CHAPTER - 1 NATURAL NUMBERS by G.S Shashidhara., DME, BE(Mech -General), MBA

 

NATURAL NUMBERS

The counting numbers 1, 2, 3, 4, ... etc. These counting numbers are called natural numbers. The smallest natural number is 1.

Natural numbers are used in many different contexts and in many ways. Natural numbers help us in counting concrete objects. We can count objects in large numbers. For example, the number of students of your school, the number of people of your city (town or village). As the process of counting is endless, there is no largest natural number.

NUMBER SYSTEM (INDIAN SYSTEM)

You have already learnt in previous class the method of writing and reading the numerals up to 99, 999 using the digits 1,2,3,4,5,6,7,8,9 and 0 and also by the help of place values.

 

Digits or figures:

Any number (howsoever large) can be written with the use of ten symbols 0,1,2,3,4,5,6,7,8 and 9. Each of these symbols is called a digit or a figure.

A number system involves counting in tens. When we speak of counting in tens, it simply means that we are thinking of collections by tens. Ten is called the base of the system.

In our number system:

Ten ‘units or ones’ make one ten i.e. 10 × 1 = 10.
Ten ‘tens’ make one hundred i.e. 10 × 10 = 100.
Ten ‘hundreds’ make one thousand i.e. 10 × 100 = 1000 and so on.

Example 1: Examine the following number 54239

In this there are five digits. They are 9,3,2,4 and 5. Hence this number contains five places. These places are read from right to left.

The place value of this number is represented by a table.

Term (Group)	In Thousands		In Units
Place value	Ten Thousand	Thousand	Hundred	Ten	Unit
Number	5	4	2	3	9

Let us consider the number 54239.

As the digit 5 occupies ten thousand’s place, the place (or local) value of the digit 5 = 5 × 10,000 = 50,000.

Similarly, the place value of the digit 4 = 4 × 1,000 = 4,000,

the place value of the digit 2 = 2 × 100 = 200.

the place value of the digit 3 = 2 × 10 = 30.

the place value of the digit 9 = 9 × 1 = 9.

 

The method of reading the above number is

Fifty-four-thousand-two-hundred-and-thirty-nine

The method of writing is 54,239.

Note: Counting from right side, after 3 digits (Three places) a comma is placed.

Let us consider the number 7302.

As the digit 7 occupies thousand’s place, the place (or local) value of the digit 7 = 7 × 1000 = 7000.

Similarly, the place value of the digit 3 = 3 × 100 = 300,
the place value of the digit 0 = 0 × 10 = 0 and the place value of the digit 2 = 2 × 1 = 2.

Thus, the place (or local) value of a (non-zero) digit in a number depends upon the place it occupies in the given number; and the place value of the digit 0 is always 0 regardless of the place it occupies in the given number.

Let us consider the number 5354.

The place value of the digit 5 at ten’s place = 5 × 10 = 50 and the place value of the digit 5 at thousand’s place = 5 × 1000 = 5000 but the face (true or intrinsic) value of both the fives is 5.

Thus, the face (true or intrinsic) value of a digit in a number is the digit itself, regardless of the place it occupies in the number.

 

Example 2: Observe the following number 267385. The place value of the number is represented by a table.

 

Term (Group)	In Lakhs	In Thousands		In Units
Place value	Lakh	Ten Thousand	Thousand	Hundred	Ten	Unit
Number	2	6	7	3	8	5

 

The method of reading the above number is

Two-lakh-sixty-seven-thousand-three-hundred-and-eigthy-five.

The method of writing is 2,67,385.

Note: Counting from right, after three number, a comma is put, and afterwards another comma is put after 2 numbers.

 

Example 3: Observe the following number

89564270

In this number, how many digits are there? What are those? This number contains eight digits.

Name of the eight place is Crore, hence it is the place of Crore. If the above contains eight digits.

Name of the eight place is Crore, hence it is the place of crore. If the above number represented by a table.

Term	In Crores	In Lakhs		In Thousands		In Units
Place Name	Crore	Ten Lakhs	Lakh	Ten Thousand	Thousand	Hundred	Ten	Unit
Place Value	1,00,00,000	10,00,000	1,00,000	10,000	1,000	100	10	1
Number	8	9	5	6	4	2	7	0

Method of reading the above number

Eight-crores-ninety-five-lakhs-sixty-four-thousand-two-hundred-and-seventy.

Method of writing this number (convenient to read)

8,95,64,270

Observe: Counting from right a comma after three numbers is put, then another comma after two number, further another after two numbers. This system is called Indian system.

Use of comma

To read it conveniently, place values are grouped. In the Indian system the groups are from right.

Units group: Unit, Ten and Hundred.

Thousands group: Thousand and Ten thousand.

Lakhs group: Lakh and Ten lakhs.

Crores group: Crore and Ten Crores and so on.

While writing and separating these groups commas are put from tight after hundred, ten thousand, ten lakhs, ten crores.

 

Exercise problems:

1. Write the following numbers using symbols.

a) Thirteen-thousand-five-hundred-eighty-six – 13,586.

b) Fifty-four-thousand-nine-hundred-and-seven – 54,907.

c) Eighteen-thousand-seven-hundred – 18,700.

d) Six-lakhs-fifty-six-thousand-three-hundred-forty-two – 6,56,342.

e) Eighty-three-lakhs-fifteen-thousand-two-hundred-and twenty-five – 83,15,225.

f) One-crore-sixty-two-lakhs-thirteen-thousand – 1,62,13,000.

 

2. Using commas (according to Indian system) write the following numbers and then write in words.

a) 95830 – 95,830 – Ninety five thousand, eight hundred and thirty.

b) 26394 – 26,394 – twenty six thousand, three hundred and ninety four.

c) 426570 – 4,26,570 – Four Lakhs, twenty six thousand, five hundred and seventy.

d) 305784 – 3,05,784 – Three lakhs, five thousand, seven hundred and eighty four.

e) 1965204 – 19,65,204 – Nineteen lakhs, sixty five thousand, two hundred and four.

f) 5847921 – 58,47,921 – Fifty eight lakhs, forty seven thousand, nine hundred and twenty one.

g) 80000000 – 8,00,00,000 – Eight Crore.

h) 55555555 – 5,55,55,555 – Five Crore, fifty five lakhs, fifty five thousand, five hundred and fifty five.

i) 70000070 – 7,00,00,070 – Seven crores and seventy.

 

3. Mention the place and place value of 5 in the following numbers.

a) 24,572  (Hundred; 500)

b) 85,692 (Thousand; 5000)

c) 47,653 (Ten; 50)

d) 97,435 (Unit; 5)

e) 5,36,710 (Lakhs; 5,00,000)

f) 67,58,200 (Ten thousand; 50,000)

g) 3,58,62,438 (Ten lakhs; 50,00,000)

h) 5,86,24,179 (Crore; 5,00,00,000)

i) 5,00,36,001 (Crore; 5,00,00,000)

 

4. Some numbers are given below. Write the succeeding number for each (Example: Suceeding number to 49 is 50).

a) 99 (Suceeding number to 99 is 100)

b) 999 (Suceeding number to 999 is 1,000)

c) 9,999 (Suceeding number to 9,999  is 10,000)

d) 99,999 (Suceeding number to 99,999  is 1,00,000)

e) 9,99,999 (Suceeding number to 9,99,999  is 10,00,000)

f) 99,99,999 (Suceeding number to 99,99,999  is 1,00,00,000)

 

5. Some numbers are given below. Write the preceeding number for each. (Example: Preceeding number to 80 is 79).

a) 4,000 (Preceeding number to 4,000 is 3,999)

b) 3,800 (Preceeding number to 3,800 is 3,799)

c) 57,640 (Preceeding number to 57,640 is 57,639)

d) 10,00,000 (Preceeding number to 10,00,000 is 9,99,999)

e) 7,01,60,499 (Preceeding number to 7,01,60,499 is 7,01,60,498)

f) 1,00,00,000 (Preceeding number to 1,00,00,000 is 99,99,999)

 

6. Write the following numbers in the expanded form.

a) 689 – (600 + 80 + 9)

b) 5,027 – (5000 + 0 + 20 + 7)

c) 3,27,594 – (3,00,000 + 20,000 + 7,000 + 500 + 90 + 4)

d) 87,65,432 – (80,00,000 + 7,00,000 + 60,000 + 5,000 + 400 + 30 + 2)

e) 2,58,061 – (2,00,000 + 50,000 + 8,000 + 0 + 60 + 1)

f) 94,76,285 – (4,00,000 + 70,000 + 6,000 + 200 + 80 + 5)

 

NUMBER SYSTEM (MILLION SYSTEM)

As in the Indian system, even in the million system (English system of counting) using the numbers, 9,8,7,6,5,4,3,2,1 and 0 and by the help of place value the numbers are read and written. Here also places are read from right to left. The place value increases by 10 times as we go from right to left.

Observe the place, names equivalent and differences.

Place	Place Name (Indian System)	Place Name (Million System)
(from Right First)	Unit	Unit
Second	Ten	Ten
Third	Hundred	Hundred
Fourth	Thousand	Thousand
Fifth	Ten Thousand	Ten Thousand
Sixth	Lakh	Hundred Thousand
Seventh	Ten Lakh	Million
Eighth	One Crore	Ten Million

 

1. No change either in reading or in writing in the first five places.

2. In the indian system the sixth place is read as one lakh, and in the million system it is read as hundred thousand.

3. The seventh place is read as ten lakhs in the indian system and the same is read as one million in the million system.

4. The Eighth place is read as one crore in the Indian syatem and the same is read as ten million in the million system.

 

Method of writing and reading in the million system

Example 1: Observe  the following number 267385

In this there are six digits. Hence there are six places from left to right.

Method of reading the above number

Two-hundred-and-sixty-seven-thousand-three-hundred-and-eighty-five.

267,385 OR 267 385

Note: Reading from right to left, the first three digits are separated from the rest by a comma or by leaving some space between the two groups.

 

Example 2: Observe the following number:

3859216

This contains seven digits. From right sixth place is hundred thousandth place. Similarly from right seventh place name is million; million place.

Method of reading this number

Three-million-eight-hundred-fifty-nine-thousand-two-hundred-and-sixteen

To read this number easily the method of writing is

3,859,216 OR 3 859 216

Observe: From right three places are to be grouped. To separate it commas are to be inserted or space should be left.

 

Example 3: Observe the following number

78395126

This number contains eight places. From right the seventh place is million place. Similarly from right the name of the eighth place is Ten million; Ten million place.

The method of reading this number is

Seventy-eight-million-three-hundred-ninety-five-thousand-one-hundred-and-twenty-six.

To read this number conveniently, the method of writing this number is 78,395,126 OR 78 395 126.

Note: Using commas is an old system, and this can be used. But according to the SI System (convention) commas are not used to separate the groups and instead of this spaces are left.

System International:

When we go from right to left in the place value graph, we count unit’s place, ten’s place, hundred’s place, thousand’s place, ten thousand’s place, hundred thousand’s place, million place, and this system of counting is called  Million Sytem.

To read it conveniently, in this system from right to left every three places are grouped and commas are put OR sufficient space may be left.

 

In crores		In lakhs		In thousands		In units			Term (Group)
	Crore	Ten Lakhs	Lakh	Ten Thousand	Thousand	Hundred	Ten	Unit	Place Name
	1,00,00,000	10,00,000	1,00,000	10,000	1,000	100	10	1	Place Value
	1,00,00,000	10,00,000	1,00,000	10,000	1,000	100	10	1	Place Value
	Ten Million	Million	Hundred thousand	Ten thousand	Thousand	Hundred	Ten	Unit	Place Name
	In Million		In thousands			In units			Term (Group)

Place value table showing the relationship between Indian system and Million Sytem.

 

Example: 6389462231

The digits are grouped in threes from the ones place and a comma is put for every group of 3 digits as follows:

6,389,462,231

 

Billions	Millions	Thousands	Ones
HTO	HTO	HTO	HTO
6	389	462	231

 

 6,389,462,231 is read as six billion three hundred eighty-nine million four hundred sixty-two thousand two hundred thirty-one.

 

Remember:

One Lakh means 100 Thousands.

Ten Lakh means One Million

One Crore means Ten Million.

Ten Crores means 100 millions.

Hundred Crores means One Billion.

 

Note:

10 = 10 ones

100 = 10 tens

1000 = 10 hundreds

10000 = 10 Thousands

100000 = 10 Ten thousands.

Exercises

1. Write the following numbers in International system and then write in words.

a) 95830 – 95,830 (Ninety five thousand, eight hundred and thirty).

b) 62934 –  62,934  (Sixty two thousand, nine hundred and seventy five).

c) 642075 – 642,075 (Six hundred forty two thousand and seventy five).

d) 530478 – 530,478 (Five hundred thirty thousand, four hundred and seventy eight).

e) 9162504 – 9,162,504 (Nine million, one hundred and sixty two thousand, five hundred and four).

f) 1297485 – 1,297,485 (One million, two hundred ninety seven thousand, four hundred and eighty five).

g) 30000000 – 30,000,000 (Thirty million).

h) 22222222 – 22,222,222 (Twenty two million, two hundred twenty two thousand, two hundred and twenty two).

i) 80008000 – 80,008,000 (Eighty million, eight thousand).

2. Write the following numbers in Indian system.

a) 205 937 – 2,05,937

b) 638 504 – 6,38,504

c) 5 134 260 – 51,34,260

d) 3 202 330 – 32,02,330

e) 16 616 616 – 1,66,16,616

f) 69 340 785 – 6,93,40,785

 

3. Write the following numbers in the million system.

a) 8,59,10,347 – 85,910,347

b) 7,05,21,694 – 70,521,694

c) 6,00,00,000 – 60,000,000

d) 4,41,66,144 – 44,166,144

 

4. Write the following in numerals.

a) Seventy-eight-million-fifty-thousand-eight-hundred-and-forty-two. – 78,050,842.

b) Sixty-six-million-two-thousand-and-three. – 66,002,003.

c) Fifty-two-million-one-hundred-seventeen-thousand-five-hundred. – 52,117,500.

d) Seven-hundred-thousand-eight. – 700,008.

 

5. Write the following using million system.

a) The distance between Earth and Moon is about Three-hundred-Eighty-four-thousand-four-hundred kilometers. – 384,400.

b) The area of Karnataka land is One-hundred-ninety-one-thousand-seven-hundred-and-ninety-one square kilometers. – 191,791 .