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Chapter 01 : Number Theory > Topic 1 : Types of Numbers

 

The number theory or number systems happens to be the back bone for CAT preparation. Number systems not only form the basis of most calculations and other systems in mathematics, but also it forms a major percentage of the CAT quantitative section. The reason for that is the ability of examiner to formulate tough conceptual questions and puzzles from this section. In number systems there are hundreds of concepts and variations, along with various logics attached to them, which makes this seemingly easy looking topic most complex in preparation for the CAT examination. The students while going through these topics should be careful in capturing the concept correctly, as it’s not the speed but the concept that will solve the question here. The correct understanding of concept is the only way to solve complex questions based on this section.

Real numbers: The numbers that can represent physical quantities in a complete manner. All real numbers can be measured and can be represented on a number line. They are of two types:

Rational numbers: A number that can be represented in the form p/q where p and q are integers and q is not zero. Example: 2/3, 1/10, 8/3 etc. They can be finite decimal numbers, whole numbers, integers, fractions.

Irrational numbers: A number that cannot be represented in the form p/q where p and q are integers and q is not zero. An infinite non recurring decimal is an irrational number. Example: √2, √5 , √7 and Π(pie)=3.1416.

The rational numbers are classified into Integers and fractions

Integers: The set of numbers on the number line, with the natural numbers, zero and the negative numbers are called integers, I = {…..-3, -2, -1, 0, 1, 2, 3…….}


Fractions:
A fraction denotes part or parts of an integer. For example 1/6, which can represent 1/6th part of the whole, the type of fractions are:

1. Common fractions: The fractions where the denominator is not 10 or a multiple of it. Example: 2/3, 4/5 etc.
2. Decimal fractions: The fractions where the denominator is 10 or a multiple of 10. Example 7/10, 9/100 etc.
3. Proper fractions: The fractions where the numerator is less than the denominator. Example ¾, 2/5 etc. its value is always less than 1.
4. Improper fractions: The fractions where the numerator is greater than or equal to the denominator. Example 4/3, 5/3 etc. Its value is always greater than or equal to 1.
5. Compound fraction: A fraction of a fraction is called a compound fraction

Example 3/5 of 7/9 = 3/5 x 7/9 = 21/45

6. Complex fractions: The combination of fractions is called a complex fraction.

Example (3/5)/ (2/9)

7. Mixed fractions: A fraction which consists of two parts, an integer and a fraction. Example 3 ½, 6 ¾

Example: Express 27/8 as a mixed fraction

Ans. Divide the numerator by denominator; note the multiplier, whatever remainder is left divide it with the original denominator. For 27/8, 24/8 = 3, and remainder left is 3, therefore 3 3/8 is the mixed fraction

Example: Express 35 7/17as an improper fraction.

Ans. Here we need to multiply the denominator with the non-fraction part and add it to numerator and using same denominator.
For 35 7/17= = 602/17

The integers are classified into negative numbers and whole numbers

Negative numbers: All the negative numbers on the number line, {…..-3, -2, -1}

Whole numbers: The set of all positive numbers and 0 are called whole numbers, W = {0, 1, 2, 3, 4…….}.

Natural numbers: The counting numbers 1, 2, 3, 4, 5……. are known as natural numbers, N = {1, 2, 3, 4, 5…..}. The natural numbers along with zero make the set of the whole numbers.

Even numbers: The numbers divisible by 2 are even numbers. e.g., 2, 4, 6,8,10 etc. Even numbers can be expressed in the form 2n where n is an integer other than 0.

Odd numbers: The numbers not divisible by 2 are odd numbers. e.g. 1, 3, 5, 7, 9 etc. Odd numbers are expressible in the form (2n + 1) where n is an integer other than 0.

Composite numbers: A composite number has other factors besides itself and unity .e.g. 8, 72, 39 etc. A real natural number that is not a prime number is a composite number.

Prime numbers: The numbers that has no other factors besides itself and unity is a prime number. Example: 2, 23,5,7,11,13 etc. Here are some properties of prime numbers:


• The only even prime number is 2
• 1 is neither a prime nor a composite number
• If p is a prime number then for any whole number a, ap – a is divisible by p.
• 2,3,5,7,11,13,17,19,23,29 are first ten prime numbers (should be remembered)
• Two numbers are supposed to be co-prime of their HCF is 1, e.g. 3 & 5, 14 & 29 etc.
• A number is divisible by ab only when that number is divisible by each one of a and b, where a and b are co prime.
• To find a prime number, check the rough square root of the given number and divide the number by all the prime number lower than the estimated square root
• All prime numbers can be expressed in the form 6n-1 or 6n+1, but all numbers that can be expressed in this form are not prime

Example: If a, a + 2 and a + 4 are prime numbers, then the number of possible solutions for a is: (CAT 2003)

(a) 1 (b) 2
(c) 3 (d) more than 3

Ans. (a) a, a + 2, a + 4 are prime numbers. The number fits is 1, 3, 5 and 3, 5, 7 but post this nothing will fit. Now 1, 3, 5 are not prime numbers as 1 is not prime number.
So, only one possibility is there 3, 5, 7 for a = 3.

Prime Factors: The composite numbers express in factors, wherein all the factors are prime. To get prime factors we divide number by prime numbers till the remainder is a prime number. All composite numbers can be expressed as prime factors, for example prime factors of 150 are 2,3,5,5.

A composite number can be uniquely expressed as a product of prime factors.

e.g. 12 = 2 x 6 = 2 x 2 x 3 = 22 x 31
20 = 4 x 5 = 2 x 2 x 5 = 22 x 51 etc

Note : The number of divisors of a given number N ( including one and the number itself ) where N = am x bn x cp ……. Where a, b, c are prime numbers
are = ( 1 + m ) ( 1 + n ) ( 1 + p ) …………..

e.g. 90 = 2 x 3 x 3 x 3 x 5 = 21 x 32 x 51

Hence here a = 2 b = 3 c = 5
m = 1 n = 2 p = 1
then the number of divisors are = ( 1 + m ) ( 1 + n ) ( 1 + p ) = 2 x 3 x 2 = 12
the factors of 90 = 1 , 2 , 3 , 5 , 6 , 9 , 10 , 15 , 18 , 30 , 45 , 90 = 12


the sum of divisors of given number N is ( am+1 – 1 ) ( bn+1 - 1 ) ( cp+1 – 1 ) ……..
___________________________________
( a – 1 ) ( b – 1 ) ( c – 1 ) ……….

Perfect number: If the sum of the divisor of N excluding N itself is equal to N , then N is called a perfect number. e.g. 6, 28, 496

Finding a perfect number through Euclid’s method

Euclid's method makes use of the powers of 2, which are numbers obtained by multiplying by 2 by itself over and over again, which are 1, 2, 4, 8, 16, 32, 64, 128….

Note that the sum of the two numbers in this series (in ascending order) is equal to the third number minus 1:

1+2 = 3 = 4 - 1,
1+2+4 = 7 = 8 - 1,

STARTING FROM THE NUMBER 1, IF YOU ADD THE POWERS OF 2 AND IF THE SUM IS A PRIME NUMBER, THEN YOU GET A PERFECT NUMBER BY MULTIPLYING THIS SUM TO THE LAST POWER OF 2.

If you add 1+2, the sum is 3, which is a prime number. Therefore 3 x 2 = 6 is a perfect number.

If you add 1+2+4, the sum is 7, a prime number. Therefore 7 x 4 = 28 is a perfect number.

If you add 1+2+4+8, the sum 15 is not a prime number, so you can't use Euclid's method here.

If you add 1+2+4+8+16, the sum is 31, a prime number. Therefore 31 x 16 = 496 is another perfect number.

Absolute value of a number:
The absolute value of a number a is | a | and is always positive.

Fibonacci numbers: The Fibonacci numbers is a sequence where
X (n+2) = X (n+1) + X (n), X (1) = 1, X (2) = 1
Example1,1,2,3,5,8,13,21,34,55,89,144.., it can be clearly seen that any number in the series is the addition of the last two numbers, other than the first two numbers

Example: The price of pens has increased over the years. Each year for the last 7 years the price has increased, and the new price is the sum of the prices for the two previous years. Last year a pen cost 60 rupees. How much does a pen cost today? How much did a pen cost 7 years ago?

Let this year price be x. Last year it was 60, so the previous year it must have been x-60, continuing this process backwards gives us a sequence of expressions:

x, 60, x-60, 120-x, 2x-180, 300-3x, 5x-480, 780-8x, 13x-1260

All of these increases must be positive as every year prize has gone up. That gives us a sequence of inequalities, each of which can be solved to find a range for x:

x > 0
60 > 0
x-60 > 0 x > 60
120-x > 0 x < 120
2x-180 > 0 x > 90
300-3x > 0 x < 100
5x-480 > 0 x > 96
780-8x > 0 x < 97.5
13x-1260 > 0 x > 96.92

Looking at this, we can say

96.92 < x < 97.5

The whole number value x can have is 97, with which we get

x = 97
60 = 60
x-60 = 37
120-x = 23
2x-180 = 14
300-3x = 9
5x-480 = 5
780-8x = 4
13x-1260 = 1

Seven years ago, the price was 4 rupees

In the CAT/MCQ format, where you have the four answers, you can check it by working forward and seeing if the results are correct. You can try putting the given answers for original price and see which one fits in the equation.

Golden ratio: The golden ratio is a special number approximately equal to:
1.6180339887498948482...

Golden ratio = (1 + √ 5)/2

To find the golden ratio, we define the golden ratio as the ratio between x and y if

x y
--- = -----
y x+y

Let's say x is 1. Then we have 1/y = y/(y+1). If we solve this equation to find y, we'll find that it is the value given above, about 1.618
A golden rectangle is a rectangle in which the ratio of the length to the width is the golden ratio.

The concepts like Fibonacci and golden ratio are reference concepts, students are advised not to cram them but just understand the concepts as they are.


Other topics covered

Number Theory