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 and Place value
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
We often need to compare the two or more numbers. Here are some of the ways it can be performed easily
1) If two numbers have unequal number of digits, then the number with the greater number of digits is greater.
2) 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.
Example a) Comparison between 358 and 4567 Answer
Here 4567 has four digits and 358 has three digits, so clearly 4657 is greater than 358
b) Comparison between 1345 and 2456 Answer
Here both the number have same digit, So we need start looking at the extreme left digit
1345 -> 1
2456 -> 2
Now 2 > 1
So we can clearly state 2456 > 1345 c) Comparison between 4345 and 4656 Answer
Here both the number have same digit, So we need start looking at the extreme left digit
4345 -> 4
4656 -> 4
As they are same, we need start looking at second extreme left digit
4345 -> 3
4656 -> 6
Now 6> 3
So we can clearly state 4656 > 4345
Some important terms
The arrangements of numbers from the smallest to the greatest is called ascending order.
The arrangement of numbers from the greatest to the smallest is called descending order.
Example
Arrange the following in ascending order: 5392, 5782, 5789, 5654 Answer
The above rules of comparison can be applied here also.
Here all the number are of same number of digit and extreme left digit is also same. So we need to look at second extreme left digit
So 5782,5789 > 5654 > 5392
Now in 5782,5789, unit place makes the comparison easier
5789> 5782 > 5654 > 5392
So ascending order would be
5392 < 5654 < 5782 < 5789
Place Value of digit
Let’s discuss the place value of digits in the number and how a number can be written in that form 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. Example
5,46,851 = 5 × 1,00,000 + 4 × 10,000 + 6 × 1,000 + 8 × 100 + 5 × 10 +1 × 1
This number has 1 at one’s place, 5 at tens place, 8 at hundreds place, 6 at thousands place, 4 at ten thousands place and 1 at lakh place.
Number Name are also written based on the place value name. So Its number name is Five lakh forty-six thousand eight hundred fifty-one
We can use below table format for easily reading and writing the Number
Use of Commas
Commas added to numbers help us read and write large numbers easily. As per Indian Numeration, Commas are used to mark thousands, lakhs and crores. The first comma comes after hundreds place (three digits from the right) and marks thousands. The second comma comes two digits later (five digits from the right). It comes after ten thousand place and marks lakh. The third comma comes after another two-digits (seven digits from the right). It comes after ten lakh place and marks crore
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. Use of Commas
As per International Numeration, Commas are used to mark thousands and millions. It comes after every three digits from the right. The first comma marks thousands and the next comma marks millions. For example, the number 10,101,592 is read in the International System as tem million one hundred one thousand five hundred ninety-two. In the Indian System, it is 1 crore one lakh one thousand five hundred ninety-two.
Estimation of the Numbers
This topic is not included in Rationalized syllabus
A reasonable guess of the actual value is called an estimate.
A quick, rough estimate of the result of number operations can be done by rounding off the numbers is involved. Rules of Estimation
1)Estimating numbers to the nearest tens is done by rounding off numbers 1, 2, 3 and 4 to 0 and number 6, 7, 8, 9 to 10.
2) Estimating numbers to the nearest hundreds is done by rounding off numbers 1 to 49 to 0 and numbers 51 to 99 to 100.
3) Estimating numbers to the nearest thousands is done by rounding off numbers 1 to 499 to 0 and the numbers 501 to 999 to 1000.
Estimation involves approximating a quantity to an accuracy required. We can apply the above rules depending on the accuracy required.
We can estimate Sum, difference and Multiplication by applying the rules of estimation also. We can apply the above rules depending on the accuracy required and how quickly answer can be find out
Roman Numerals
This topic is not included in Rationalized syllabus
Roman Numerals system is another system used apart of Hindu-Arabic system.
The Roman numerals are
I
1
II
2
III
3
IV
4
V
5
VI
6
VII
7
VIII
8
IX
9
X
10
X1
11
X11
12
XX
20
L
50
C
100
D
500
M
1000
Rules of the system
1) In Roman numerals a symbol is not repeated more than three times, but the symbols V, L and D are never repeated.
2) Roman numerals are read from left to right and the letters of Roman numerals are arranged from the largest to the smallest.
3) If a symbol of smaller value is written to the right of a symbol of greater value, then its value gets added to the value of greater symbol.
VI = 5 + 1 = 6
4) If a symbol of smaller value is written to the left of a symbol of greater value, its then value is subtracted from the value of the greater symbol.
IV = 5 – 1 = 4
5)The symbol I can be subtracted from V and X only.
The symbol X can be subtracted from L, M and C only.
Importance of Brackets
This topic is not included in Rationalized syllabus
Brackets help in simplifying an expansion with more than one mathematical operation.
In an expression that includes brackets, the numbers inside the brackets must be simplified into a single
