Chapter 1: Knowing our Numbers (Class VI NCERT Maths)
Numbers
Numbers are used for counting and to keep record of data.
Comparison of Numbers

Out of two numbers, one having more number of digits is greater. For example if we take two numbers as 4544 and 544. Here first number (4544) is having 4 digits while second number (544) is having 3 digits. Hence 4544 is greater.
4544 (4 digit) > 544 (3 digit)

If number of digits are same, then we will check the first digit from left. For example if we take two numbers as 4544 and 5666. Here number of digits are 4 for both the numbers. But second number (5666) is having first digit from left as 5 while the first number (4544) is having 4 at same place.
5666 > 4544 (number of digits are same, digits at the first place from left to be compared)

If first digit from the left is same for both the numbers, then we will check the next digit and will repeat this procedure till we find the greater number out of two.
5666 > 5467 (number of digits and first digit from left are same, digits at the 2nd place from left to be compared)
5666 > 5643 (number of digits and 1st, 2nd digit from left are same, digits at the 3rd place from left to be compared)
5666 > 5661 (number of digits and 1st, 2nd, 3rd digit from left are same, digits at the 4th place from left to be compared)
Forming Greatest and Smallest numbers from the given digits
If we have four digits and want to form greatest and smallest numbers without repetition of any digit. Greatest number will be formed by putting all the digits left to right in decreasing order and smallest number will be formed by putting all the digits left to right in increasing order.
Example: Suppose we have four digits 7, 0, 5, 4 and we want to form greatest and smallest 4 digit numbers from these digits.

Greatest number  7540 (formed by putting the digits in decreasing order left to right)

Smallest number  4057 (formed by putting the digits in increasing order left to right i.e. 0457 but since 457 is a 3 digit number, smallest four digit number is formed by putting next greatest digit i.e. 4 at first place and 0 next to it)
Ascending and Descending order

Ascending order: Numbers are arranged smallest to greatest in increasing order.

Descending order: Numbers are arranged greatest to smallest in decreasing order.
Example: 466, 455, 7654, 987

Ascending order  455,466,987,7654

Descending order  7654,987,466,455
Ten Thousand and Larger Numbers
Smallest 5 digit number is 10000 (Ten Thousand) which is obtained by adding 1 in greatest four digit number i.e. 9999.
In any 5 digit number first digit from the left is called at ten thousands place, second digit at thousands place, third digit at hundreds place, fourth digit at tens place and fifth digit at ones place. These are called place value of digits.
Example: Suppose a 5 digit number is given as 98578.
In expanded form it may be written as
98578 = 9 x 10000 + 8 x 1000 + 5 x 100 + 7 x 10 + 8 x 1 (place value of each digit)
Largest 5 digit number is 99999.
Smallest 6 digit number is 100000 (1 Lakh) which is obtained by adding 1 in largest 5 digit number i.e. 99999.
Example: Suppose a 6 digit number is given as 865984.
In expanded form it may be written as
865984 = 8 x 100000 + 6 x 10000 + 5 x 1000 + 9 x 100 + 8 x 10 + 4 x 1 (place value of each digit)
Largest 6 digit number is 999999.
Smallest 7 digit number is 1000000 (Ten Lakh) , 8 digit number is 10000000 (1 Crore) and 9 digit number is 100000000 (Ten Crore).
Indian System of Numeration
Commas are used to aid in reading large numbers.
In Indian system of numeration, first comma is used after three digit from right (for reading thousands and ten thousands) and thereafter commas are put after every two digit (for reading lakhs and ten lakhs; crores and ten crores etc).
5,000 (Five Thousand)
50,000 (Fifty Thousand)
2,50,000 (Two Lakh Fifty Thousand)
22,50,000 (Twenty Two Lakh Fifty Thousand)
2,22,50,000 (Two Crore Twenty Two Lakh Fifty Thousand)
52,22,50,000 (Fifty Two Crore Twenty Two Lakh Fifty Thousand)
International System of Numeration
In international system of numeration large number are represented in millions, billions and trillions.
After ones, tens and thousands comes million, billion and trillion.
1 million = 1000 thousands
1 billion = 1000 million
1 trillion = 1000 billion
Accordingly commas are put after every digit from the right.
65,550,652,550 (Sixty Five Billion Five Hundred Fifty Million Six Hundred Fifty Two Thousand Five Hundred Fifty)
Writing number using placement boxes
Suppose you want to write forty two lakh fifty thousand seventy two in numeric form. Make placement boxes for one, tens, thousands, ten thousands, lakhs and ten lakhs place. With the help of these boxes, you can write the number easily without mistake.
TL(4) L(2) T Th(5) Th(0) H(0) T(7) O(2)
So the number is 42,50,072.
Exercise 1.1 (NCERT)
Measurement and Large Numbers
1) Length measurement
1 centimeter = 10 millimeters
millimeter is used for very small distances.
1 meter = 100 centimeters = 1000 millimeters
1 kilometer = 1000 meters = 1000 x 1000 = 1000000 mm
kilometer is used for large distances.
2) Weight measurement
1 kilogram = 1000 gram
1 gram = 1000 milligram
1 kilogram = 1000000 milligram
3) Volume Measurement
1 liter = 1000 milliliter
EXERCISE 1.2 (NCERT)
Estimation
An estimation is done by approximate numbers in cases where we can not accurately predict the actual number or answer is quickly required for making a dicision. A good estimation should be as close as possible to the exact number.
For example, if somebody asks that how much time you will take to finish all the exercises of Chapter 1 of mathematics. You can not tell the exact time needed for the same but approximately how much time will be required that you can estimate.
For estimation numbers are rounded off to the nearest tens, hundreds or thousands etc depending on the size of number and accuracy requirements. Rounding off is done in such a way that results are reasonably near to the actual value as well as estimation is quick.
Rounding off to nearest tens
14 is rounded off to 0 and 69 rounded off to 10. 5 being equidistant from 0 and 10 is rounded off to 10 as a practice.
Rounding off to nearest hundreds
149 is rounded off to 0 and 5199 is rounded off to 100. 50 is rounded off to 100 as a practice.
Rounding off to nearest thousands
1499 is rounded off to 0 and 501999 is rounded off to 1000. 500 is rounded off to 1000 as a practice.
Use of Brackets
Brackets are used to avoid confusion in counting. Suppose we want to do sum and multiplication of number respectively as follows:
3 x (70 + 40) = 3 x 110 = 330
Now if brackets are not put here and same is given as 3 x 70 + 40, it will give a totally different result (= 210 + 40 = 250).
Roman Numbers
The number system which we commonly use is called HinduArabic numeral system (1,2,3,4,5,6,7,8,9,0).
Roman numeral is one other system of numbers where numbers are represented as:
1  I
5  V
10  X
50  L
100  C
500  D
1000  M
Following rules are followed for formulation of numbers:

If a symbol is repeated, its value gets added. For example XX = 10 + 10 = 20.

No symbol is repeated more than 3 times.

V, L and D are never repeated.

If a symbol with smaller value is on the right of symbol with greater value, both gets added. For example XV = 10 + 5 = 15.

If a symbol with smaller value is on the left of symbol with greater value, its value gets substracted. For example IX = 10  1 = 9.