Introduction to Numbers

Numbers play an important role in our life. We use numbers in our day to day life to count things.

While counting we use numbers to represent any quantity, to measure any distance or length.

The counting numbers starting from 1, 2, 3, 4, 5, ……… are termed as ‘natural numbers’.

The set of counting numbers and zero are known as ‘whole numbers’.

Whole numbers are 0, 1, 2, 3, 4, 5, 6, 7, ……. and so, on

The symbols used by different civilizations to represent numbers are:

Types of Numbers

Even Natural Numbers: Numbers which are divisible by 2 are called even numbers. e.g. 2, 4, 6, 8, 10, ....

Odd Natural Numbers: Numbers which are not divisible by 2 are called odd numbers. e.g. 1, 3, 5, 7, 9, ....

Prime Numbers: Numbers which is only divisible by 1 and itself are called prime numbers. e.g. 2, 3, 5, 7, 11, 13, ….

Composite Numbers: Numbers which are not prime are called composite numbers as they can be divided by more than two numbers. e.g. 4, 6, 8, 9, 10,…

Ordinal Numbers: Numbers that indicate the exact position or rank of an object or a person. e.g. First, second, third, …….

Cardinal Numbers: The counting numbers which are used to represent the number of objects or people are called cardinal numbers. e.g. 1, 2, 3, 4, 5, ….

Factors and Multiples: When two or more numbers are multiplied, each number of the product is called a factor and the product is a multiple of each of its factors. e.g. 4 × 6 = 24

Here, 4 and 6 are factors of 24 and 24 is a multiple of 4 and 6.

Perfect Numbers: Numbers whose sum of its factors is twice the number are called perfect numbers. e.g. 6, 28, 496,……

Integers: Integers are the collection of whole numbers and negative numbers. e.g. –3, –2, –1, 0, 1, 2, 3,

Fractions: A fraction is a number representing a part of a whole. e.g. ,,…..

Decimals: Numbers which has a whole number and the fractional part separated by a decimal point. e.g. 4.56, 5.78, ….

1.0Comparing and Building Numbers

To put large numbers in order, you must check the number of digits in them first. If the number of digits vary in each number, the smallest number is the one which is having the least number of digits and the greatest number is the one which is having the maximum number of digits.

Comparing Numbers with the Same Number of Digits

Comparison of the numbers with the same number of digits starts from the left-hand side. You must compare the face values of the digits having the same place value in the numbers until you come across unequal digits.

Note:

The successor of a given number is obtained by adding 1 to the given number.

The predecessor of a given number is obtained by subtracting 1 from it.

Building Numbers

Now, you will learn to build numbers, under different conditions.

2.0Introducing 6-Digit, 7-Digit and 8-Digit Numbers

Till now you have learnt up to 5 digit numbers and you know that the greatest 5 digit number is 99,999. On adding 1 to it, we get the smallest 6-digit number.

99999 + 1 = 1,00,000, read as one lakh. 

The largest 6-digit number is 9,99,999. On adding 1 to it, we get the smallest 7-digit number.

9,99,999 + 1 = 10,00,000, read as ten lakh.

The largest 7-digit number is 99,99,999. On adding 1 to it, we get the smallest 8-digit number.

99,99,999 + 1 = 1,00,00,000, read as one crore.

Ascending Order

When the numbers are arranged from the smallest to the largest number, those numbers are said to be in an ascending order. The numbers are arranged from left to right in increasing order.

Descending Order

When the numbers are arranged from the largest to the smallest number, those numbers are said to be in descending order. The numbers are arranged from left to right in decreasing order.

Ascending order is represented by < (less than) symbol, whereas descending order is represented by > (greater than) symbol.

Shifting Digits

Changing the position of digits in a number, changes magnitude of the number.

Example:

Take a number 257.

The condition here is to exchange its hundreds and unit digit and form the new number.

That is, exchange 2 to 7 and 7 to 2.

Here comes a question.

Which is greater and which is least among the numbers?

To find that express the numbers formed in both ascending and descending order.

The number before shifting is 257. Exchanging the hundreds and the unit digits, the number after shifting is 752.

That is, if we exchange the hundreds and unit digits, the resultant number becomes greater.

Place Value and Face Value

Every digit has two values i.e. the place value and the face value. The face value of a digit does not change while its place value changes according to its position and number.

3.0Expanded Form of a Number

Indian and International System of Numeration

Suppose a newspaper report state that Rs. 2500 crore has been allotted by the government for National Highway construction. The same amount of Rs. 2500 crore is sometimes expressed as 25 billion. In the Indian system, we express it as Rs. 2500 crore and in the international system, the same number is expressed as 25 billion. Hence, you need to understand both the systems and their relationship.

Indian System of Numeration

The Indian system of numeration or Hindu–Arabic numeral system is a positional decimal numeral system developed between the 1st and 5th centuries by Indian mathematicians, adopted by Persian and Arabian mathematicians and spread to the western world by the High Middle Ages. It uses ten basic symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (called digits) and the idea of place value.

For a given numeral, we start from the extreme right as : Ones, Tens, Hundreds, Thousands, Ten Thousands, Lakhs, Ten Lakhs, etc. Each place represents ten times the one which is immediately to its right.

Indian system of numbers

Indian place-value chart

1 C = 1 crore = 1,00,00,000

International System of Numeration

International system of numeration is adopted by all the countries throughout the world.

International system of numbers

International place-value chart

1 TM = 10 million = 10,000,000

Note:

One million = ten lakhs 

Ten millions = one crore

Hundred million = ten crores

1 billion = Hundred crores

1 trillion = 1000 billions = 1 million millions = 10 lakh millions = 1 lakh crore

Use of Commas

Commas help us in reading and writing large numbers. In our Indian system of numeration, commas are used to mark thousands, lakhs and crores. The first comma comes after hundreds place and marks thousands. The second comma comes after ten thousands place and marks lakh. The third comma comes after ten lakh place and marks crore.

In International system of numeration, commas are used to mark thousands and millions. It comes after every three digits from the right.

Fun Facts 

• Every odd number has an ‘e’ in it.
• Four (4) is only one number which can be spelled with the same number of letters as itself.
• Forty (40) is the only number that is spelt with letters arranged in alphabetical order.
• One is the only number that is spelt with letters arranged in descending order.
• In the Indian number system, when we write numbers from 0 to 1000, letters A only appears first in 1000 (one thousand).   

Solved Example

 1. Compare 45967 and 45861.

Solution

As the number of digits are the same so starting from the left hand side, we notice that 2 digits are the same.

45967    and    45861

On comparing the digits at the hundred places in both the numbers we find that 9 in 45967 is greater than 8 in 45861.

∴ 45967 > 45861

2. Make the greatest and the smallest four-digit numbers by using different digits such that digit 6 is always in the tens place.

Solution

We know that the digits written in the descending order are 9, 8, 7, 6, 5, 4, 3, 2, 1, 0.

Keeping 6 in the tens place, we have

Greatest number = 9 8 6 7

Smallest number = 1 0 6 2

3. Express in expanded form.

(i) 3,54,039  

(ii) 3,85,00,386

Solution

(i) Place value of 3 = 3 × 100000 Place value of 5 = 5 × 10000

Place value of 4 = 4 × 1000 Place value of 0 = 0 × 100

Place value of 3 = 3 × 10 Place value of 9 = 9 × 1

∴ The expanded form of 3,54,039 is

3 × 100000 + 5 × 10000 + 4 × 1000 + 0 × 100 + 3 × 10 + 9 × 1.

(ii) Likewise, the expanded form of 3,85,00,386 is

3×10000000 + 8×1000000 + 5×100000 + 0×10000 + 0×1000 + 3×100 + 8×10 + 6×1.