Number Systems

Real Numbers

All real numbers can be classified into two types: Rational and Irrational.

Rational & Irrational Numbers

A number n is called a rational number, if it can be written in the form p/q. where p and q are integers and q≠0. Irrational numbers are just opposite of these.

All integers and fractions are rational numbers. Rational numbers are classified as Positive and Negative rational numbers. When the numerator and denominator, both, are positive integers, it is a positive rational number. When either the numerator or the denominator is a negative integer, it is a negative rational number. The number 0 is neither a positive nor a negative rational number. There are unlimited number of rational numbers between two rational numbers. 

Addition, Subtraction, Multiplication, Division

Two rational numbers with the same denominator can be added by adding their numerators, keeping the denominator same. Two rational numbers with different denominators are added by first taking the LCM of the two denominators and then converting both the rational numbers to their equivalent forms having the LCM as the denominator.

To multiply two rational numbers, just multiply their numerators and denominators separately. To divide one rational number by the other non-zero rational number, multiply the rational number by the reciprocal of the other.

Decimal Expressions

The decimal expansion of a rational number is either terminating or non-terminating recurring. A number whose decimal expansion is terminating or non-terminating recurring is rational. The decimal expansion of an irrational number is non-terminating non-recurring. A number whose decimal expansion is non-terminating non-recurring is irrational.

Integers

Integers are a bigger collection of numbers which is formed by whole numbers and their negatives.

I = {..., -3, -2, -1, 0, 1, 2, 3, ...}

Product of even number of negative integers is positive, whereas the product of odd number of negative integers is negative.

Number Line

 

Whole Numbers

The smallest whole number is 0.

W = {0, 1, 2, 3, 4, 5,…}

Natural Numbers

All positive integers are natural numbers. Examples: 1, 2, 3, 4, 7, 8, etc.

N = {1, 2, 3, 4, 5, 6,…}

Number 1 is the least natural number and there are infinite natural numbers. Based on the divisibility, there are two types of natural numbers.

  1. Prime
  2. Composite

Prime Numbers

A natural number larger than unity is a prime number if it does not have other divisors except for itself and unity. 1 is not a prime number.

Properties of Prime Numbers

  • The lowest prime number is 2.
  • 2 is also the only even prime number.
  • The lowest odd prime number is 3.
  • The remainder when a prime number greater than 5 is divided by 6 is 1 or 5. However, if a number on being divided by 6 gives a remainder 1 or 5 need not be prime.
  • The remainder of division of the square of a prime number greater than 5 divided by 24 is 1.
  • For prime numbers greater than 3, p- 1 is divided by 24.
  • If a and b are any 2 odd primes then, a- b2 is composite. Also a+ b2 is composite.
  • The remainder of the division of the square of a prime number greater than 5 divided by 12 is 1.

How to Check A Number is Prime or Not

Take the square root of the number. Round of the square root to the next highest integer. Lets call this number as Z. Check for divisibility of the number N by all prime numbers below Z. If there is no number below the value of Z which divides N, then the number will be prime.

Twin Primes: Two prime numbers which differ by 2 are called twin primes. Examples: (3, 5), (5, 7), (11, 13), etc.

Composite Numbers

The numbers which are not prime are known as composite numbers. Examples: 4, 6, 8, 9, 10, 12, 14, 15, etc. 1 is neither prime nor composite.

Co-Primes

Two numbers a an b are said to be co-primes, if their HCF is 1. In other words, two numbers are said to be relatively prime if they do not have any common factor other than 1. Examples: (2, 3), (4, 5), (7, 9), (8, 11), etc.

Even and Odd Numbers

Even Numbers

Numbers which are exactly divisible by 2 are called even numbers. For example, -4, -2, 0, 2, 4, etc.

Sum of first n even numbers = n (n + 1)

Odd Numbers

Numbers which are not exactly divisible by 2 are called odd numbers. For example, -5, -3, -1, 0, 1, 3, 5, etc. 

Sum of first n odd numbers = n2

Perfect Number

A number is said to be a perfect number, if the sum of all its factors excluding itself is equal to the number itself. For example, factors of 6 are 1, 2, 3 and 6.

Sum of factors excluding 6 = 1 + 2 + 3 = 6. Thus, 6 is a perfect number.

Other examples of perfect numbers are 28, 496, 8128, etc.

Euclid’s Division Lemma

Given positive integers a and b, there exist whole numbers q and r satisfying a= bq+ r, 0 ≤ r< b.

Euclid’s division algorithm is based on Euclid’s division lemma. According to this, the HCF of any two positive integers a and b, with a > b, is obtained as:

  • Step 1: Apply the division lemma to find q and r where a= bq+ r, 0 ≤ r< b.
  • Step 2: If r= 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r.
  • Step 3: Continue the process till the remainder is zero. The divisor at this stage will be HCF (a, b). Also, HCF(a, b) = HCF(b, r)

Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic says that every composite number can be expressed (or factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

Factors and Multiples

Factor: A number which divides a given number exactly is called a factor of the given number. For example, 1, 2, 3, 4, 6, 8, 12 and 24 are factors of 24.

  • 1 is a factor of every number
  • A number is a factor of itself
  • The smallest factor of a given number is 1 and the greatest factor is the number itself
  • If a number is divided by any of its factors, the remainder is always zero
  • Every factor of a number is either less than or at the most equal to the given number
  • Number of factors of a number are finite

Number of Factors of a Number

If N is a composite number such that N = am bn where a, b are prime factors of N and m, n are positive integers, then the number of factors of N is given by (m+1)(n+1)

Multiple: A multiple of a number is a number obtained by multiplying it by a natural number. For example, multiples of 5 are 5, 10, 15, 20, etc.

  • Every number is a multiple of 1
  • The smallest multiple of a number is the number itself
  • We cannot find the greatest multiple of a number
  • Number of multiples of a number are infinite