Fascinating Ancient Numbers : Greek Numerals (Attic and Ionian)

The history of numbers is a fascinating journey through human civilization.

Early Beginnings: In prehistoric times, the earliest evidence of counting came from tally marks on bones and stones, dating back to around 35,000 BCE. The Ishango Bone, discovered in Africa, is a notable example. Also, humans likely used physical objects like pebbles, sticks, or knots in a string to keep track of numbers.

Ancient Civilizations: Around 3000 BCE, the Sumerians developed one of the earliest number systems, using a base-60 (sexagesimal) system. The Babylonians inherited and expanded this system, influencing their mathematics and astronomy. It inspired ancient Egyptians to create a decimal (base-10) system around 3000 BCE. Their hieroglyphs represented numbers and were used in administrative and architectural contexts.

Greeks: The Greeks used a variety of systems, including the Attic and the later alphabetic (Ionian) numerals, which assigned values to letters of their alphabet.

Attic: 

• 1: | (a vertical stroke)
• 5: Π (the first letter of "pente," meaning five)
• 10: Δ (the first letter of "deka," meaning ten)
• 50: ΠΔ (a combination of the symbols for 5 and 10)
• 100: Η (the first letter of "hekaton," meaning hundred)
• 500: ΠΗ (a combination of the symbols for 5 and 100)
• 1,000: Χ (the first letter of "chilioi," meaning thousand)
• 5,000: ΠΧ (a combination of the symbols for 5 and 1,000)
• 10,000: Μ (the first letter of "myrioi," meaning ten thousand)
  
  

    This system was additive, meaning that the values of the symbols were added together to form numbers. This system was additive, meaning that the values of the symbols were added together to form numbers. 

For example:  49 = 40 + 9 = ΔΔΔΔ + ΠΙΙΙΙ = ΔΔΔΔΠΙΙΙΙ

Ionian (Alphabetic):

• 1-9:
• 1: Α (alpha)
• 2: Β (beta)
• 3: Γ (gamma)
• 4: Δ (delta)
• 5: Ε (epsilon)
• 6: Ϛ (stigma)
• 7: Ζ (zeta)
• 8: Η (eta)
• 9: Θ (theta)
• 10-90:
• 10: Ι (iota)
• 20: Κ (kappa)
• 30: Λ (lambda)
• 40: Μ (mu)
• 50: Ν (nu)
• 60: Ξ (xi)
• 70: Ο (omicron)
• 80: Π (pi)
• 90: ϙ (koppa)
• 100-900:
• 100: Ρ (rho)
• 200: Σ (sigma)
• 300: Τ (tau)
• 400: Υ (upsilon)
• 500: Φ (phi)
• 600: Χ (chi)
• 700: Ψ (psi)
• 800: Ω (omega)
• 900: ϡ (sampi)

In this system, numbers were written by combining these letters. 

For example: the number 241 = ΣΜΑ (200 + 40 + 1).

The Ionian system was more efficient and flexible than the Attic system, leading to its widespread adoption throughout the Greek world. Thereafter, the Roman numeral system, developed around 500 BCE, used combinations of letters from the Latin alphabet (I, V, X, L, C, D, M) to represent numbers.

        The Attic and Ionian numeral systems played a crucial role in various aspects of ancient Greek life, contributing to their economic prosperity, scientific advancements, cultural achievements, and administrative efficiency.

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Credits:

• OpenAI. (2024). ChatGPT [Large language model]. https://chatgpt.com/c/de7a198b-bf21-4493-84d4-35227eb07f58
• Attic numerals. (2023, June 1). In Wikipedia. https://en.wikipedia.org/wiki/Attic_numerals
• Apostol, Tom M. Introduction to analytic number theory. Springer Science & Business Media, 2013.

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Thank You

Alok Pradhan

30.07.2024

Co-prime numbers

Set of  numbers that have only common divisor as 1 are called co-prime numbers.

Co-prime numbers are also known as Mutually prime number or Relatively prime numbers.

How to Check weather two numbers are Prime or not?

Consider any two numbers, if no positive integer divide both the numbers except 1 then the two numbers are called Relatively Prime or Co-prime.

Example1: are 15 and 22 co-prime?

Answer: 

Divisors of 15: 1, 3, 5, 15

Divisors of 22: 1,2,11,22

Since 15 and 22 don't have any common divisor they are co-prime.

Example2: are 20 and 34 co-prime?

Answer: 

Divisors of 20 : 1,2,4,5,10,20

Divisors of 34 : 1,2,17,34

These two numbers have 2 as the common divisors so they are not co-prime.

FACTS:

• Every number is co-prime with 1.
• Two even number can never be co-prime because every even number is divisible by 2.
• Every Prime number is co-prime to each other. 
• Any two consecutive number are always co-prime. Take any two number like (1,2), (2,3), (3,4)... (501,502).... are always co-prime.

                                                            

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There Are Infinite Numbers Of Primes

Proof:

let us assume that there are finite numbers of primes,                                                         
       say it be  P₁,P₂,P₃,_{.............}P_K

       let  N=  P₁P₂P_{3.............}P_K + 1
       If N is a prime number and  N > P_i,
                    where i=1,2,3,4.............k
then it contradicts our assumption that we have finite number of primes.

So if N is a composite number, 
let P_j will divide N for some 1<j<k
=>P_j|N
=>N=P_j*t , where "t" is an integer
=>P₁,P₂,P₃,_{.............}P_K =P_j*t
=>P₁P₂P_3.....P_j......P_K + 1 = P_j*t

Devide both sides by P_j
=>(P₁P₂P_3.....P_j.......P_K + 1)/P_j  = t

=>P₁P₂P_3.....Pj-1.P_j+1.......P_K + 1/P_{j =   t}

since "t" is an integer so
P₁P₂P_3.....P_j-1.P_j+1.......P_K + /P_j will be an integer. 
which is not possible
So N must be a prime number
this is a contradiction to our assumption that N is composite.

=> N = P₁P₂P_{3.............}P_K + 1 is also a prime.
Hence there are infinite number of primes.


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1-1+1-1+1-1+1-1+......... =1/2

           Let
    S = 1-1+1-1+1-1+1......._____(1)

    Add (-1) on both sides
    S-1=-1+1-1+1-1+1-.......

         Multiply both sides by (-1)
    -1(S-1)=-1(-1+1-1+1-1+......)
    => -S+1=1-1+1-1+1-1+.......
    => -S+1= S   ( from_1 )
    => S+S= 1
    => 2S=1
    => S=1/2
    => 1-1+1-1+1-1.....=1/2

                                             
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Prime Number

A whole number greater than 1 which has only two factors, 1 and the number itself is called Prime number.

Example:- 2,3,5,7,11,13......

FACTS:

• 1 is neither a prime nor a composite number.
• 2 is the only even prime number.
• There are 25 prime numbers between 1 and 100.
• Every Prime number except 2 is an odd number.
• Every number greater than 1 can be divided by at least one prime number.
• Every even positive integer greater than 2 can be expressed as the sum of two primes.
• There are infinite number of Primes.
• Till date the largest known prime number is   (2^82,589,933 − 1) which has 24,862,048 digits.

                                                                                           
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