Knowing our Numbers
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Indian System: 73, 32, 54,329
International System: 733,254,329
Note: A Comma is inserted each period in both the systems.
Indian system

1 lakh

10 lakhs

1 Crore

10 Cores

100
Cores

International System

Hundred Thousands

1 Million

10 Millions

100 Millions

1 Billon

 To write numbers in the Indian System, we put a comma after three digits from the right. and then after every two digits.
e.g., 5340769 = 53, 40,769
 To write numbers in the International System, we put a comma after every three digits, starting from the right.
e.g., 5340769 = 5,340,769
The face value of a digit is the value of the digit itself.
e.g. In 7308, the face value of 7 is 7.
Place value = Face value \[\times \]Value of its place in the place value chart
e.g., (1) In 39065, the face value of 9 is 9 and the place value of 9 is 9 x 1000 = 9000.
e.g., (2) In 7308, the face value of 3 is 3; while its place value is 300.
Note: Place value of zero(0) at any place is always zero. Thus, the face value and the place value of zero are 0.
 Expanded form of a number:
To write a number in expanded form, start from the leftmost digit. Write the digit in the given number followed by as many zeroes as the number of digits to its right. Place a + sign before writing the next digit. Continue this until the rightmost digit is written.
e.g.. The expanded form of
906352146 = 900000000 + 00000000 + 6000000 + 300000 + 50000 + 2000 + 100 +40+6
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Whole Numbers
 Counting numbers are called natural numbers. Natural numbers are represented by N
\[\therefore \]N\[=\]{1,2,3,4,.........}
The least natural number is 1.
The greatest natural number does not exist as there is no end for natural numbers.
 Natural numbers along with 0 are called whole numbers.
Whole numbers are represented by W.
\[\therefore \]W= {0, 1, 2, 3, 4,……}
The least whole number is 0.
The greatest whole number does not exist as there is no end for whole numbers.
 Successor: A number which comes immediately after a given number is called its successor.
Successor = given number + 1
 Predecessor: A number which comes immediately before a given number is called its
Predecessor= given number – 1
 Number line: A line on which we locate numbers at equal intervals.
 Properties of Addition of Whole Numbers:
(i) Closure property: If 'a' and 'b' are any two whole numbers, then a + b is also a whole number.
e.g., 5 and 13 are whole numbers. Their sum 5 + 13 = 18 is also a whole number.
Hence whole numbers are closed under addition.
(ii) Commutative property: If 'a' and 'b' are any two whole numbers, then a + b = b + a. The order of addition of two whole numbers does not affect their sum.
e.g., 3+11 =14=11+3
Hence whole numbers satisfy the commutative property under addition.
(iii) Associative property: If 'a', 'b' and 'c' are any three whole numbers, then
(a + b) + c = a + (b + c) = (a + c) + b
e.g., (2 + 3) + 4 = 9 = 2 + (3 + 4) = (2 + 4) + 3
Hence whole numbers satisfy the associative property under addition.
(iv) Additive identity: If 'a' is any whole number then a+0=0+a=a. Since the sum is the same as the number considered, 0 is called the identity element for addition of whole numbers.
e.g., 9+0=0+9=9
Note: For subtraction of whole numbers closure property. Commutative. Property, identity and associative properties are not applicable.
 Properties of Subtraction of Whole Numbers:
(i) If 'a' and 'b' are two whole numbers such that a > b (or) a = b then a  b is a whole number. If a < b, then a  b is not a whole number.
(ii) If 'a' and 'b' are two whole numbers such that a\[\ne \] b, then a  b\[\ne \] b  a.
(iii) If 'a' is any whole number, then a  0 = a, but 0 more...
Playing with Numbers
 A factor of a number is an exact divisor of that number.
 A number is said to be a multiple of any of its factors.
e.g.. We know that 35 = 1\[\times \]35 and 35 = 5 \[\times \] 7.
This shows that each of the numbers 1, 5, 7 and 35 divides 35 exactly.
Therefore 1, 5, 7 and 35 are all factors of 35 and 35 is a multiple of each one of the numbers
1, 5, 7 and 35.
 All multiples of 2 are called even numbers.
e.g., 2, 4, 6,8,10, etc.
 Numbers which are not multiples of 2 are called odd numbers.
e.g., 1,3,5,7,9,11, etc.
 Each of the numbers which has exactly two distinct factors, namely 1 and itself is called a
prime number.
e.g., 2,3,5,7,11,13,17,19,23,29 etc.
 Numbers having more than two factors are known as composite numbers.
e.g., 4, 6, 8,9,10 etc.
Note: (i) 1 is neither prime composite. (since 1=1, the two factors are not distinct.)
(ii) 2 is the lowest prime numbers.
(iii) 2 is the only even prime number. (All other even number are composite numbers.)
 Two consecutive prime numbers differing by 2 are known as twinprimes.
e.g., (i) 3, 5 (ii) 5, 7 (iii) 11, 13 etc.
A set of three consecutive prime numbers, differing by 2, is called a prime triplet.
An example of prime triplet is (3, 5, 7).
 If the sum of all the factors of a number is twice the number then the number is called a perfect number,
e.g., 6 is a perfect number, since the factors of 6 are 1,2,3,6 and (1 + 2 + 3 + 6) = (2 x 6).
 Two numbers are said to be coprime if they do not have a common factor other than 1.
e.g., (i) 2, 3 (ii) 3, 4 (iii) 8, 15
Note: (i) Two prime numbers are always co – prime.
(ii) Two co – prime need not be prime numbers.
e.g., 6, 7 are coprimes, while 6 is not a prime number
 Every even number greater than 4 can be expressed as the sum of two odd prime numbers.
e.g., (i) 6 = 3 + 3 (ii) 8 = 3+5
Tests of divisibility of numbers
 Test of divisibility by 2:
 A number is divisible by 2, if its units digit is 0,2,4,6 or 8.
e.g., 42,84,120,1456,568 etc. are divisible by 2.
 Test of divisibility by 3:
A number is divisible by more...
Basic Geometrical Ideas
 Point: A dot marked on a paper with the sharp tip of a pencil is called a point.
 A point is denoted by a single capital letter such as A, B, C etc. A point has only a position.
 Line: A countless number of points placed closely form a line. It extends indefinitely on both sides.
Two points on a line determine it. A line is denoted as\[\overleftrightarrow{~AB}\].
 Line segment: If A and B are two points on a line, the part AB of the line is called the line segment AB, denoted as\[\overline{AB}\]. It has 2 end points  A and B. Sometimes a line is denoted by small case letters such as /, m etc.
 The line AB (\[\overleftrightarrow{AB}\]) is the same as the line BA (\[\overleftrightarrow{BA}\] ). The line segment AB is same as the line segment BA.
Two line segments \[\overline{AB}\]and \[\overline{CD}\]are said to be congruent if they are of equal length.
 Ray: The part of a line that extends indefinitely in one direction from a given point 0 is called a ray. A ray has only one end point, 0. Ray \[\overline{OA}\]is denoted as o/T. It extends in the direction of A.
 \[\overleftrightarrow{OA}\]and \[\overleftrightarrow{AC}\] are two different rays. The first letter in the representation gives the initial point of the ray.
 A ray has no definite length, as it can be extended indefinitely in a direction.
 An unlimited number of rays can be drawn with the same initial point.
 Angle: The part of the plane between two rays with the same initial point is called an angle. The common end point is called the vertex and the two rays are called the arms of the angle. An angle is denoted with the vertex in between two points on the arms of the angle.
\[\therefore \]The angle shown in the figure is \[\angle \]AOB •
It is same as more...
Integers
 The set of positive numbers, 0 and negative numbers is called the set of integers.
The set of integers is denoted by I or Z.
Z ={.....,3,2,1, 0,1,2,3,......}
 Natural numbers are contained in the whole numbers. Therefore, N (= W.)
 Whole numbers are contained in the integers.
Therefore, We Z.
 Negative numbers are placed at the bottom of 0 on the vertical number line. Zero is neither positive nor negative.
 To represent quantities like profit, income, increase, rise, high, north, east, above, depositing climbing and so on, positive numbers are used.
 To represent quantities like loss, expenditure, decrease, fall, low, south, west below withdrawing, sliding and so on, negative numbers are used.
(a) Every positive integer is larger than every negative integer and zero.
(b) Zero is less than every positive integer.
(c) Zero is larger than every negative integer.
(d) Zero is neither negative nor positive.
(e) Farther a number from zero on the right, larger is its value.
(f) Farther a number from zero on the left, smaller is its value.
(g) When no sign is given before a number, it is considered as a positive number.
(h) It is compulsory to write a '' (minus) sign before a negative number.
(h) Usually, negative numbers are placed within simple brackets to avoid confusion arising due to the operator before it.
(a) Comparing numbers of different signs: A positive number is always greater than a negative number and zero.
e.g., 6 >  12; 6 > 0 etc.
(b) Comparing two positive integers: The positive number with larger value is larger and that with the smaller value is smaller.
e.g., 73 > 57; 45 > 36 etc.,
 Comparing two negative integers: The negative number with larger value is smaller and that with smaller value is larger.
e.g.,6 <  2;  12 > 18 etc.,
Note: If a a > b then – a <  b.
eg., 8 > 5 => 8 <5
 Addition on the number line:
(a) Adding a positive number to a given number is represented by moving to the right on the number line.
e.g.,  4 + 5 = 1
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Fractions
 A fraction is a part of a whole.
 In the fraction \[\frac{6}{7}\] ,6 is called the numerator and 7 is called the denominator.
 The denominator is the number of equal parts into which a whole is divided.
 The numerator is the number of parts considered of the whole.
Like fractions: Fractions with the same denominators.
e.g., \[\frac{4}{5},\frac{6}{5},\frac{3}{5}\]
Unlike fractions: Fractions with different denominators,
e.g.,\[\frac{1}{2},\frac{9}{4},\frac{3}{7}\]
Proper fractions: Fractions in which the denominator is greater than the numerator.
e.g.,\[\frac{2}{9},\frac{5}{6},\frac{2}{3}\]
Improper fractions: Fractions in which the numerator is greater than or equal to the denominator.
e.g., \[\frac{9}{2},\frac{6}{5},\frac{3}{2},\frac{7}{7}\]
Mixed fractions: Fractions with a whole number part and a fractional part are called mixed fractions.
e.g.,\[1\frac{1}{2},2\frac{2}{3},3\frac{1}{4}\]
 Fractions can be represented on the number line. Every fraction has a point associated with it on the number line.
 Conversion of an improper fraction to a mixed fraction
\[\frac{13}{5}=2\frac{3}{5}\left[ Q\frac{R}{D}\,from \right]\]
 Conversion of a mixed fraction to an improper fraction
De = Denominator, Nu = Numerator, WN = Whole Number
\[3\frac{1}{4}=\frac{\left( De\times WN \right)+Nu}{De}=\frac{\left( 4\times 3 \right)+1}{4}=\frac{12+1}{4}=\frac{13}{4}\]
 Equivalent fractions: All fractions that have the same value are called equivalent fractions.
 Equivalent fractions of a given fraction can be written by multiplying (or dividing) the numerator and the denominator by the same number.
 Simplification of fractions: Reducing a fraction to its lowest terms is called simplification of the fraction. Dividing the numerator and the denominator of a fraction by a common factor reduces it into its lowest terms. The H.C.F of the numerator and denominator of a fraction in its simplest form is 1.
 (a) Comparing like fractions: Among like fractions. The fraction with greater numerator is greater.
(b) Comparing unlike fractions: Unlike fractions are first converted into like fractions by writing their equivalent fractions and then compared.
e.g., Compare \[\frac{a}{b\,}\,and\,\frac{c}{d}\]
(i) If ad > bc, then \[\frac{a}{b\,}\,>\,\frac{c}{d}\].
(ii) If ad < bc, then \[\frac{a}{b\,}\,<\frac{c}{d}\]
(iii) If ad = be, then \[\frac{a}{b\,}\,=\frac{c}{d}\]
 Add or subtract like fractions: To add or subtract two or more like fractions, we add or subtract the numerators and write the result over the common denominator.
e.g.,\[\frac{7}{8}+\frac{3}{8}\frac{5}{8}=\frac{7+35}{8}=\frac{105}{8}=\frac{5}{8}\]
 Add or subtract unlike fractions:
To add or subtract two or more unlike fractions, we change them to like fractions and then add or subtract. To add or subtract unlike fractions, follow the steps given below.
(i) Change the mixed fractions (if any) to improper fractions.
(ii) Change all the fractions into like fractions (by taking L.C.M. of the denominators).
(iii) Add or subtract the numerators more...
Decimals
 The number expressed in decimal form are called decimal numbers or simply decimals.
e.g., 5.8, 19.63, 7.269, 0.058, etc.,
 A decimal number has two parts, namely
(i) Wholenumber part and (ii) Decimal part
These parts are separated by a dot (.) called the decimal point.
 Place value chart of a decimal number:
e.g., Write 236.784 in place value chart.

No. of digits

Smallest

Largest

1

0

9

2

10

99

3

100

999

100

10

1

0.1

0.01

0.001

Hundreds

Tens

Ones

Tenths

Hundredths

Thousandths

2

3

6

7

8

4

 Representing decimals on number line: Let us represent 0.6 on the number line.
0.6 lies between 0 and 1.
0.6 is 6 tenths.
So, divide the unit length between 0 and 1 into 10 equal parts & take 6 parts as shown.
 The number of digits in the decimal part of a decimal number gives the number of decimal places.
e.g., 5.39 has two decimal places and 9.368 has three decimal places.
 Decimals having the same number of decimal places are called like decimals.
e.g., 9.82, 5.03, 13.85 etc.,
 Decimals having different number of decimal places are called unlike decimals.
e.g, 3.4, 5.98, 111.035 etc.,
Note: Adding and number Zeros to the extreme right of the decimal part of a decimal number does not change its value.
e.g., 2.64 = 2.640 = 2.6400 etc.
Step 1 : Convert the given decimal numbers into like decimals.
Step 2 : First compare the wholenumber part.
The decimal number with the greater whole number part is greater.
Step 3: If the whole number parts are equal, compare the tenths digit. The decimal number with the greater digit in the tenths place is greater.
Step 4: If the tenths digits are also equal, compare the hundredths digits, and so on.
e.g., (i) 63.87 > 59.87
(ii) 24.85 > 24.65
(iii) 111.035 < 111.038
 Conversion of decimals to fractions:
Express 2.35 as a fraction.
Step 1: In the numerator, write the given decimal number without the decimal point.
\[2.\underset{\leftarrow }{\mathop{35}}\,=\frac{(235)}{(\,\,\,\,\,)}\]
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