Place value is one of the fundamental concepts in mathematics.It is important as it helps students to understand the meaning of a number. Place value is needed to understand the order of numbers as well. The concept that numbers can be broken apart and put back together gives the student a better understanding of how different mathematical operations work. It will be easy for the student to carry out operations such as addition, subtraction, multiplication, division, expanded notation, etc.
Place value of any digit is the value of digit according to its position in the number.
Place value of a digit depends upon the position it occupies in the number.
Largest number of n digit + 1 = smallest number of (n + 1) digits.
Smallest number of n digit – 1 = Largest number of (n – 1) digits.
A concrete number is a number which refers to a particular unit and is meaningful such as 8 meters, 12 kg etc.,
An abstract number is a number which does not refer to any particular unit such as 8, 12 etc.,
Let us consider some examples:
Example 1:
Write the place value of both the six in the number 36268 and find the sum of these values.
Solution:
In 36268, place value is 6000.
The other 6 at ten’s place, so its place value is 60.
Sum = 6000 + 60 = 6060
Example 2:
Write the place value of both the five in number 9,45,582 and find the difference of these place values.
Solution:
In 9,45,582, place value is 5000. The other 5 at hundred’s place, so its place value is 500.
Required sum = 5000 – 500 = 4500.
Example 3:
Find the place value of 7 in number 5731?
Solution:
Place value of 7 is 700.
Example 4:
Write the largest 4 digit number having 3 in tens place?
Before the invention of numbers, counting was done using some sort of physical objects such as pebbles or sticks. The numbers came into existence, eventually and then the need for adapting to a standard system of counting.
The Hindu–Arabic numeral system also known as the Arabic numeral system or Hindu numeral system, is a positional decimal numeral system. It is the most common system for the symbolic representation of numbers in the world. It was invented between the 1st and 4th centuries by Indian mathematicians. The system was adopted in Arabic mathematics by the 9th century. The system later spread to medieval Europe.
The system is based on ten different symbols. The symbols in actual use are descended from Brahmi numerals and have split into various typographical variants.
Today, this numerical system is still used worldwide.
Hindu Arabic system of numeration:
In Indian number system, ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are used to write numeral. Each of this number is called a digit.
Values of the places in the Indian system of numeration are Ones, Tens, Hundreds, Thousands, Ten thousand, 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.
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.
Periods
Crores
lakhs
Thousands
Ones
Places
Tens
Ones
Tens
Ones
Tens
Ones
Hundreds
Tens
Ones
Example 1:
Using Hindu-Arabic system, read the number 850746
Solution:
850746 – Eighty Lakh Fifty Thousand Seven Hundred Forty Six. Place value chart is as shown in the picture below on the left:
Example 2:
Write four crores fifteen lakh fifty thousand five hundred twenty seven in the numeral form using the Hindu-Arabic system.
Solution:
Number – 4,15,50,527
The international system of numeration:
A numeral system (or system of numeration) is a system for expressing numbers; using digits or other symbols in a consistent manner. The number the numeral represents is called its value. The most commonly used system of numerals is the Hindu–Arabic numeral system which was invented by Indian mathematicians.
The International number system is another method of representing numbers. In the International numbering system also, different periods are formed to read the large numbers easily. The periods used here are ones, thousand and millions, etc.
The international system of numeration:
As per the International numeration system, the first comma is placed after the hundreds place. Commas are then placed after every three digits.
The 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 thousand, 1 billion = 1000 millions.
Shown on the left side is a chart with the International number system. Shown on the right side is a chart with a comparison between Indian and International system:
Let us consider some examples.
International system – table
Indian system – table
Let us consider some examples:
Example 1:Using the International system, write the number; Six million, four hundred and eleven thousand, two hundred and sixty.
Solution: 6,411,260.
Example 2: Using the international system, read the number 7456123.
Solution: 7456123- Seven million Four Hundred Fifty six thousand one hundred twenty three.
Example 3: Write the number in words: 12,367,169.
Solution: Twelve million, three hundred and sixty-seven thousand, one hundred and sixty-nine.
Example 4: Write seven hundred forty three million eight hundred thirteen thousand two hundred fifty six in the numeral form using the international system.
Solution: 743,813,256.
Similarly, 48670002 can be read as
48,670,002 – Forty eight million, six hundred and seventy thousand and two.
Data: A collection of numbers gathered to give some information.
Recording Data: Data can be collected from different sources.
Pictograph: The representation of data through pictures of objects. It helps answer the questions on the data at a glance.
Bar Graph: Pictorial representation of numerical data in the form of bars (rectangles) of equal width and varying heights.
We have seen that data is a collection of numbers gathered to give some information.
To get a particular information from the given data quickly, the data can be arranged in a tabular form using tally marks.
We learnt how a pictograph represents data in the form of pictures, objects or parts of objects. We have also seen how to interpret a pictograph and answer the related questions.
We have drawn pictographs using symbols to represent a certain number of items or
things.
We have discussed how to represent data by using a bar diagram or a bar graph. In a
bar graph, bars of uniform width are drawn horizontally or vertically with equal
spacing between them. The length of each bar gives the required information.
To do this we also discussed the process of choosing a scale for the graph.
For example, 1 unit = 100 students. We have also practiced reading a given bar graph. We have seen how interpretations from the same can be made.
We have discussed multiples, divisors, factors and have seen how to identify factors and multiples.
We have discussed and discovered the following:
(a) A factor of a number is an exact divisor of that number.
(b) Every number is a factor of itself. 1 is a factor of every number.
(c) Every factor of a number is less than or equal to the given number.
(d) Every number is a multiple of each of its factors.
(e) Every multiple of a given number is greater than or equal to that number.
(f) Every number is a multiple of itself.
We have learnt that –
(a) The number other than 1, with only factors namely 1 and the number itself, is a prime number. Numbers that have more than two factors are called composite numbers. Number 1 is neither prime nor composite.
(b) The number 2 is the smallest prime number and is even. Every prime number other than 2 is odd.
(c) Two numbers with only 1 as a common factor are called co-prime numbers.
(d) If a number is divisible by another number then it is divisible by each of the factors of that number.
(e) A number divisible by two co-prime numbers is divisible by their product also. We have discussed how we can find just by looking at a number, whether it is divisible by small numbers 2,3,4,5,8,9 and 11. We have explored the relationship between digits of the numbers and their divisibility by different numbers.
(a) Divisibility by 2,5 and 10 can be seen by just the last digit.
(b) Divisibility by 3 and 9 is checked by finding the sum of all digits.
(c) Divisibility by 4 and 8 is checked by the last 2 and 3 digits respectively.
(d) Divisibility of 11 is checked by comparing the sum of digits at odd and even places. We have discovered that if two numbers are divisible by a number then their sum and difference are also divisible by that number.
We have learnt that –
(a) The Highest Common Factor (HCF) of two or more given numbers is the highest of their common factors.
(b) The Lowest Common Multiple (LCM) of two or more given numbers is the lowest of their common multiples.
When someone gets you a surprise gift you are likely to estimate its approximate cost. Thus, in English estimation is the rough calculation of the value or the extent of something. Similarly, an approximation is a value that is nearly correct but not exact.
In mathematics, you will come across many situations where in you need to estimate.
Estimation means to make a judgment of quantities, approximate calculation of size, cost, population etc.
Approximation means almost correct amount. The word approximation is derived from Latin approximatus, from proximus meaning very near and the prefix, ap- meaning to.
In this chapter, you will learn about estimation and approximation.
The procedure of estimation depends upon the following:
Fractional numbers whose denominators ar 10 , 100, 1000 etc, are called decimal fractions or decimals. The dot “ . “ is called the decimal point. For Example : (i) 3 10 is expressed as 0.3 (ii) 1 100 is expressed as 0.01 II. Conversion of Decimals Example : convert the following fraction into decimals (i)
Solutions : 8÷3 = 0.375
= 5.375
III . Conversion of Units : 1 kilometre = 1000 metres 1 metre = 100 centimetres 1 centimetre = 10 milimetres 1 decimetre = 10 centimetre 1 metre = 10 decimetres 1 hectomere = 10 decametres 1 Kilometre = 10 hectometres
III . Conversion of Mass : 1 kilometre = 1000 grams 1 grams = 100 centigrams 1 centigram = 10 miligrams 1 decigram = 10 centigrams 1 gram = 10 decigram 1 hectogram = 10decagrams 1 Kilogram = 10hectograms
A fraction means a part of a whole (group or region). Every fraction has a numerator and a denominator. In the fraction 3/5 , 3 is the numerator part and 5 is the denominator part.
Types of Fraction:- (i) Proper Fraction: Fraction in which the numerator is less than the denominator, is called proper fraction. For eg: 4/5 , 6/11, 999/1000. (ii) Improper Fraction: Fraction in which the numerator is either equal to or greater than the denominator, is called the improper fraction. For eg: 3/5 , 6/5, 1000/999. (iii) Like Fraction: Fraction having the same denominator are called like fractions. For eg: 3/5, 5/5, 6/5 (iv) Unlike Faction: Fraction having different denominators are called unlike fractions. For eg: 3/5, 4/3, 4/7 (v) Mixed Fraction : Mixed Fraction like 1 4 1 (vi) Equivalent Fraction:Two or more factions having the same value or representing the same part of whole are called equivalent fraction. Example. Write an equivalent fraction of 4/5 with numerator as 12. Solution:- 1 st Method : To get 12 as the numerator, we have to multiply 4 by 3. Therefore, denominator 5 should also be multiplied by 3.
So, 4/5 = 4 x 3 5 x 3 = 12/15 2 nd Method : 4 12 5 ? = 4 x ? = 12 x 5 12 x 5 4 = 15 So, 4/5 = 12/15 Hence, 12 / 15 is an equivalent fraction of 4/5 Example : Meera cuts 54 m of cloth into some pieces, each of length 3 3/8 meters. How many pieces does she get. Solution : Total length of clothe = 54m Length of each piece = 3 3/8 m = 27/8 m No. of pieces formed = 54 / 27/8 = 54 x 8 /27 = 16 Hence , 16 pieces each of length 3 3/8 m can be cut down from the cloth of 54 m length.
Conceptual division:- • Introduction of Set : A set is a collection or group of objects/elements which have a similar characters.
• Way of Representing Set a. Elements of a Set b. Properties of Set
• Representation of Set a. Listing Method (Roster form) b. Rule Method ( Set-builder form)
• Types of Set a. Equal Sets b. Empty Set or Null Set or Void Set c. Non-Empty Set or Overlapping Set d. Singleton Set e. Finite Set f. Infinite Set g. Null or Empty or Void Set h. Equivalent Set i. Disjoint Set j. Cardinality of Set
• Examples : • Disjoint Set
Q.1Two sets are given by A = x , y, z and B = 1,2,3,4,5
Solution : Clearly , sets A and B have no element common to both . Therefore sets A and B are disjoint sets.
Q.2 Find the cardinality of the set A = 2 , 4,8,10,12
Solution : Since , set A = 2, 4, 8, 10, 12 has total 5 elements. Hence , the cardinality of the set A is 5.
