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Number Realms

Classifications of Numbers

  • Real numbers

  • Complex numbers

  • Rational numbers

  • Integers

A fundamental division of numbers is the rational or irrational classification. It seems that nature does not like to use digital mathematics to relate fundamental ratios, such as $π$, $e$, the Golden Ratio, etc. Irrational numbers cannot be defined by the ratio of two integers.

Mathematics relies on a common understanding of what symbols represent, and what the limitations and scope of these symbols are. Some of the complexity of the naming system derives from the historical sequence of discovery. Real numbers were the assumed only realm of mathematics, until the 18th century discovery of 'imaginary' numbers.

Number theory

What was generally understood as 'arithmetic' evolved by the turn of the 20th century to the more all-embracing 'Number Theory', which is defined as the study of whole numbers (integers). These can be interpreted as entities of their own, or as solutions to geometric problems. Number theorists have a particular obsession with prime numbers.

Pi

π (pi) is the ratio of the circumference of a circle to its diameter, and is very approximately: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 .... (the current record for the definition of pi is 12,100,000,000,050 decimal places!!!!!!!)

Euler's No.

$e$ is named after Leonid Euler, who Simon de lapace called the 'Father of all we mathematicians'.

$e$ is defined as:

$$e=∑↙{n=0}↖{∞}1/{n!} = 1 + 1/1 + 1/{1⋅2} + 1/{1⋅2⋅3} + ....$$

Jakob Bernoulli discovered $e$ as the limit of compound interest: the maximum yield on 1 euro invested at 100% interest for one year, if compounded an infinite number of times, is $e$ dollars. (Not that Swiss banks have ever offered 100% interest...).

Euler's 'beautiful' equation:

$$e^{iφ}=cosφ+isinφ$$

Real numbers ℝ are the realm of numbers which are either 'rational' ℚ or 'irrational':

Rational does not refer to the common usage of 'making sense', although admittedly there is some element of that. Instead, 'rational' refers to 'ratio', which in mathematics means these numbers can be expressed as a quotient of two other real numbers. For example, the integers ℤ 1, 2, 3, 4, etc., are real numbers, but so are any ratio of these integers. So, fractions, such as ½, ⅓, and ¾ are also real and rational numbers.

Rational numbers can always be written in the form $a/b$, where a and b are whole numbers.

A square root of a squared number is rational. Otherwise the root is irrational. i.e. $√{a^2} = a$, but $√{b⋅a}$ does not give a rational number solution.

Irrational numbers, therefore, are numbers that cannot be expressed as a fraction, or quotient, of integers. Examples are π and e. Neither of these may be expressed as a ratio of two integers. Their values are defined as a geometric series, or for more practical purposes a open-ended series of decimals:

π is the ratio of the circumference of a circle to its diameter, and is very approximately: 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679 .... (the current record for the definition of pi is 12,100,000,000,050 decimal places).

Euler's number, e, is the base for the natural logarithm, and is very approximately e = 2.71828 18284 59045 23536 02874 71352 66249 77572 47093 69995 95749 66967 62772 40766 30353 54759 45713 82178 52516 64274 27466 39193 20030 59921 81741 35966 29043 57290 03342 95260 59563 07381 32328 62794 34907 63233 82988 07531 95251 019...

Other common examples of irrational numbers include the square root of two √2 and the golden ratio.

Transcendental Numbers

Transcendental numbers are those, for example e and π, which cannot be the root of an algebraic expression with rational coefficients.

Rules of surds

Surds (or radicals) are irrational numbers which contain square roots. The rules of surds are:

$(√a)^2 = a$ ; $√{a⋅b} = √a ⋅ √b$ ; $√{a/b} = {√a}/{√b}$

Sicne surds are irrational, it is more accurate to leave the root part, and not use a calculator to convert it to a decimal approximation.

LCD, least common multiplier

The least (or lowest) common multiplier of a set of integers is the lowest integer which can be divided by all the integers in the set without leaving a remainder.

For example, the LCM of 8 and 12 is 24, of 5 and 7 it is 35.

$$lcm(a,b) = {|a⋅b|}/{gcd(a,b)}$$

where gcd is the greatest common divisor (the largest integer that will divide exactly into all members of the set.

e.g. lcm(9,21) = ${9⋅21}/3 = 63$.

HCF, highest common factor

The highest common factor is an integer which divides into all members of a set exactly, leaving no remainder.

To determine the HCF, find the prime factors of the numbers in the set, and multiply all the factors that are common to both numbers.

e.g. the HCF of 18 and 24:
18 = 3 × 3 × 2;
24 = 3 × 2 × 2 × 2
The common prime factors are: 3 and 2, so the HCF = 3 × 2 = 6

Content © Andrew Bone. All rights reserved. Created : December 25, 2014

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