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Negative Numbers

• The numbers with a negative sign and which lies to the left of zero on the number line are called negative numbers.

To know more about Application of Negative Numbers in Daily Life

Introduction to Zero

The number Zero

• The number zero means an absence of value.

The Number Line

Integers

• Collection of all positive and negative numbers including zero are called integers. ⇒ Numbers …, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, … are integers.

Representing Integers on the Number Line

• Draw a line and mark a point as 0 on it
• Points marked to the left (-1, -2, -3, -4, -5, -6) are called negative integers.
• Points marked to the right (1, 2, 3, 4, 5, 6) or (+1, +2, +3, +4, +5, +6) are called positive integers.

Absolute value of an integer

• Absolute value of an integer is the numerical value of the integer without considering its sign.
• Example: Absolute value of -7 is 7 and of +7 is 7.

Ordering Integers

• On a number line, the number increases as we move towards right and decreases as we move towards left.
• Hence, the order of integers is written as…, –5, –4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5…
• Therefore, – 3 < – 2, – 2 < – 1, – 1 < 0, 0 < 1, 1 < 2 and 2 < 3.

Addition of Integers

 Positive integer + Negative integer

• Example: (+5) + (-2) Subtract: 5 – 2 = 3 Sign of bigger integer (5): + Answer: +3
• Example: (-5) + (2) Subtract: 5-2 = 3 Sign of the bigger integer (-5): – Answer: -3

Positive integer + Positive integer

• Example: (+5) + (+2) = +7
• Add the 2 integers and add the positive sign.

Negative integer + Negative integer

• Example: (-5) + (-2) = -7
• Add the two integers and add the negative sign.

Properties of Addition and Subtraction of Integers

Operations on Integers

Operations that can be performed on integers:

• Addition
• Subtraction
• Multiplication
• Division.

Subtraction of Integers

• The subtraction of an integer from another integer is same as the addition of the integer and its additive inverse.
• Example: 56 – (–73) = 56 + 73 = 129 and 14 – (8) = 14 – 8 = 6

Properties of Addition and Subtraction of Integers

Closure under Addition

• a + b and a – b are integers, where a and b are any integers.

Commutativity Property

• a + b = b + a for all integers a and b.

Associativity of Addition

• (a + b) + c = a + (b + c) for all integers a, b and c.

Additive Identity

• Additive Identity is 0, because adding 0 to a number leaves it unchanged.
• a + 0 = 0 + a = a for every integer a.

Multiplication of Integers

• Product of a negative integer and a positive integer is always a negative integer. 10×−2=−20
• Product of two negative integers is a positive integer. −10×−2=20
• Product of even number of negative integers is positive. (−2)×(−5)=10
• Product of an odd number of negative integers is negative. (−2)×(−5)×(6)=−60

Properties of Multiplication of Integers

Closure under Multiplication

• Integer * Integer = Integer

Commutativity of Multiplication

• For any two integers a and b, a × b = b × a.

Associativity of Multiplication

• For any three integers a, b and c, (a × b) × c = a × (b × c).

Distributive Property of Integers

• Under addition and multiplication, integers show the distributive property.
• For any integers a, b and c, a × (b + c) = a × b + a × c.

Multiplication by Zero

• For any integer a, a × 0 = 0 × a = 0.

Multiplicative Identity

• 1 is the multiplicative identity for integers.
• a × 1 = 1 × a = a

Division of Integers

• (positive integer/negative integer)or(negative integer/positive integer)
  ⇒ The quotient obtained is a negative integer.
• (positive integer/positive integer)or(negative integer/negative integer)
  ⇒ The quotient obtained is a positive integer.

Properties of Division of Integers

For any integer a,

• a/0 is not defined
• a/1=a

Integers are not closed under division.

Example: (–9)÷(–3)=3 result is an integer but (−3)÷(−9)=−3−9=13=0.33 which is not an integer.
To know more about Number Lines

PLACE VALUE

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?

Solution:

Largest 4 digit number is 9837.

Hindu–Arabic numeral system 

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.

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.

Estimation And Approximation

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:

• The degree of accuracy required.
• The simplicity of computation.
• How quickly is the estimation completed?
• How quickly would the guessed answer be obtained?

Let us consider some of the examples:

Example 1:

Estimate the sum (69 + 73) as nearest ten.

Solution:

          Round off each given number to nearest ten.

69 = 70

73 = 70

Required sum = 70+ 70 = 140

Introduction to Decimals

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

Introduction to Fraction

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.1 Two 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.