Knowing our numbers

1. COMPARING NUMBERS
Natural numbers
Counting numbers 1, 2, 3, 4, …… etc. are called Natural numbers. The smallest natural number is 1 and there is no largest natural number.
Digits
Numbers are formed using the ten symbols 0, 1, 2, 3, 4,  5, 6, 7, 8, 9. These symbols are called digits or figures. To find the place value of a digit in a number, multiply the digit with the value of the place it occupies.
Comparison of numbers
If two numbers have an unequal number of digits, then the number with the greater number of digits is greater. If two numbers have equal number of digits then, the number with greater valued digit on the extreme left is greater. If the digits on extreme left of the numbers are equal then the digits to the right of the extreme left digits are compared and so on.

The greatest single-digit number is 9. When 1 is added to the greatest single-digit number, we get 10, which is the smallest two-digit number. Therefore, the greatest single-digit number + 1 = the smallest two-digit number.

The greatest two digit-number is 99. When 1 is added to the greatest two-digit number, we get 100, which is the smallest three-digit number. Therefore, the greatest two-digit number + 1 = the smallest three-digit number.

The greatest three-digit number is 999. When 1 is added to the greatest three-digit number, we get 1000, which is the smallest four-digit number. Therefore, the greatest three-digit number + 1 = the smallest four-digit number.

The greatest four-digit number is 9999. When 1 is added to the greatest four-digit number, we get 10,000, which is the smallest five-digit number. Therefore, the greatest four-digit number + 1 = the smallest five-digit number.

The greatest five-digit number is 99999. When 1 is added to the greatest five-digit number, we get 1,00,000, which is the smallest six digit number. Therefore, the greatest five-digit number + 1 = the smallest six-digit number. The number, that is, one with five zeroes (100000), is called one lakh. 

The greatest six-digit number is 999999. When 1 is added to the greatest six-digit number, we get 10,00,000, which is the smallest seven-digit number. Therefore, the greatest six-digit number + 1 = the smallest seven-digit number. The number, that is, one with six zeroes (1000000), is called ten lakh.

The greatest seven-digit number is 9999999.  When we add 1 to this seven-digit number, we get 10000000, which is the smallest eight-digit number. Therefore, the greatest seven-digit number + 1 = the smallest eight-digit number. The number, that is, one with seven zeroes (10000000),is called one crore.
Ascending order
The arrangement of numbers from the smallest to the greatest is called ascending order.   
e.g. 2789, 3560, 4567, 7662, 7665
Descending order
The arrangement of numbers from the greatest to the smallest is called descending order. 
e.g. 7665, 7662, 4567, 3560, 2789.
Example 1
Which number is greater, 426 or 4378?
Solution:
The first number is a three-digit number and the second number is a four-digit number. Therefore. 4378 is greater than 426.
Example 2
Find the greatest and the smallest number among the following numbers.
1018,1081,1801,1011,1065
Solution:
Here, all the given numbers are four-digit numbers. First digit is same for all the numbers. Second digit from the left is the greatest in third number (which is 8). Therefore. 1S01 is the greatest number.
The third digit is same in the first and fourth numbers (which is 1). while the second and fifth numbers have S and 6 as their third digit. Among 1018 and 1011. the fourth digit is smaller for the latter number. Therefore. 1011 is the smallest number.
Formation of numbers
Numbers can be formed using the given digits with or without repetition of digits.
Example 1
What will be the greatest and the smallest five-digit number that can be formed using the digits 5, 7,0,2, and 9 without repetition?
Solution
If we arrange the numbers in descending order, then we will obtain the greatest number. Therefore, the greatest number using the given digits is 97520.
If we arrange the numbers in ascending order, then we will obtain the smallest number. Therefore, the smallest number is 20579 (since a number cannot beam with 0. it cannot be placed at the first place).
Example 2
Write the greatest and the smallest four-digit number.
Solution
The greatest four-digit number should contain the maximum number of nines (as 9 is the largest digit among the numbers 0-9).
Therefore, the greatest four-digit number is 9999.
The smallest four-digit number should contain the maximum number of zeroes. However, we cannot put zero at the first place from the left. Therefore. 1 should come at the first place followed by three zeroes.
Therefore, the smallest four-digit number is 1000.
Conversion of units of measurement
Units of length:
1 kilometre = 1,000 metres
1 metre = 100  centimetres
1 centimetre = 10 millimetres

Units of weight
1 kilogram = 1,000 grams
1 gram = 1000 milligrams

Units of capacity
1 kilolitre = 1000 litres
1 litre = 1,000 millilitres

Indian system of numeration
Values of the places in the Indian system of numeration are Ones, Tens, Hundreds, Thousands, Ten thousands, Lakhs, Ten Lakhs, Crores and so on.

The following place value chart can be used to identify the digit in any place in the Indian system.

Periods

Crores

lakhs

Thousands

Ones

Places

Tens

 Ones

Tens

Ones

Tens

Ones

Hundreds

Tens

Ones

International system of numeration
Values of the places in the International system of numeration are Ones, Tens, Hundreds, Thousands, Ten thousands, Hundred thousands, Millions, Ten millions and so on.

1 million = 1000 thousands,
1 billion = 1000 millions

Following place value chart can be used to identify the digit in any place in the International system.

Periods

Billions

Millions

Thousands

Ones

Places

100’s

 10’s

 1’s

100’s

 10’s

 1’s

100’s

 10’s

 1’s

100’s

 10’s

 1’s

Comparison of the Indian and the International numeration systems

Indian Numeration

Crore

Ten Lakh

Lakh

Ten Thousand

Thousand

Hundred

 Ten

One

 Numbers

 10000000

 1000000

100000

 10000

 1000

 100

 10

 0

 International

Numeration

Ten  Million

 Million

Hundred Thousand

Ten Thousand

 Thousand

 Hundred

 Ten

 Ones

Commas in Indian system of numeration
Commas are placed to the numbers to help us read and write large numbers easily. As per Indian system of numeration, the first comma is placed after the hundreds place. Commas are then placed after every two digits.
e.g. (i) 88,76,547

The number can be read as eighty-eight lakh seventy-six thousand five hundred and forty-seven. 
(ii) 5,67,89,056

The number can be read as five crore sixty-seven lakh eighty-nine thousand and fifty-six.
Commas in International system of Numeration
As per International numeration system, the first comma is placed after the hundreds place. Commas are then placed after every three digits.
e.g. (i) 8,876,547
The number can be read as eight million eight hundred seventy-six thousand five hundred and forty-seven. 
(ii) 56,789,056
The number can be read as fifty-six million seven hundred eighty-nine thousand and fifty-six. 
Example 1
Write 39784012 in words using both the Indian and the International number system.
Solution
In the Indian number system, the number can be written according to the place value of each digit as follows.

Crore

Ten lakh

Lakh

Ten thousand

Thousand

Hundred

Ten

One

3

9

7

8

4

0

1

7

Therefore, the number can be written as three crore ninety seven lakh eighty four thousand and twelve.
In the International number system, the number can be written according to the place value of each digit as follows.

Ten million

Million

Hundred thousand

Ten thousand

Thousand

Hundred

Ten

One

3

9

7

8

4

0

1

7

Therefore, the number can be written as thirty nine million seven hundred eights four thousand and twelve.
Example 2
Insert commas between the numbers according to both the Indian and the International number system.
(a) 88500784      (b) 32098175
Solution
(a) According to the Indian number system, the number will be written as 8.85.00.78.
      According to the International number system, the number will be written as
      88.500.784.
(b) According to the Indian number system, the number will be written as
      3.20.98.175.
      According to the International number system, the number will be written as 32.098.175.

2. LARGE NUMBERS IN PRACTICE
Estimation Of The Numbers
The estimation of a number is a reasonable guess of the actual value. Estimation means approximating a quantity to the accuracy required. This is done by rounding off the numbers involved and getting a quick and rough answer. 
Rounding off a number to the nearest tens
The numbers 1, 2, 3 and 4 are nearer to 0. So, these numbers are rounded off to the lower ten. The numbers 6, 7, 8 and 9 are nearer to 10. So, these numbers are rounded off to the higher ten. The number 5 is equidistant from both 0 and 10, so it is rounded off to the higher ten. 
e.g.
(i) We round off 31 to the nearest ten as 30
(ii) We round off 57 to the nearest ten as 60
(iii) We round off 45 to the nearest ten as 50
Rounding off a number to the nearest hundreds
The numbers 201 to 249 are closer to 200. So, these numbers are rounded off to the nearest hundred i.e. 200. The numbers 251 to 299 are closer to 300. So, these numbers are rounded off to the higher hundred i.e. 300. The number 250 is rounded off to the higher hundred.
e.g.
(i) We round off 578 to the nearest 100 as 600.(ii)  We round off 310 to the nearest 100 as 300.
Rounding off a number to the nearest thousands
Similarly, 1001 to 1499 are rounded off to the lower thousand i.e.1000, and 1501 to 1999 to the higher thousand i.e. 2000. The number 1500 is equidistant from both 0 and 1000, and so it is rounded off to the higher thousand i.e.2000. 
e.g.
(i) We round off 2574 to the nearest thousand as 3000.
(ii) We round off 7105 to the nearest thousand as 7000.
Estimation of sum or difference:
When we estimate sum or difference, we should have an idea of the place to which the rounding is needed.
e.g.
(i) Estimate 4689 + 19316
We can say that 19316 >  4689
We shall round off the numbers to the nearest thousands.
19316 is rounded off to 19000
4689 is rounded off to 5000

Estimated sum:
19000 + 5000 = 24000
(ii) Estimate 1398 – 526
We shall round off these numbers to the nearest hundreds. 
1398  is rounded off to 1400
526 is rounded off to 500

Estimated difference:
1400 – 500 = 900

Estimation of the product:
To estimate the product, round off each factor to its greatest place, then multiply the rounded off factors. 
e.g.
Estimate 92 × 578
The first number, 92, can be rounded off to the nearest ten as 90.
The second number, 578, can be rounded off to the nearest hundred as 600.
Hence, the estimated product =90 × 600 = 54,000.
Estimating the outcome of number operations is useful in checking the answer.

Example 1
Round off 234 to the nearest ten.
Solution
The number 234 lies between 230 and 240. However. 234 is closer to 230. Therefore. 234 is rounded off to 230.

Example 2
Estimate 4504 by rounding off to the nearest thousand. Solution:
4000 and 5000 are two multiples of thousand between which the given number lies. Now. 4504 is closer to 5000. Therefore, the rounded off value of 4504 is 5000.

3. ROMAN NUMERALS
Hindu – Arabic number system:
Many years ago, Hindus and Arabs developed a number system called the Hindu–Arabic number system. It is the name given to the number system that we use today.

Roman numerals
It is the numeral system that originated in ancient Rome. This numeral system is based on certain letters, which are given values and are used as numerals. The following are the seven number symbols used in the Roman numeral system, and their values: 

Symbols

I

V

X

L

C

D

M

Value

1

5

10

50

100

500

1000

Seven letters of English alphabet, i.e. I, V, X, L, C, D and M, are used to represent Roman numerals.  Roman numerals do not have a symbol for zero. Roman numerals are read from left to right, and are arranged from the largest to the smallest. Multiplication, division and other complex operations were difficult to perform on Roman numerals. So Hindu-Arabic numerals were used. The Roman numerals are used in some clocks.
The Roman numerals for the numbers 1 – 15 are shown below:

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

I

II

III

IV

V

VI

VII

VIII

IX

X

XI

XII

XIII

XIV

XV

Rules for Roman numerals:

  • In Roman numerals, a symbol is not repeated more than thrice. If a symbol is repeated, its value is added as many times as it occurs. For example, if the letter I is repeated thrice, then its value is three.
  • The symbols V, L and D are never repeated.
  • If a symbol of smaller value is written to the right of a symbol of greater value, then its value gets added to the symbol of greater value. For example, in case of VI, I is written to the right of It means that 1 should be added to 5. Hence, its value is 6.
  • If a symbol of smaller value is written to the left of a symbol of greater value, then its value is subtracted from the symbol of greater value. For example, in case of IV, I is written to the left of V.  It means that 1 should be subtracted from 5.  Hence, its value is 4. 
  • The symbols V, L and D are never written to the left of a symbol of greater value, so V, L and D are never subtracted. For example, we write 15 as XV and not VX.
  • The symbol I can be subtracted from V and X only. For example, the value of IV is four and the value of IX is nine. 
  • The symbol X can be subtracted from L, M and C only. For example, X is subtracted from L to get 40, which is represented by XL.
  • If a symbol of smaller value is written between the symbols of greater value, then its value is subtracted from the symbol of the greater value which is written immediately after the symbol of the smaller value. For example the value of XIV is 10 + (5 – 1) i.e. 14.
  • If bar is written over a symbol, it’s value gets multiplied by 1000. For example, the value of  is 5 × 1000 i.e. 5000.

Example 1
Write the following numbers in Roman numerals.
(a) 29 (b) 45
Solution
(a) 29 = 20 + 9= 10+10 + 9
        = XX + IX = XXIX
(b) 45 = 40 + 5= (50-10) + 5
       = XL + V =XLV

Example 2
Write the following Roman numerals in Hindu-Arabic numerals
(i) LXXVII (ii) CXCIX (iii) DCLXIV
Solution
(i) LXXVII = L + XX + VII = 50 + 20 + 7 = 77
(ii) CXCIX = C + XC + IX = 100 + 90 + 9 = 199
(iii) DCLXIV = D + C + L + X + IV = 500 + 100 + 50 + 10 + 4 = 664

4. IMPORTANCE OF BRACKETS
Using brackets
Brackets help in simplifying an expression that has more than one mathematical operation.
If an expression that includes the brackets is given, then perform the operation inside the bracket and change everything into a single number. Then carry out the operation that lies outside the bracket.
e.g.
1. (6 + 8) × 10 = 14 × 10 = 140
2. (8 + 3) (9 – 4) = 11 × 5 = 55

Expanding brackets
The use of brackets allows us to follow a certain procedure to expand the brackets systematically.
e.g.
1. 8 × 109 = 8 × (100 + 9) = 8 × 100 + 8 × 9 = 800 + 72 = 872
2. 105 × 108 = (100 + 5) × (100 + 8)
     =  (100 + 5) × 100 + (100 + 5) × 8     
     = (100 × 100) + (5 × 100) + (100 × 8) + (5 × 8)                    
     = 10000 + 500 + 800 + 40= 11340.

Example 1
Write a situation for the expression 3× (8 – 5) where brackets are necessary.
Solution
The situation of the given expression can be written as follows.
Sahil bought 8 apples. Out of them. 5 apples were rotten. The cost of one apple was Rs 3. What was the cost of good apples?

Example 2
4 is multiplied with the difference of 100 and 70. Write the expression for the given statement using brackets and solve that expression.
Solution
The expression is 4 × (100 – 70). Now. 4 × (100-70) = 4× 30 = 120

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