Number System

Number Systems


Real Numbers

Real numbers include natural numbers, whole numbers, integers, rational numbers and irrational numbers.

Natural numbers
: 1, 2, 3 ...

Whole Numbers
: 0, 1, 2, 3 ...

Integers : -3, -2, -1, 0, 1, 2, 3 ...


Rational Numbers:



Rational numbers can be expressed as p/q where p and q are integers and q≠0
Examples: 1/12, 42, 0, −8/11 etc.

All integers, fractions and terminating or recurring decimals are rational numbers.
How to find rational numbers between two given rational numbers?

If m and n be two rational numbers such that m < n then 1/2 (m + n) is a rational number between m and n.

Question: Find out a rational number lying halfway between 2/7 and 3/4.

Solution:

Required number = 1/2 (2/7 + 3/4) 

= 1/2 ((8 + 21)/28) 

= {1/2 × 29/28) 

= 29/56

Hence, 29/56 is a rational number lying halfway between 2/7 and 3/4.

Question: Find out ten rational numbers lying between -3/11 and 8/11.

Solution:

We know that -3 < -2 < -1 < 0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8

Therefore, -3 /11< -2/11 < -1/11 < 0/11 < 1/11 < 2/11 < 3/11 < 4/11 < 5/11 < 6/11 < 7/11 < 8/11

Hence, -2/11, -1/11, 0/11, 1/11, 2/11, 3/11, 4/11, 5/11, 6/11 and 7/11 are the ten rational numbers lying between -3/11 and 8/11.

Irrational Numbers

Any number which is not a rational number is an irrational number. In other words, an irrational number is a number which cannot be expressed as p/q, where p and q are integers.

For instance, numbers whose decimals do not terminate and do not repeat cannot be written as a fraction and hence they are irrational numbers.

Example: π, √2, (3+√5), 
4√3 (meaning 4×√3), 6√3 etc

Please note that the value of π = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679...

We cannot write π as a simple fraction (The fraction 22/7 = 3.14.... is just an approximate value of π)

Complex Numbers:


A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i= −1. In this expression, a is the real part and b is the imaginary part of the complex number.

Even NNumbers: Divisible by 2 (2, 4, 6, 8, 22, 44, 68, 34234... )

Odd NNumbers: NOT divisible by 2 (3, 5, 7, 9, 23, 45, 67, 34235... )

Prime Numbers: 

A number greater than 1 is called a prime number, if it has only two factors, namely 1 and the number itself.

Prime numbers up to 100 are: 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

Procedure to find out the prime number

Suppose A is the given number.

Step 1: Find a whole number nearly greater than the square root of A. 
Let K is nearby square root of A 
Step 2: Test whether A is divisible by any prime number less than K. If yes A is not a prime number. If not, A is a prime number.

Example:

Find out whether 337 is a prime number or not?

Step 1: 19 is the nearby square root (337) Prime numbers less than 19 are 2, 3, 5, 7, 11, 13, 17 
Step 2: 337 is not divisible by any of them

Therefore, 337 is a prime number

Co-Prime Numbers: 

In number theory, two integers a and b are said to be co-prime if the only positive integer that evenly divides both of them is 1. That is, the only common positive factor of the two numbers is 1. This is equivalent to their H.C.F. being 1. e.g. (2,3), (6,13), (10,11), (25,36) etc

Composite Numbers:


A natural number greater than 1 that is not a prime number is called a composite number. e.g. 4, 6, 8, 9, 10, 12 etc

Perfect Numbers:


A perfect number is a positive integer that is equal to the sum of its positive divisors excluding the number itself (proper positive divisors). 

The first perfect number is 6, because 1, 2, and 3 are its proper positive divisors, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 + 6 )/2= 6. The next perfect number is 28 = 1 + 2 + 4 + 7 + 14.

There are total 27 perfect numbers.

Surds :

Let a be any rational number and n be any positive integer such that a√n is irrational. Then a√n is a surd.

Example: √3, 10√6, √43 etc

Every surd is an irrational number. But every irrational number is not a surd. (E.g.: Π, e etc are not surds though they are irrational numbers.)




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