__Number System__

Quantitative aptitude deals mainly with the different topics in arithmetic,which is the science which deals with the relations of numbers to one another.it includes all the methods that are appreciable to numbers.

Numbers are expressed by means of figures – 1,2,3,4,5,6,7,8,9, and 0 are called digits.out of these , 0 is called insignificant digit whereas the others are called significant digits.

**Numerals** – A group of figures , representing a number is called a numeral. Numbers are separated into the following types :

**Natural Number** – Numbers which we use for counting the objects are known as natural numbers.They are denoted by ‘N’ N = {1,2,3,4,5….}

**Whole Number** – These are the numbers that consist of zero and all the positive integers. They are represented by the symbol ‘W’.W = {0,1,2,3,4,5,…..}

**Prime Numbers** – A number other than 1 is called a prime number if it is divisible only by 1 and itself.

**Ex. 1. Is 349 a prime number ?**

Ans – The square root of 349 is approximately 19. The prime numbers less than 19 are 2,3,5,7,11,13,17.

Noticeably, 349 is not dividable by any of them. Therefore, 349 is a prime number.

**Ex. 2. I s 881 a prime number ?**

Ans – The approximate sq root of 881 is 30.

Prime numbers less than 30 are 2,3,5,7,11,13,17,19,23,29. 881 is not dividable by some of the above numbers, thus it is a prime **number.**

**Ex. 3. Is 979 a prime number ?**

Ans – The approximate sq foot of 970 is 32. Prime number less than 32 are 2,3,5,7,11,13,17,19,23,29,31.

We observe that 979 is divisible by 11, so it is not a prime numbers.

**Composite Numbers** – A number , other 1 , which is not a prime number is called a composite number.

e.g – 4,6,8,9,12,14

**Even Number – **The number which is dividable by 2 is recognized as an even number.

e.g – 2,4,8,12,24,28…..

**Odd Number** – The number which is not dividable by 2 is recognized as an odd number.

e.g – 3,9,11,17,19,….

**Consecutive Numbers** – A series of numbers in which each is greater than that which precedes it by 1 is called a series of consecutive numbers.

e.g – 6,7,8 or, 13,14,15,16 or 101,102,104,104

**Integers – **The set of numbers which contains of whole numbers and negative numbers is recognized as a set of integers. It denoted by I.

e.g – I = {-4,-3,-2,-1,0,1,2,3,,4,5..}

**Rational Numbers – **When the numbers are written in fractions, they are known as rational numbers.

e.g – ½, 3/4, 8/9, 13/15 are rational numbers.

**Irrational Numbers** – These are real numbers that cannot be expressed as a fraction of two integers, that is, p/q, where p and q are both integers.

e.g – √3 = 1.732….,√2 =1.414…

**Real Numbers** – Real numbers contain both rational as well as irrational numbers.

__Rule 1__

If we add the same number to both the top and bottom parts of the fractions, the fraction that comes at the end is the largest.

**Ex.1. – Which one of the following fractions is the greatest ?**

Ans – We see that numerator as well as denominators of the above fractions increase by 1, so the last fraction ,i.e 5/6 is the greatest f**raction.**

**Ex.2. – Which one of the following fractions is the greatest ? 2/5, 2/7 and 6/9**

Ans – We see that numerator as well as denominators of the following fractions increase by 2. So the last fraction i.e 6/9 , is the greatest fraction.

**Ex.3.- Which one of the following fractions is the greatest ? 1/8 , ,4/9 , 7/10 **

Ans – We see that the numerator increase by 3 ( a constant value ) and the denominator also increases by a constant value (1) , so the last fraction i.e 7/10 is the greatest fraction.

__Rule 2 __

**The segment whose numerator afterward cross-multiplication gives the greater value is greater.**

**Ex.1 – Which is greater 5/8 or 9/14 ?**

Ans – Students generally solve this question by changing the fractioṇ into decimal values or by equating the denominators. But we advise you a better technique for receiving the answer more quickly.

1) cross – multiply the two given fractions.

5/8 – 9/14 we have 5×14 = 70 and 8×9 = 72

2) As 72 is greater than 70 the numerator involved with the greater value is 9, the fraction 9/14 is the greater of the two.

**Ex.2 – Which is greater : 13/15 or 20/23 ?**

Ans – 1) 13×23<15×20

2) 20/23 is greater

You can see how quickly this method works.After good practice , you wont need to calculate before answering the question.

The arrangement of fractions into the ascending or desending order becomes easier now.Choose two fraction at a time.See which one is greater.This way you may get a quick arrangement of fractions.

**Ex.3 – Arrange the following in ascending order. 3/7 , 4/5 , 7/9, and 3/5**

Ans – The LCM of 7,5,9,2,5,is 630.

Now to equate the denominators, we divide the LCM by the denominators and multiply the quotient by the respective numerators.

Like , for 3/7,630÷7=90 so multiply 3 by 90

Thus the fraction change to 270/630, 504/630 , 490/630. 315/630 and 378/630

The fraction which has large numerator is naturally large. So 504/630>490/630>378/630>315/630>270/630 or 4/5>7/9>3/5>1/2>3/7

__Some Rules on Coun,ting Numbers__

1) Sum of all the first a natural numbers = n(n+1)/2

For example 1+2+3+….. +105 = 105(105+1)/2 = 5565

2) Amount of 1st n odd numbers = n2

For example 1+3+5+7 = 42 = 16 ( as there are four odd numbers)

For example 1+3+5+….+20^{th} odd number (i.e 20×2-1=39) = 202 = 400

3) Sum of squares of first n natural numbers = n(n+1)(2n+1)/6

For example 12+22+32+ …….+102 = 10(10+1)(2×10+1)/6 =10×11×21/6 = 385