Concise introduction to number theory: rational, irrational and transcendental numbers

Number theory, a major branch of mathematics, introduces the quintessential concept of natural, integer, fractions, mixed numbers, algebraic, rational, irrational and transcendental numbers, collectively referred as real (Re) numbers. Pushing the envelope further, imaginary (Im) numbers extends the theoretical boundary, adding the super-set of complex numbers, which combines the best of real and imaginary worlds.

Integer numbers or just integers represent a set of “whole” numbers, either positive, or negative, and zero: {…,-3, -2, -1, 0, 1, 2, 3,…} or in more compact form: {0, ±1, ±2, ±3,…}

Natural numbers represent a subset of integers. There are two possible definitions of natural numbers: traditional definition, which is the most restrictive one, refers just to all positive integers {1,2,3,…}, while alternative definition specifies natural numbers as all non-negative integers {0,1,2,3,…}, thus including zero into the set. In regards to the latter one, the inter-relations between integers and natural numbers is quite simple: integers could be considered as all naturals with their algebraic negatives; note that negative zero is equal zero: 0=(-0).

Fractions are essentially the numbers, expressed as the quotient of two integers: n/d, where n is called a numerator and d is called denominator. Fractions represent part of the whole; for example 5 slices of pizza cut in 8 pieces have numeric interpretation as a fraction 5/8. If the numerator is less than denominator, then the fraction is called proper fraction; otherwise, it’s called improper fraction. In case numerator and denominator are equal, then the fraction is simply transformed to number 1. Also, you could think of natural numbers as being a subset of fractions with denominators equal 1, for example: 5 = 5/1

Mixed number is the sum of a whole number and a fraction. For example 2 1/3 actually means 2+1/3; there is implicit plus sign between the whole and the fractional part in mixed number conventional form. Any mixed number could be converted into improper fraction and vice versa: 2 1/3 = 7/3

Decimals have multiple definition, thus some disambiguation is needed;
• Decimal is the number in base ten numeral system (the system itself is called a decimal system)
• Decimal fraction is a special type of fractions, whose denominator is a power of 10
In most cases decimals actually refer to the decimal fractions, either proper (containing the number 0 to the left of the decimal point), or improper (in case of non-zero integer standing to the left of decimal point).

Quick summary: natural numbers, integers, fractions and mixed numbers belong to the group of rational numbers. Any rational number could be presented as the quotient (ratio) of two integers, and also as either terminated or non-terminated, but periodic decimal (i.e., ending with some repeated numbers). Correspondingly any fraction could be converted into either exact (terminated) decimal, or non-terminated periodic decimals, e.g.: 2/5 = 0.4 (exact conversion); 2/3 = 0.6666... (non-terminated periodic decimal). Any decimal could be precisely converted into fraction, e.g.: 3. 27 = 3 27/100

Irrational numbers cannot be presented neither as a quotient of two integers, nor terminated or periodic decimals. Irrational numbers contain algebraic and transcendental numbers.

Algebraic numbers represents the solution of algebraic polynomial equations (for example, quadratic equation), that cannot be presented as the quotient of two integers (in other words, they cannot be written as fractions). As we have already mentioned the quadratic equation, then the first most intuitive example of irrational number will be square root of 2: (√2), which is approximately 1.41…. and so on.

Transcendental numbers are defined as irrational non-algebraic numbers, which cannot be presented neither as the quotient of two integers (which is the fundamental condition of numeric rationality), nor as the roots of algebraic polynomial equation with rational coefficients. The most commonly known transcendental numbers are: number pi, sometimes referred as "Ludolf number", which is a constant defined as the ratio of circle's circumference to its diameter, and number e, which is the base of natural logarithm function. Transcendental numbers, representing various mathematical abstractions, could be calculated using the infinite power series and corresponding computer algorithms.

Note: there is a fundamental difference between rational and irrational numbers: while rational numbers could be presented as either exact (finite) decimal or as an infinite decimal with repeated numeric group at the end, irrational numbers contain infinite line of decimals without repetition. Irrational numbers could be approximated by decimal or fractions with very high level of accuracy, but not exactly: the truncation error always applies.

All above mentioned numbers belong to the real (Re) domain. Further extension of the number theory is complex numbers comprise the real (Re) and imaginary (Im) parts; the latter ones are defined as square root of algebraic negatives, for example: 2 + 5i, where i= √(-1). Real numbers are graphically interpreted as points on the infinite length number line; complex numbers represent the points on a plane.

Wrapping up the topic with short historical passages: natural numbers emerged as the tool to count things. It’s really hard to accurately trace their origin, which is going back about 5 thousand years to ancient Egypt, Babylon and other ancient civilizations. The earliest known use of fractions, presumably by ancient Indus valley civilization, could be traced back in history to the 2800 BC. Fractions were extensively used by ancient Egyptian (presumably as early as 1000 BC) and Greeks. Introduction of null (zero) concept around 700 BC by Babylonians marked another milestone on the number theory evolutional timeline. The concept of irrationality was implicitly mentioned by Indian mathematician presumably in 7th century BC, but the first mathematically strong proof of irrationality is usually attributed to Pythagoreans (the Greek mathematicians), who discovered that the square root of 2 cannot be presented as fraction. The next giant leap in number theory was marked by the introduction of negative numbers, presumably in 3rd century AD, by Chinese and Indian mathematicians. Negatives in that distant past were commonly used to represent the debt value (which is very close to the modern asset/liability accounting principles). Finally, in XVI century AD Italian mathematician Girolamo Cardano defined complex numbers, which essentially brought the whole number theory to its contemporary stage.

The most famous transcendental number pi deserves the special coverage. Through the history, many mathematical geniuses of the past tried to find the best approximation for pi either in fractional or decimal forms. In 1596, German mathematician Ludolph van Ceulen published the pi approximation with 20-digit accuracy, and later extended it to astonishing 35 digits. In recognition of his work, number pi is also referred as “Ludolph number” (this number was engraved on his tombstone). Greek philosopher Archimedes introduced the simple approximation of pi as fraction 22/7. In XVII century Dutch mathematician Adriaan Metius published the approximate value of pi expressed as 355/113, which is accurate to 6 decimal digits (this fraction presumably was in use by Chinese mathematician since V century). Pi Day (March 14) is observed in many schools. To feel the spirit of the historical “pi adventure”, follow the steps of the mathematical titans and try to find the approximate value, given by ancient fraction 355/113, by using modern-time online fraction calculator (see the image attached to the article).

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