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Introduction

1. Ancient method of counting.
2. Why numbers are required.

#### Ancient method of counting :

• Facts clearly state that counting sense in humans emerged long before the names of the numbers 1,2,3,4......
• In ancient times,the method of counting was based on distinct,uniform objects like fingers,stones,knots and lines.

#### For Example : Kharosthi Numerals : Indian writing found in third century B.C.

Here, 1 is represented by I line,2 is represented by II,3 is represented by III and so on.... this is more of a symbolic representation to counting rather than a numerical representation.

#### Why numbers are required :

• Numbers are required to count concrete objects.
• Numbers help to create sequence by arranging smaller to bigger and bigger to smaller.
• Helps in creating order.

### Shifting digits

Definition : Shifting given set of numbers from one place to another.

Let us consider set of certain numbers :

#### Examples : Exchange digit at hundreds place to the digit at ones place ### 1.2.3.Introducing 10000

As we all know that 99 is the largest two digit number,1 added to 99 gives us the smallest 3 digit number.

99+1 =100 ( smallest three digit number )

Similarly,

999+1 =1000 (smallest four digit number )
Largest 3 digit number + 1 = smallest 4 digit number.

9999+1 =10000 (smallest five digit number )
Largest 4 digit number + 1 = smallest 5 digit number.

 10 → 1×10 100 → 10×10 1000 → 10×100 10000 → 10×1000 10000 is actually 10 times 1000.so it is 10000

So,we concluded that

 Greatest single digit number+1=Smallest 2 digit number Greatest 2 digit number  +1=Smallest 3 digit number Greatest 3 digit number  +1=Smallest 4 digit number Greatest 4 digit number  +1=Smallest 5 digit number

### 1.2.4 :Revisiting place value

Let us consider few examples.

• 28 = 20 + 8 = 2 × 10 + 8
• 528 = 500 + 20 + 8 = 5 × 100 + 2 × 10 + 8 × 1
• 4528 = 4000 + 500 + 20 + 8 = 4 × 1000 + 5 × 100 + 2 × 10 + 8 × 1
• 64528 = 60000 + 4000 + 500 + 20 + 8 = 6 × 10000 + 4 × 1000 + 5 × 100 +2 × 10 + 8 × 1.

#### Let us illustrate above example using a table :

NumberTen ThousandThousandHundredsTensOnes
2828
528528
45284528
6452864528

### Expansion of numbers:

#### Example :

 Number Number Name Expansion 50000 Fifty Thousand 5 × 10000 28000 Twenty Eight Thousand 2 × 10000 + 8 × 1000 68250 Sixty Eight Thousand Two Fifty 6 × 10000 + 8 × 1000 + 2 × 100 + 5 × 10 89264 Eighty Nine ThousandTwo Hundred andSixty Four 8 × 10000 + 9 × 1000 + 2 × 100 + 6 × 10 + 4 × 1

### 1.2.5 Introducing 100000:

Greatest 5 digit number.
Adding 1 to the greatest 5 digit number gives the smallest 6 digit number.

#### Example :

99,999 + 1 =100000 ( This number is one lakh )

10 × 10000 = 100000

Let us consider 6 digit number in the expanded form for example.

8,56,243 = 8 × 100000 + 5 × 10000 + 6 × 1000 + 2 × 100 + 4 × 10 + 3 × 1.

This number has 3 in one's place,4 in ten's place,2 in hundred's place,6 in thousand's place,5 in ten thousands place and 8 in lakhs place and the number is "eight lakhs fifty six thousand two hundred and forty three".

#### Few more examples of expansion :

 Number Number Name Expansion 500000 Five Lakh 5 × 100000 450000 Four Lakh Fifty Thousand 4 × 100000 + 5 × 10000 398029 Three Lakh Ninety EightThousand and Twenty Nine 3 × 100000 + 9 × 10000 + 8 × 1000 + 2 × 10 + 9 × 1