Factor and Multiples
Synopsis

- When two or more numbers are multiplied, a product is obtained. Each number is called a factor of the product. The product is called

- Multiples of a multiple of each of the numbers.

- The multiples which appear in both the lists of multiples of the given numbers are called their common multiples.

- A number is said to be the factor of another if it can divide the other completely. So, a factor is also called the divisor. The factors of a number are less than or equal to the number.
- 1 is a factor of every number. Every number is a factor of itself.
- A number is the greatest factor of itself and 1 is the least factor of a number.
- Every multiple of a number is greater than or equal to the number itself.
- Every number is a multiple of 1.
- Every number is a multiple of itself.
- There is no greatest multiple of a number.

- A number that has only 2 factors, that is, 1 and the number itself is known as a prime number, g., 2, 3, 5, 7, 11, ...

- A number which has more than two factors is called a composite number.
- g., 4, 6, 8, 9, . ..
- Number 1 has only one factor. It is neither prime nor composite. It is called a unique number.
- 2 is the only even prime number and also the smallest prime number.
- All prime numbers except 2 are odd numbers.
- All even numbers except 2 are composite numbers.

- 4 Two numbers which have no other common factor except 1 are called co-prime numbers. g., 3, 4; 4,7; etc.,

- Consecutive prime numbers with a difference of 2 between them are called twin primes.

Fractions
Synopsis
Fraction:

Unit fractions: Fractions with 1 or unity as the numerator are called unit fractions
e.g., \[\frac{1}{3},\frac{1}{5}\]etc.,
Mixed fraction: A fraction which is a combination of a whole number and a fraction is called a mixed fraction or mixed number
e.g., \[1\frac{3}{4},7\frac{1}{11}\]
Equivalent fractions: The fractions that represent the same part are called equivalent fractions.
Both \[\frac{2}{8}\] and \[\frac{1}{4}\] represent the same part of a whole. So \[\frac{2}{8}=\frac{1}{4}\].
Comparing fractions:
(a) Fractions with the same numerators: Of two fractions with a common numerator, the fraction that has a smaller denominator is greater.
e.g., \[\frac{3}{5}\]and \[\frac{3}{7}\]
Since, the numerators are the same, comparing their denominators, we get \[\text{5}<\text{7}\].
\[\therefore \] \[\frac{3}{5}>\frac{3}{7}\]
(b) Fractions with the same denominators: Of two fractions with a common denominator, the fraction that has a larger numerator is greater
e.g., \[\frac{2}{9}\]and \[\frac{7}{9}\]
Since the denominators are the same, comparing their numerators, we get \[7>2\].
\[\therefore \] \[\frac{7}{9}>\frac{2}{9}\]
(c) Fractions with different numerators and denominators: To compare fractions with different numerators and denominators, first convert them to get the same denominator by writing their equivalent fractions and then compare.
e.g., \[\frac{3}{4}\] and \[\frac{5}{6}\]
Step 1: Check if numerators or denominators are the same. The given fractions do not have the same numerators or denominators.
Step 2: Find equivalent fractions with common denominators. \[\frac{3\times 3}{4\times 3}=\frac{9}{12}\] and \[\frac{5\times 2}{6\times 2}=\frac{10}{12}\]
Step 3: Compare more...

- A fraction represents a part of a whole.
- In a fraction, two numerals are written one below the other separated by a line. The written above the line is called the numerator and the numeral below the line is denominator.

- In a fraction, the denominator tells us how many equal parts the whole has been divided into and the numerator tells us how many parts of the whole are being considered.

Note: if the numerator is equal to the denominator, the fraction represents a whole number, i.e., 1 |

Number Sense and Numeration
Numbers
Numbers are mathematical objects by which we express date, time, position, quantity etc.
Writing and Reading Numbers
Numbers are written using symbols or digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) called numerals.
For example, 3564 is a numeral in which four digits (3, 5, 6 and 4) are used. In this section, we will study two types of numeration.
(i) Indian system of numeration.
(ii) International system of numeration.
Indian System of Numeration
This method of numeration is based on the place value chart. This system is also known as Hindu - Arabic system.
Given below is the Indian place value chart

Period | Kharab | Arab | Crores | Lakhs | Thousands | Ones | ||||||

Places | ten Kharab (T-kh) 1000000000000 | Kharab (kh) 00000000000 | Ten Arab (T-A) 1000000000 | Arab (A) 100000000 | Ten Crores (T-C) 10000000 | Crores (C) 1000000 | Ten lakhs (T-L) 100000 | Lakhs (L) 10000 | Ten thousands (T-TH) 10000 | Thousands (TH) 1000 | Hundred (H) 100 | Ones (0) 0 |

- Example

Period | Arab | Crores | Lakhs | Thousands | Ones | |||||||||||||

Places | Ten Arab (T-A) (10000000000) | more...
Roman Numerals
Introduction
The numerals we use is commonly known as Indo-Arabic Numerals. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ______ etc., are example of indo -Arabic numerals in ancient time Romans developed a system of numerations (numbering) which is known as Roman Numerals. I, II, III, IV, V, VI, VII, VIII, IX, _______ etc. are example of Roman Numerals.
Roman numerals are formed by using the following symbols:
Operation on Numbers: Addition and Subtraction
Introduction
In our daily life, we come across many activities when we need to apply the method of addition and subtraction. We are aware of numbers and number system. Now we will discuss two simple algebraic operations, that is, addition and subtraction.
Addition
Addition is one of the very common arithmetic operation used in mathematics. Addition is the operation to know the total quantity, when two or more than two quantities are taken together.
- Example
- Example
- Example
- Example
Operation on Numbers - Multiplication and Division
Introduction
In this chapter we will study two important arithmetic operations "multiplication and division". Multiplication is repeated addition of a specific quantity, whereas division is a distribution of a quantity into some equal parts. Let us study them.
Multiplication
When a quantity is added to itself for a number of times, we use operation of multiplication to find the resulting quantity,
- Example
- Example
Fractions and Decimals
Fraction
Fraction is used to indicate a part of a whole. Fraction is written, for example, as\[\frac{a}{b}\]. The top number in a fraction is called numerator and the bottom number is called denominator of the fraction. Hence in the given example 'a' is numerator and 'b' is denominator.
Look at the shaded part in the following figures which has been represented by fractions:
- Example
- Example
- Example
- Example
Unitary Method
Unitary Method
Unitary method is a method under which a calculation is carried out to find the value of the number of items, by first finding the value of one item.
From daily life experience, we know that when we increase the quantity of articles, their cost increases and when we decrease the quantity of articles, their cost decreases. In other words, more articles have more value and less articles have less value.
Note: In unitary method:
(i) To get more value we multiply.
(ii) To get less value we divide.
- Example
- Example
- Example
- Example
- Example
Money
Introduction
We require a number of things in our day to day life. We buy these things from the market and in return we pay money as per the rate of the article. So money is of great importance to us.
Different countries use different currencies. Indian currency is known as rupees.
Short form of the rupees is Rs, written by the symbol Rs. We write 78 rupees as Rs. 78.
Conversion of Rupees into Paise
One rupee is equal to 100 paise, so to convert rupees into paise, we multiply the Rs. by 100.
- Example
- Example
- Example
- Example
- Example
- Example
- more...
Geometrical Figures
Introduction
In our day to day life we come across a number of objects. All the objects has a specific shape and size.
We recognize a number of objects by their shape. Therefore, to know about the objects and of their shapes is very important. In this chapter we will study about the shapes of different geometrical figures.
Point
To show a particular location, a dot (.) is placed over it, that dot is known as a point.
A is a point
Line Segment
Line segment is defined as the shortest distance between two fixed points. It has fixed length.
- Example
- Example
- Example
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