Decimal Fractions

Decimal Fractions :

Fractions in which denominators are powers of 10 are called decimal fractions.
1/10, 1/100, 1/1000, … etc., are respectively the tenth, the hundredth and the thousandth part of 1.

5/10 is 5 tenths, can be written as 0.5, called decimal five;

4/100 is 4 hundredths, can be written as 0.04, called decimal zero-four;

15/1000 is 15 thousandths, can be written as 0.015, called decimal zero-one-five; and so on.

Rule – Converting a Decimal into a Vulgar Fraction :
Put 1 in the denominator, under the decimal point and annex with it as many zeros as is the number of digits after the decimal point. Remove the decimal point and reduce the fraction to its lowest terms.

Note 1 : Annexing zeros to the extreme right of a decimal fraction does not change its value.
(i.e.)0.2 = 0.20 = 0.200 etc.,
Note 2 : If numerator and denominator of a fraction contain the same number of decimal places, can remove the decimal sign.
(i.e.) 2.5/1.5 =25/15

Rule –
For Addition and Subtraction of Decimal Fractions :
The given Numbers are so placed under each. Other than the decimal point lies in one column. The numbers so arranged can be Added or Subtracted in a usual way.

For Multiplication :
Shift the decimal point to the right by as many places of decimal, as is the power of 10.
(i.e) 25.20369 X 100 = 2520.369

Multiply the given numbers considering them without the decimal point. In the product, the decimal point is marked off to obtain as many places of decimal as is the sum of the number of decimal places in the given numbers.

For Division :
Dividing a decimal fraction by a counting number :
Divide the given decimal numeral without considering the decimal point, by the given counting number. In the quotient, put the decimal point to give as many places of decimal as are there in the dividend.

Dividing a decimal fraction by a decimal fraction :
Multiply both the dividend and the divisor by a suitable multiple of 10 to make divisor a whole number. Then proceed as decimal fraction by a counting number.

Recurring Decimal :
In a decimal fraction, If a figure or set of figures is repeated continuously, such a number is called a Recurring Decimal.

If a single figure is repeated, it is expressed by putting a dot on it. If a set of figures is repeated, it is expressed by putting a bar on it.

Pure Recurring Decimal : A Decimal fraction in which all the figures after the decimal point are repeated, is called a Pure Recurring Decimal.

For Converting a Pure Recurring Decimal into a Vulgar fraction, write the repeated figure only once in the numerator, and take as many nines in the denominator as is the number of repeating figures.

Mixed Recurring Decimal : A Decimal fraction in which some figures do not repeat and some of them repeat, is called a Mixed Recurring Decimal.

For Converting a Mixed Recurring Decimal into a Vulgar fraction, in the numerator, take the difference between the number formed by all the digits after decimal point (taking the repeated digits only once) and the formed by non – repeating digits. In the denominator, take the number formed by as many nines as there are repeating digits, followed by as many zeros as is the number of non-repeating digits.

NUMBERS – IMPORTANT FACTS

NUMBERS: We use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 & 9 are called Digits, to represent any Number.

A Group of figures representing a number is called a Numeral.

Representation of a number in figures is called notation and expressing a number in words is called numeration.

Place value or Local value of a Digit in a Numeral:

Example: 657891232, in this Place value of 2 is (2X1) = 2; Place value of 3 is (3X10) = 30; Place value of 2 is (2X100) = 200; and so on.. then the place value of 6 is (6X10⁸ ) = 600000000

Face value: The face value of a digit in a numeral is the value of the digit itself at whatever place it may be. In the above example, the face value of 2 is 2, the face value of 3 is 3 and so on.

TYPES OF NUMBERS:

Natural Numbers : Counting numbers 1, 2, 3, …. are called Natural Numbers.

Whole Numbers: All Counting numbers together with zero form the set of Whole Numbers. Thus,

• 0 is the only whole number which is not a natural number.
• Every natural number is a whole number.

Integers: All Natural numbers, 0 and negatives of counting numbers (i.e.){…..,-3, -2, -1, 0, 1, 2, 3,….} together form the set of integers.

‘0’ is neither positive nor negative. So {0, 1, 2, 3,…} represents non-negative integers, while {0, -1, -2, -3 ….} represents the set of non-positive integers.

Even Numbers: A number divisible by 2 is called an even number. (Eg.) 2, 4, 6, …..

Odd Numbers: A number not divisible by 2 is called an odd number. (Eg.) 1, 3, 5, …..

Prime Numbers: A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself. Two numbers are said to be Co-Primes, if their H.C.F. is 1. Eg. : (2,3), (7,9), etc.,

Composite Numbers: Numbers greater than 1 which are not prime, are known as composite numbers. (Eg.) 4, 6, 8, 9, 10

• 1 is neither prime nor composite.
• 2 is the only even number which is prime.
• There are 25 prime numbers between 1 and 100.

Prime numbers upto 100 : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

How can we find prime numbers Greater than 100? :

Let p>100, First find a whole number nearly greater than the square root of p. Let k be the number which is greater than square root of p. Test whether p is divisible by any prime number less than k. If yes, then p is prime. Otherwise p is not prime.

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