Number System Tutorial I: Integers, Fractions, Prime & Composite

Integers and its properties

The set of integers is denoted by the letter 'Z'. Z = { .....-4, -3, -2, -1, 0, 1, 2, 3, 4.....}.

Subsets of Z (set of integers)

• Odd numbers: {......-5, -3, -1, 1, 3, 5......}
• Even numbers: {......-4, -2, 0, 2, 4, 6, 8......}
• Negative integers: {......-4, -3, -2, -1}
• Non-negative integers (Whole Numbers): {0, 1, 2, 3, 4......}
• Positive integers (Natural Numbers): {1, 2, 3, 4, 5......}
• Prime Numbers: {2, 3, 5, 7, 11, 13, 17......}
• Composite numbers: {4, 6, 8, 9, 10, 12, 14, 15......}

Properties of odd and even numbers

Property I: Addition

• Sum of odd number of odd numbers is odd. Eg: 1 + 3 + 5 = 9.
• Sum of even number of odd numbers is even. Eg: 3+ 11 + 7 + 5 = 26.
• Sum of any number of even numbers is even.
• In a certain summation of odd and even numbers, then the sum is:
• Odd: when there is odd number of odd numbers in the selected group of integers
• Even: when there is even number of odd numbers in the selected group of integers

Property II: Subtraction

What are the properties considered in the addition of integers, the same properties are applicable in the subtraction in integers.

• If a+b${\displaystyle a+b}$ is odd for any two integers a and b, then; a−b${\displaystyle a-b}$, −a+b${\displaystyle -a+b}$, −a−b${\displaystyle -a-b}$ are all odd.
• If a+b${\displaystyle a+b}$ is even, then a−b${\displaystyle a-b}$, −a+b${\displaystyle -a+b}$, −a−b${\displaystyle -a-b}$ are all even.

Property III: Multiplication

If a group of integers consist at least one even number then the product of the given integers should be even, otherwise the product become odd (ie. all integers are odd).

Property IV: Division

Consider a rational number xy${\displaystyle \frac{x}{y}}$, where x and y are two integers and y≠0${\displaystyle y\ne 0}$.

Case I: Numerator is even and denominator is odd:

In a division in the form xy${\displaystyle \frac{x}{y}}$, if the numerator is an even multiple of the odd denominator y, then the result of xy${\displaystyle \frac{x}{y}}$ is always an even number. Example: 369=4${\displaystyle \frac{36}{9}=4}$ (it is even).

Case II: Both the numerator and denominator are even:

In xy${\displaystyle \frac{x}{y}}$, where x and y are integers and x is divisible by y, then it is not possible to make a conclusion of the nature of the value of xy${\displaystyle \frac{x}{y}}$. Example: 105=2${\displaystyle \frac{10}{5}=2}$ (it is even)

Case III: Dividing an odd number by an even number:

In xy${\displaystyle \frac{x}{y}}$ , if x is odd and y is even, then x can't be any multiple of y. Hence the value of xy${\displaystyle \frac{x}{y}}$ in this situation can't be an integer.

Case IV: Divide an odd number by another odd number:

In the division xy${\displaystyle \frac{x}{y}}$, both x and y are odd numbers and x is divisible by y, then the value of xy${\displaystyle \frac{x}{y}}$ is always odd Example: 153=5${\displaystyle \frac{15}{3}=5}$ (it is odd)

Some interesting properties of odd numbers

• Cube of all natural numbers greater than '1' can be expressed as the sum of a certain number of consecutive odd numbers.
  n3=[(n−1).n+1]+[(n−1).n+3]
  Examples:
  • 23=3+5${\displaystyle 23=3+5}$
  • 33=7+9+11${\displaystyle 33=7+9+11}$
  • 43=13+15+17+19${\displaystyle 43=13+15+17+19}$
• +[(n−1).n+5]+...+[(n−1).n+(2n−1)]
• 
  
• All odd natural numbers except '1' can be one of the three values of a Pythagorean triplet.
  • 32 + 42 = 52
  • 52 + 122 = 132
  • 72 + 242 = 132
  • 92 + 402 = 412
  • 112 + 602 = 612
  • 132 - 122 = 52
  • 152 + 82 = 172
  • 172 - 152 = 82

Prime and Composite numbers

 

Prime number

A natural number which doesn't have any factor other than 1 and itself is a prime number. Example: 2, 3, 5, 7, 11, 13, 17 etc.

Properties of prime numbers

• 2 is the least prime number
• 2 is the one and only one even prime.
• All primes greater than 3 can be expressed in either of the form 6k−1${\displaystyle 6k-1}$ or 6k+1${\displaystyle 6k+1}$, where k${\displaystyle k}$ is any natural number, but all the numbers which are in the above mentioned forms are not necessarily prime always. Eg when k = 4, 6k+1=25${\displaystyle 6k+1=25}$, it is not a prime.
• Goldbach's Conjecture: Every even natural number greater than 2 can be expressed as the sum of two primes ( not necessarily distinct). Eg: 4=2+2${\displaystyle 4=2+2}$, 6=3+3${\displaystyle 6=3+3}$, 8=3+5${\displaystyle 8=3+5}$, 10=3+7${\displaystyle 10=3+7}$ and so on.
• Relatively prime / co-prime numbers: If two natural numbers doesn't have a common factor other than 1 are called relatively prime/co-prime numbers. Eg. 8 and 9 are relatively prime. 4 and 5 are co-prime.
• 1 is relatively prime to all the other natural numbers.
• Twin Primes: If any two consecutive odd numbers are prime, ten they are called twin primes. Eg. (3,5), (5,7), (11,13), (17,19), etc.
• Prime triplet: If three consecutive odd numbers are prime, then they are called prime triplets. There is only one prime triplet and that is (3,5,7).
• Fermatt's theorem: If 'a' and 'b' are any two prime numbers, then ab−a${\displaystyle a^b-a}$ is always divisible by b. Eg: 23−2=6${\displaystyle 2^3-2=6}$ is divisible by 3.
  35−3=240${\displaystyle 3^5-3=240}$ is divisible by 5
• Wilson's theorem: For any prime 'p', (p−1)!+1${\displaystyle (p-1)!+1}$ is divisible by p. Eg: Let p = 5, (p−1)!+1=4!+1=25${\displaystyle (p-1)!+1=4!+1=25}$ is divisible by 5.
  Let p = 11, (p−1)!+1=10!+1=3628800+1=3628801${\displaystyle (p-1)!+1=10!+1=3628800+1=3628801}$ is divisible by 11.
• Progression of prime numbers: Some set of prime numbers are forming an Arithmetic Progression. Eg: 11,17,23,29
  199,409,619,829, 1039, 1249, 1459, 1669, 1879, 2089.
• How to identify a prime number? Consider 173. For checking 173 a prime or not, consider the square of any prime number which is immediately less tan 173. Here 132=169${\displaystyle {13}^2=169}$ is immediately less than 173.
  Check the divisibility of all the prime numbers up to 13 with 173. None of the prime numbers 2, 3, 5,7, 11 and 13 is not a factor of 173. Therefore 173 is a prime number.
  Example: Is 391 a prime number?
  Square of a prime number, which is immediately less than 391 is 192=361${\displaystyle {19}^2=361}$. While checking the divisibility of all primes up to 19, we will get 17 is a factor of 391 (Ie. 391=17×23${\displaystyle 391=17\times 23}$). Therefore 391 is not a prime number.
• In the set of natural numbers from 1 to 50 there are exactly 15 prime numbers and from 51 to 100, there are 10 prime numbers.
• For any prime p>3${\displaystyle p>3}$, p2−1${\displaystyle p^2-1}$ is a multiple of 24.

Distribution of prime numbers in the first thousand natural numbers

Range of natural numbers	Number of prime numbers
1 to 100	25
101 to 200	21
201 to 300	16
301 to 400	16
401 to 500	17
501 to 600	14
601 to 700	16
701 to 800	14
801 to 900	15
901 to 1000	14

Composite number

If a natural number has at least one factor other than 1 and itself is called a composite number. Eg: 4, 6, 8, 9, 10, 12, 14, 15 etc.

Properties of composite numbers

• 4 is the least composite number.
• 
  
• Wilson's theorem: For any composite number c>4${\displaystyle c>4}$, (c−1)!${\displaystyle (c-1)!}$ is divisible by c.
  Eg: Let c = 6
  (c−1)!=(6−1)!=5!=24${\displaystyle (c-1)!=(6-1)!=5!=24}$ is divisible by 6.
  Let c = 12
  (c−1)!=11!=39916800${\displaystyle (c-1)!=11!=39916800}$ is divisible by 12.

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