Numbers

Numbers are an abstraction from the inexplicable perception that there is something rather than nothing.

Various animals have demonstrated they recognize when a certain number has been increased or decreased by small amounts.

A Sketchy History

Time is an epiphenomenon emerging from number and cycle. Humans can count repetitions in cycles and beats in rhythms, thus they can put events and beats/periods into one-to-one correspondence. We call this ability time.

About 43,000 years ago, a human living near the Lebombo Mountains used a sharpened stone to carve ||||||||||||||||||||||||||||| into a baboon fibula: the Lebombo bone was found during the 1970's Border Cave excavation. The tally just happens to be the closest prime to the lunar cycle. [2]

A human who lived near Moravia roughly 30,000 years ago notched ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| ||||| || into a young wolf's shinbone with an edged stone. The Wolf bone was unearthed at Dolní Věstonice in 1924. [1]

Someone living circa 17,500 BCE scratched ||||| |||||||||||||| ||||||||||||||||| ||||||||||||| ||||||||||| down one side, ||||||| ||||| |||| | | ||||||||| |||||||| |||| |||||| ||| to the right of that, and ||||||||| ||||||||||||||||||| ||||||||||||||||||||| ||||||||||| down the side opposite the first, making separate tallies in each of three columns of the Ishango bone, along one of the Nile's headwaters, Lake Edward. The groupings of notches may suggest unitary incrementation of 10 and 20, doubling, primality, and lunar counting. The themes of primality and lunar cycles seem to be common to these artifacts. [1]

Let's take a moment to examine what has been revealed so far. Certainly, as of the time of the Ishango bone, people were concerned with multiplication and the lunar cycle. This cannot be unless they were also concerned with addition, subtraction, and division. Thus, they must have been studying number through the lens of rectangles, at least. Let's rebuild what must have been elementary to them by this time.

8000 - 3000 BCE: The people in the ancient Near East develop a system of 16 different tokens for counting various commodities, such as herd size or grain measures. [1]

3500 BCE - Egyptians had a well-developed system of decimal pictographs for representing indefinitely large numbers, which their priests made more efficient with a system of ciphered numerals for use in Hieratic script. [1]

500 BCE - The Greek historian Herodotus reports that king Darius of Persia gave the Ionians a leather thong tied with 60 knots to count the days until he returned. [1]

4th Century BCE - The Ionians cipher numbers as Greek and old Phoenician letters, introducing additional symbols as needed. [1]

12th Century CE - Present: Tally sticks were accepted as loan and transaction documents in Europe. The burning of the Exchequer Tallies in 1834 destroyed the British House of Parliament. [1]

16th Century CE: Spanish conquistadores report the Incas of Peru have an "official of the knots" in each city, responsible for keeping quipus as transaction records in decimal which employed knots whose values were understood relative to their positions with respect to other knots and twists within them. Zero was represented by the absence of a knot. [1]

Santal headsmen in India used knotted strings in 4 colors to keep census. [1]

1200 BCE - 1800 CE: Mayan civilization creates a calendrical year composed of eighteen 20-day months, which leads to vigesimal numeration. [1]

Motivating Modular Arithmetic

We can accomplish the same shift by numbering the letters in the English alphabet -- 0 = A through 25 = Z -- and adding 3 to each value. In other words, the Caesar cipher is x + 3, but you go back to 0 for 26, 1 for 27, and 2 for 28 to ensure that X --> A, Y --> B, and Z --> C.

In mathematics, this wrap-around is accomplished by modular congruence. We say that a ≡ b mod c if and only if a ≡ b + kc for some integer k. Thus, 26 ≡ 0 mod 26, 27 ≡ 1 mod 26, and 28 ≡ 2 mod 26 and so the last three letters of the English alphabet become the first three letters of the encrypted version of the Caesar cipher as a consequence of the definition rather than a manual operation. What are 57 mod 26 and -4 mod 26?

How many solutions are there to the congruence x ≡ 57 mod 26? We refer to the solution set of x ≡ 57 mod 26 as the equivalence class of the remainder when 57 is divided by 26 (or the residue of 57 modulo 26.)

Let's consider a simpler example: modulo 2. How many equivalence classes are there modulo 2? What are they?

The Caesar cipher is an example of a shift cipher, x + s mod 26, where x is the letter value and s is the shift. Encrypt "The West is the best" with the shift cipher x + 5 mod 26. Decipher the following message encrypted with a shift cipher.

Encryption and decryption require keys. Determine a key for decrypting the cipher x + a mod 26? Is the decryption key unique for every a?

The affine cipher, ax + b mod 26 is another way to encrypt messages. Is an affine cipher more or less secure than a shift cipher? Does every affine cipher have a unique decryption key?