The Echoes of the Mind (48-3) Numbers

Numbers

It is the civilized world’s facility with manipulating the representations of reality provided by numbers that has led to such awesome material progress in the past few centuries. ~ English statistician David Hand

There are numerous sets of numbers. The numbers used for tallying are natural numbers {1, 2, 3,…}. Adjoining zero to the natural numbers gives the set of whole numbers.

Extending whole numbers into negative territory yields integers {…, -3, -2, -1, 0, 1, 2, 3,…}.

It took mathematicians quite a while to accept negative numbers. The Greeks, so interested in geometry, had no need for negatives.

By the 7th century Indian bookkeepers were using negative numbers to represent debt. Italian polymath Gerolamo Cardano used negative numbers in his 1545 algebra textbook Ars Magna.

Among the natural numbers, a number greater than 1 which has only itself and 1 as factors is a prime number {2, 3, 5, 7, 11, 14, 17, 19, …}. A natural number that is not a prime is called a composite. Euclid demonstrated ~300 bce that there are an infinite number of primes.

Prime numbers have long been a fascination for mathematicians. Many questions about them remain open. One of the oldest and best-known issues is Goldbach’s conjecture: that every even number greater than 2 can be expressed as the sum of 2 primes. (Goldbach’s conjecture arose from an aside that German mathematician Christian Goldbach wrote in a letter to Leonhard Euler on 7 June 1742.) Generally assumed to be true, and shown so far to 4 × 1018, the conjecture remains unproven despite strenuous effort.

The set of real numbers includes integers, rational, irrational, algebraic, and transcendental numbers.

Any number that can be expressed as a fraction of 2 integers is a rational number (e.g. 3/4). A rational number may have an infinitely repeating decimal fraction (e.g. 82 / 111 = 0.738738…).

A real number that is not rational is irrational (e.g. √2, which equals 1.41421356…). An irrational number has decimal representations which are neither terminating nor repeating.

An algebraic number is one that can be expressed as the root of a non-zero polynomial with 1 variable and rational coefficients. (It is easier to understand that algebraic numbers are real numbers that are not transcendental.)

A polynomial is an algebraic expression consisting of variables and coefficients, where the expression (aka function) is limited to addition, subtraction, multiplication, and non-negative integer powers (aka exponents). An exemplary algebraic expression is:

3y2–5x + 7

In the above expression, x and y are variables (aka indeterminates): the letters represent numbers which may vary in value. A variable is a placeholder in a mathematical formula. A number before a variable is a coefficient, which stands ready to multiply whatever value a variable takes.

There are specific naming conventions for variables in the different branches of mathematics. For instance, the axes of 3d coordinate space are always x, y, and z.

An exponent signifies multiplication to a certain power (number of times). For example:

y3 = y ´ y ´ y

The irrational, real, or complex numbers that are not algebraic are transcendental. Transcendentals are numbers with decimal quantities that go on and on without repeating.

Though only a few classes of transcendental numbers are known, they are not rare. Indeed, most real and complex numbers are transcendental. But it can be extremely difficult to prove that a given number is transcendental.

π and e are the best-known transcendental numbers.

π (3.14159…) is the ratio of a circle’s circumference to its diameter. But p shows up in many situations unrelated to circles. For instance, the series 1/12 + 1/22 + 1/32 + 1/42 + 1/52… (= 1 + 1/4 + 1/9 + 1/16 + 1/25…) gets closer to the value π2/6 (= 1.645…) as more terms are added. π is also intimately (and mysteriously) involved with how prime numbers are distributed.

e (2.71828…) is the base of the natural logarithm, which is a logarithm with e as its base. In other words, the natural log is the inverse of e. Practically, e is about time under continuous growth.

The miraculous powers of modern calculation are due to three inventions: the Arabic notation, decimal fractions, and logarithms. ~ American mathematics historian Florian Cajori

A logarithm is the exponent of the power to which a base number must be raised to equal a given number. The 2 equations below are equivalent.

y=bx x=logb(y)

For example, the logarithm of 1,000 to base 10 (the decimal system) is 3.

1000=103 3=log10(1000)

e can be defined in many ways. For one, e = (1 + 1/n)n as n approaches infinity.

The natural log – written loge or ln (from the Latin logarithmus naturali) – gives the time needed to reach a certain level of growth, given constant growth.

For example, for 10 times growth of something that grows 100% annually, the wait time is 2.302 years (ln(10)).

With g as growth (at a 100% compound growth rate) and d as duration:

g=ed d=ln(g)

10 (times growth)=e2.302 2.302 (years) =ln(10)

ex is a scaling factor: the amount of continuous growth after a certain (x) duration. The inverse – ln(x) – is the duration needed to reach a certain level of growth.

ex = erate x time; with rate = 100% → e1.0 x time = etime

If the growth rate is only 5% rather than 100%, then the duration is 20 times longer (1.0 / 0.05).

Suppose we want to know how long a wait for a 30-times growth at a growth rate of 5% per period.

ln(30) = 3.4; i.e., rate x time (at 100%) = 3.4

for: (rate = 0.05) and nominally (time = 3.4)

0.05 (rate) x 3.4 (time) = 68

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The sets that comprise the all-encompassing real numbers form a nested hierarchy, with a bifurcation at the top between rational and irrational numbers. One oddity is the set of algebraic numbers, which includes all rational numbers but also selectively some irrational ones.

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Materialism drove mathematics. The earliest and essential need for math was accounting of property: both quantities of goods and tracts of land. Thus arose arithmetic and geometry.