the given number 12,345,678 the place values of each digit is:

• 8 – Ones

• 7 – Tens

• 6 – Hundreds

• 5 – Thousands

• 4 – Ten Thousand

• 3 – Hundred Thousand

• 2 – Millions

• 1 – Ten Million

The relations between them are:

• 1 hundred = 10 tens

• 1 thousand = 10 hundreds = 100 tens

• 1 million = 1000 thousand

• 1 billion = 1000 millions

Set Operations

In our everyday life, we deal with collections of objects (person, numbers or any other thing).

For example, consider the following collections:

i) Collection of all the students in your class.

ii) Collection of all the teachers at your school. 

iii) Collection of all counting numbers that are less than 10 i.e., of numbers 1, 2, 3, 4, 5, 6, 7, 8, 9.

iv) Collection of all even natural numbers less than 15 i.e., of numbers 2, 4, 6, 8, 10, 12, 14.

v) Collection of the first five natural numbers divisible by 5 i.e., of the numbers 5,10, 15, 20, 25.

vi) Collection of all vowels in the English alphabets i.e., of the letters a, e, i, o, u.

vii) Collection of all the days in a week.

viii) Collection of all the books in your bag. 

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Note: All the above collections is a well-defined collection of objects.

A ‘well-defined collection of objects’ means that if we are provided with a collection and an object, then it would be possible to assert without any doubt that if an object belongs to the collection or not. 

What is the Data Set?

A set is a defined collection of objects

The objects that belong to the set are called its members or elements. Each of the above collections is a set. 

Now consider the following collections:

i) Collection of all the intelligent students in your class. 

We cannot call it a well-defined collection because people may differ on whether a student of your class is intelligent or not.  

ii) Collection of all the competent teachers of your school.

We cannot call it a well-defined collection because people may differ on whether a teacher of your school is competent or not.

iii) Collection of four days of a week.

We cannot call it a well-defined collection because it is not known which four days of a week are to be included in the collection. 

Any of the above collections are not a set.

Notation

The sets are usually denoted by capital letters A, B, C, and so on… The members of a set are denoted by small letters x, y, z, and so on. 

If x is a member of the set A, we write x ∈ A (read as ‘x belongs to A’) and if x is not a member of the set A, we write x ∉ A (read as ‘x does not belong to A’)

If x and y are the members of the set A, we write x, y ∈ A.

Representation of A Set

A set can be represented by the following method:

1. Description method

2. Roster method or tabular form

3. Rule method or set builder form.

Types of Sets

There are Four Types of Sets

 Union of Sets

The union of two sets A and B is the set consisting of all the elements which belong to either A or B or both. We write it as A U B.

Example, if A = { a, e, i, o, u } and B = { a, b, c, d, e } the, 

             A U B = { a, e, i, o, u, b, c, d }

Intersection of Sets

The intersection of two sets A and B is the set consisting of all elements which belong to both A and B. we write it as A ∩ B.

Example Of Intersection Of Sets

 i) If A = {a, e, i, o, u} and B = {a, b, c, d, e} then, 

A ∩ B= {a,e}

ii) If A = {the colours of the rainbow} = {Violet, Indigo, Blue, Green, Yellow, Orange, Red}

And B = {Black, Red, Blue, White} the 

A ∩ B = {Red, Blue}

Difference of Sets

The difference of two sets A and B is a set with no elements in common. 

For example, 

i) A = {1, 3, 5, 7, 9} and B = {0, 2, 4, 6, 8, 10}

There is a difference of two sets A and B as there are no common elements between them. 

Properties of Set Operations

Union of Sets 

Properties of (A U B) are:

1. The commutative law holds true as (A U B) = (B U A)

2. The associative property too holds true as (A U B) U {C} = {A} U (B U C)

Intersection of Sets

Properties of – A ∩ B are:

1. Commutative law =  (A ∩ B) ∩ C = A ∩ (B ∩ C)

2. Associative law =  A ∩ B = B ∩ A 

3. Distributive law = A ∩ (B U C) = (A ∩ B) U (A ∩ C)

4. φ ∩ A = φ

5. U ∩ A = A