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maths:number_theory

# a brief summary of number theory

## Introduction

• in simple terms, numbers can be:
• either algebraic or transcedental
• rational, irrational or complex, although all numbers are part of the set of complex numbers, and real numbers are a subset of complex numbers where the imaginary scalar is zero

## Algebraic numbers

• any complex number (including real numbers) that is a root of a non-zero polynomial
• all are countable numbers
• all are computable and therefore definable and arithmetical
• includes:
• all rational numbers
• some irrational numbers
• numbers sqrt(2) and (cubed root(3))/2 are algebraic since they are roots of polynomials x2 − 2 and 8x3 − 3, respectively.
• golden ratio φ is algebraic since it is a root of the polynomial x2 − x − 1.
• some complex numbers
• for real numbers a and b, the complex number a + bi is algebraic if and only if both a and b are algebraic
• Gaussian integers, complex numbers a + bi for which both a and b are integers, are also quadratic integers.
• excludes:
• transcedental numbers such as π and e
• there is a unique monic polynomial (with rational coefficients) of least degree that has the number as a root. This polynomial is called its minimal polynomial. If its minimal polynomial has degree n, then the algebraic number is said to be of degree n. For example, all rational numbers have degree 1, and an algebraic number of degree 2 is a quadratic irrational.

## numeral systems

• documenting amounts of things has a wide range of representations throughout history, largely based upon an underlying base integer value such that a value can be represented as combinations of powers of that base integer value
• hence our decimal value of 3099 is a positional representation of 3 x 103 + 0 x 102 + 9 x 101 + 9 x 100
• the modern binary value of 01001 is a positional representation of 0 x 24 + 1 x 23 + 0 x 22 + 0 x 21 + 1 x 20 = decimal value of 9
• the sexigesimal value of 6;12;24 seconds is a positional representation of 6 x 602 + 12 x 601 + 24 x 600 = 22,344 seconds

## unary numeral systems (base 1)

• simplest counting system where each item is designated by a symbol thus 7 items may be IIIIIII
• gets cumbersome for larger values hence often modified by adding various symbols as with the Egyptian hieroglyphics and the Roman numeral system (where 12 is XII) - both of these actually use base 10 systems though

## binary numeral systems (base 2)

• although now mainly used in computing to reflect on/off states of computer bits following the development of Boolean algebra in the 19thC AD, binary type systems are also some of the oldest
• 2400-1200BC Egypt: Horus-Eye fractions are a binary numbering system for fractional quantities of grain, liquids, or other measures, in which a fraction of a hekat is expressed as a sum of the binary fractions 1/2, 1/4, 1/8, 1/16, 1/32, and 1/64. The method used for ancient Egyptian multiplication is also closely related to binary numbers.
• 9thC BC China: I Ching based on taoistic duality of yin and yang
• 1703: first introduced by Leibnitz in its present form

## decimal or denary counting system (base 10)

• 3500-2500 BC, the earliest usage of a close relative of the decimal system by the Iranian Elamites
• 2900 BC, Egyptians start counting in powers of 10
• 2600 BC, Indus Valley civilization begins the usage of decimal points with reference to measuring weight
• 1400BC, Chinese introduced a bamboo stick decimal numbering system and developed the abacus based on this system. The concept of zero represented by the Chinese as a blank space was later replaced by the clearer representation of 0 in the Hindu-Arab system
• 1st-4thC AD: the positional decimal numeral system we use today is based on the Hindu–Arabic numeral system developed by Indian mathematicians during this period

## sexagesimal counting system (base 60)

• this was invented by the Sumerian civilisation (4000-2000BC)
• papyrus was used to write on from ~3500BC
• 60 is a very useful base value as it is easily divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 whereas 10 is only divisible by 2 and 5, and 100 is only divisible by 2,4,5,10,20,25,50 which is not very useful for easy arithmetic
• use of the 3 phalanges on each of the 4 fingers counting to 12 and then the 4 fingers of other hand being a multiplier of 12 to count to 60 (no fingers for 1-12)
• 12 was thus an important number as it was the basis of phalange counting and also they had recognized there are about 12 lunar cycles (new moon to new moon) per year
• hence 60 minutes in an hour, 60 seconds in a minute, and 60 minutes in a degree
• 60 also divided nicely into their approximation for the number of days in a solar year being 360 days
• there seems to be varied opinions on who 1st ascribed 360 degrees to a circle (see below)
• the number 7 became a mystical number for many reasons including:
• 7 is the 1st natural number that did not divide into 60, the base for their sexagesimal number system
• 7 became the base for a famous mathematical problem from those days in which division by 7 is more difficult than division by 2, 3, 4, 5 or 6. The reciprocals of the numbers 2, 3, 4, 5 and 6 are 0;30, 0;20, 0;15, 0;12 and 0;10 respectively, but that of 7 is a recurring sexagesimal fraction 0;8,34,17,8,34,17,…. 1)
• the Big Dipper, Pleides and other asterisms such as Little Dipper and Northern Crown had 7 stars
• before 2600BC, Thuban (Alpha Draconis) near the Big Dipper was closer to the north celestial pole and was thus the “North Star” then and thus the Big Dipper would have rotated through a smaller circle than present day - and to find north one would look for the 7 stars of the Big Dipper
• lunar phases averaged 7.4 days
• there were “7” naked eye visible planets - Mercury and Venus having been doubly counted as both were morning and evening stars near the sun, adding to Mars, Jupiter and Saturn (uranus was discovered in 1781AD)
• hence in 26thC BC: there were Early Dynastic Proverbs, Collection One “lul-7 lu-lu “Seven lies are too numerous.” , UD-UD-7 “seven gods”, urì-gal-7 “seven (divine) standards”,
• used 7 in exorcising, created concepts of seven-branched Tree of Life, seven heavens
• later this would translate into Sumerian days of the week and the many supposedly perfect 7 concepts in many derived religions and cultures especially including in the Israelite Bible
• 2600BC: Gudea, able priest-king of Lagash, built a seven-roomed lower temple and dedicated it to a seven-day festival. He made a seven-eyed weapon with 7 meaning the fullness of power.
• the Epic of Gilgamesh mentions:
• 7 day periods as well as 30 day months and dividing a day into 12 beru, each consisting of 30 finer divisions (uš) - presumably 30 was used as a divisor to match the divisor of a month being 30 days
• in addition to the “mid-month” day 15 full moon Sabbath (when the moon rested), the Assyro-Babylonians recognised an evil or unfavorable day when restrictions were imposed on activities - these came on days 7, 14, 21 and 28th of the month as well as the 19th day of intercalary months as 7×7 = 49 and 19 is the 49th day since the start of the preceding month. The phrase new moon and sabbath recurs in the Old Testament suggesting the sabbath was still the full moon at early stages of the Bible 2)
• Sargon I, King of Akkad, in 2350BC after the conquer of UR would have the 1st documented 7 day week
• c2500BC: Sumerian priests organised the stars into the zodiac (these were later transmitted by the ancient Greeks), and inspired by the number 7, created constellations of 7 bright stars - Cassiopeia, Lyra, Cygnus, Hercules, Gemini, Virgo, etc
• Apkallu-Abgal 7 wise demigods - part man and part fish or birds, sent by the gods to impart knowledge to people.
• Egyptian usage
• c2300BC, they divided the 12 regions of the stars into 10 sections (later called decans).
• used a 360 day year.
• the Egyptians did not use equilateral triangles with equal angles of 60° in creating the edge slopes of their pyramids - the angle may have been reduced to reduce stresses on the base and to reduce construction effort - but did they have higher mathematical reasons as suggested by:
• the Great Pyramid had an angle of 5½ sekeds = 51.84° from the horizontal, the tan of which is 4/pi and interestingly, pi * tan(51.84°) * 51.84°/(180-2×51.84°) approximates to e (Euler’s number, wasn’t discovered until 1618), while 1/cos(51.84°) yields the golden ratio of 1.618! The ratio of the volume of the largest sphere which could be placed inside to the volume of the pyramid gives pi x golden mean5 3)!
• Babylonian modifications (c1500-1800BC)
• numeral system advanced into a positional system with 2 symbols (unit & ten) used to create the numbers 1 to 59 ie. used a base of 10 for documenting instead of 12
• Babylonians divided the ecliptic (the zodiacal region of the stars) into 12 equal parts (beru) each divided by 30 finer divisions (uš - or degrees) and thus a total of 360 uš (degrees) for the who circle of the ecliptic (each uš equates to 4 minutes of our modern time - in other words the earth rotates 1 degree every 4 minutes)
• 24 hour day concept invented when sundials were invented to tell the time of day and these used the important Sumerian number 12 as the divisor which created 12 “hour” divisions of daylight time and thus there became 24hrs in a day as there would be also 12 hours allocated to night time, perhaps utilising the stars of the Zodiac as markers of time
• HOWEVER, an hour of daylight in the summer would be longer than one in the winter (especially the further one was from the equator) and despite the Greek mathematician Hipparchus giving us the “Equinoctial hours” by proposing the division of a day into 24 EQUAL hours, the use of hours with a fixed length didn’t gain widespread acceptance until the invention of mechanical clocks in the 14th C AD.
• Indian Rigveda c 1500 and 1000 BCE
• at Dirghatamas, Rigveda 1.164.48 describes a wheel with 360 pegs
• Chaldean Babylonian dynasty during the reign of Nebuchadnezzar (605-562 BC)
• 360 days in the year
• basis of angular measure for the mathematicians of Babylon was the angle at each of the corners of an equilateral triangle which was assigned 60degrees as when they were placed into a circle from 6 triangles, the circle equated to 360deg.
• Euclid worked with “right angles” but as Greek geometry developed in the 2ndC BC, the Greeks adopted the Babylonian / Egyptian system of 360 degrees in the ecliptic to apply it to 360 degrees in a circle, each degree having 60 minutes, each minute having 60 seconds of angle

## Transcedental numbers

• numbers that are not solutions of any polynomial equation with rational coefficients
• such as:
• π and e
• Liouville numbers
• the number logab for rational numbers a and b provided b is not of the form b = ac for some rational c
• almost all complex numbers
• this concept arose when Euler asserted the above log statement in 1748 although the term transcedental arose in the 17th C when Gottfried Leibniz proved that the sine function was not an algebraic function.

## Real numbers R

### Rational numbers Q

• a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q
• every rational number may be expressed in a unique way as an irreducible fraction a/b, where a and b are coprime integers and b > 0.
• the decimal expansion of a rational number either terminates or eventually begins to repeat the same finite sequence of digits over and over

#### Integers

• any fraction where numerator is a rational number and the denominator = 1
• special integers:
• Prime numbers
• an integer is a prime number if the only positive integer that is a divisor is itself or 1
• two integers a and b are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1
• Fibonacci numbers
• a sequence, such that each number is the sum of the two preceding ones, starting from 0 and 1.

### Irrational numbers

• the decimal expansion of an irrational number continues without repeating
• include:
• surds such as √2
• certain constants such as π, e, and φ

## Complex (imaginary) numbers

• these were invented in the 16th century to help solve cubic equations
• z = x + yi where z = complex, x and y are real and i = sqrt(-1)