Glossary

Absolute value of a number

The notation |x| represents an absolute value of the number x. If x is positive then: |x|=x, otherwise |x|=-x.

For example:
|2|=2, |-2|=-(-2)=2

Angle

Angle is a measure of inclination between lines or segments that intersect. Positive inclination between the segments: AO and BO means that we would need to rotate AO counter-clockwise for it to coincide with BO. The angle between the segments in question is denoted with ∠AOB. Letters of Greek alphabet are another very popular way to denote angles.

Circle

A circle is a collection of all points on a surface with a fixed distance to some point O on the same surface. Point O is the circle’s centre, and distance between circle’s member points and O is the circle’s radius.

A circle with centre in point O and radius r is denoted with a shorthand: C(O,r).

Equality vs Equation

Equality means that the term on the left of “=” is always equal to the one on the right. For example:
a+b=b+a
is true no matter what numbers we insert in place of “a” and “b”.

An equation on the other hand may or may not hold. For example:
2∙x=4
is true only for x=2.

An equality is a type of equation, but an equation may or may not be an equality.

On this website, the term equality is used only when I want to put an emphasis on the fact that some equations always hold.

Equivalence

A $\boldsymbol{\Leftrightarrow}$ B
means that predicates A and B must be either both true or both false.

For example:
“All balls in the bowl are black” $\boldsymbol{\Leftrightarrow}$ “A ball randomly chosen from the bowl is always black”

Euler’s theorem

If positive and natural numbers “a” and “n” have their greatest common divisor equal to 1, then a^φ(n)-1 is divisible by “n” without a remainder. φ(n) is the quantity of natural numbers x such that 0<x<n and GCD(x,n)=1.

Factorial

Factorial of a natural number n is equal to the multiplication of all consecutive numbers starting with 1 and ending with n. It is denoted with the exclamation mark:
n! := 1∙2∙ … ∙(n-1)∙n

For example:
3! = 1∙2∙3 = 6

0! = 1 by definition

Fermat’s little theorem

If p is a prime number, then for any whole x, the number x^p-1-1 divided by p is a whole number as long as GCD(x, p)=1.

Proof for the version with natural x

Greatest common divisor (GCD)

The greatest common divisor of natural numbers x and y denoted with GCD(x,y) is the greatest number by which both x and y are divisible without the rest.

For example GCD(10,5)=5, because both 10 and 5 are divisible by 5 and their GCD cannot be any greater.

Implication

A $\boldsymbol{\Rightarrow}$ B
and
B $\boldsymbol{\Leftarrow}$ A
mean that if the predicate A is true, then B must be true as well.

For example:
“All balls in the bowl are black” $\boldsymbol{\Rightarrow}$ “A ball randomly chosen from the bowl is black”

Note that the implication in the other direction is false in this case.

Infinity

Infinity denoted with the symbol ∞ is an object greater than every number.

I said object because infinity itself is not a number. It isn’t because for example
∞ – 9 is also greater than every number so according to our definition:
∞ – 9 = ∞

On the other hand, the equation:
x – 9 = x
is not true for any number x so ∞ cannot be a number.

Irrational numbers

Contrary to rational numbers, irrational numbers cannot be expressed as a ratio of whole numbers. Every irrational number r can be approximated by some rational d and g such that:

d<r<g

where the difference g-d can be as small as we wish.

Line

Line is an infinitely long segment. It coincides with the shortest route between any two points that belong to it.

Natural exponents (powers)

Natural exponent (or power) “n” placed as a superscript after x i.e. xⁿ is used as a shorthand for x multiplied by itself n times. In the notation xⁿ, x is called the base, n – the exponent and xⁿ – the power.

Although x multiplied by itself zero times: x⁰ makes no sense, we will assume: x⁰=1 for every x apart from x=0. There is a good reason to do that.

Example:
x=2, n=3

xⁿ = 2³ = 2 ∙ 2 ∙ 2 = 8

Negative exponents (powers)

Number x raised to the negative power -r (r>0) is defined as:
x^-r:=1/x^r

Natural numbers ℕ

Numbers used to describe quantity of non-divisible objects, that is: one, two, three etc.

The set containing all the natural numbers is denoted with the symbol: ℕ. Zero is sometimes included as well.

Parallel lines

If the shortest distance between some line g and every point on some other line h is always the same, lines g and h are parallel.

PEMDAS

Pemdas is a mnemonic to remember the order in which mathematical operations are performed. It stands for: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Example:

8 + 2 ∙ (3 – 9 ÷ 3² ∙ 2 + 1)

The term in parentheses is evaluated first:
3 – 9 ÷ 3² ∙ 2 + 1 =

We don’t have any more parentheses, so power goes next:
= 3 – 9 ÷ 9 ∙ 2 + 1 =

then multiplication and division from left to right:
= 3 – 9 ÷ 9 ∙ 2 + 1 = 3 – 1 ∙ 2 + 1 = 3 – 2 +1 =

and finally addition and subtraction from left to right:
= 3 – 2 + 1 = 1 + 1 = 2

After inserting it back to the initial equation we have:
8 + 2 ∙ (3 – 9 ÷ 3² ∙ 2 + 1) = 8 + 2 ∙ 2 =

and the multiplication goes first, followed by the addition:
= 8 + 2 ∙ 2 = 8 + 4 = 12

Pentagram

A regular five-pointed star used as a symbol in many cultures, for example by the Pythagoreans in ancient Greece.

The Pythagoreans believed that the world can be described with numbers. At their time it was more a conviction than anything solid, yet in the end they were proven right and that is why a pentagram is the logo of this site.

Perpendicular lines

Lines inclined at the right angle.

Pi (Π) as a multiplication shorthand

Multiplication shorthand is useful for indexed variables e.g. x_i where index “i” is a natural number. If there is a need to multiply all x_i with the index “i” running from “n” to “m”, n<m, through all the consecutive natural indices, the following shorthand is used to describe the result:

$\prod_{i=n}^{m}x_i$

Example:

$\prod_{i=0}^{4}x_i=x_0\cdot x_1\cdot x_2\cdot x_3\cdot x_4$

Plane

A plane is constructed using any three points in space that are not on the same line. Once these are chosen, the other points of the plane can be found using a rule that all points on a line that passes through two distinct points that are already on the plane, belong to this plane as well.

Point

A point is that which has no parts (according to Euclid). It represents a location in space. It has neither length nor width nor height.

Positional numeral system (positional notation)

In the positional system, every number is expressed by a sequence of symbols taken from a set with limited quantity of members. The members of the set are called digits and they correspond to consecutive natural numbers starting with zero. The quantity of digits is called the base.

The system with the set’s base equal to ten, is called decimal and the set of all digits consists of: 0,1,2,3,4,5,6,7,8,9. Numbers greater than nine are expressed using a sequence of digits.

The system with the set’s base equal to two, is called binary, and it has only two digits: 0,1.

These two systems are the most popular, though in principle, the base can be equal to any natural number greater than one.

For every positional system the sequence of digits “d_n d_n-1… d₁d₀” is equivalent to the following sum:

$d_n d_{n-1}\ldots d_1 d_0:=\sum_{i=0}^{n}d_i\cdot \Gamma^{i}$

where Σ is a summation shorthand, Γ represents the base, and the superscript “n” the exponent.

Directly from the above it follows that:

10 = 1 ∙ Γ¹ + 0 ∙ Γ⁰ = Γ

In other words, 10 always represents whatever the base is e.g. it is interpreted as ten in the decimal system, and two in the binary. By the same token 1 followed by “n” zeros, represents Γ ⁿ :

n-th power of the base	Representation in the positional system	Meaning in the decimal system	Meaning in the binary system
Γ⁰=1	1	One	One
Γ¹=Γ	10	Ten	Two
Γ^two=Γ∙Γ	100	Hundred	Four
Γ^three=Γ∙Γ∙Γ	1000	Thousand	Eight

Example:

101 = 1 ∙ 10^two + 0 ∙ 10¹ + 1 ∙ 10⁰ = 1 ∙ 100 + 0 ∙ 10 + 1 = 1 ∙ 100 + 1
is equal to hundred and one in the decimal system and five in the binary.

For fractions i.e. numbers 0<r<1, r can be expressed as:

$r = \sum_{i=-1}^{-n} d_i \cdot \Gamma^i$

where n may be infinity (negative powers are explained here).

If we deal with some number x that has whole and fractional part, x can be written as a sum:

$x = \sum_{i=m}^{-n} d_i \cdot \Gamma^i$

where m,n>0 and n may be infinite. The shorthand for this series is:

$d_m\ldots d_2d_1d_0.d_{-1}d_{-2}\ldots d_{-n}:=\sum_{i=m}^{-n}{d_i\cdot\Gamma^i}$

where dot separates whole from the fractional part.

Predicate

Any statement with an unambigous meaning which can only be true or false. To shorten the notation true is indicated with the number 1 and false with 0.

Examples:

“lions are predators” = 1

“snails are birds” = 0

Prime number

Prime number is a natural number greater than 1 that is divisible (without a remainder) only by 1 and itself.

Properties of addition and multiplication

Consider variables “a”, “b” and “c” that can be equal to any number. With these three variables we can express properties of addition and multiplication as follows.

Commutative property

of addition: a + b = b + a

and multiplication: a ∙ b = b ∙ a

Associative property

of addition: (a + b) + c = a + (b + c)

and multiplication: (a ∙ b) ∙ c = a ∙ (b ∙ c)

Distributive property of multiplication over addition

(a + b) ∙ c = a ∙ c + b ∙ c

This last formula can be extended to:

$(\sum_{i=0}^{n}x_i) \cdot a=\sum_{i=0}^{n}(x_i\cdot a)$

where sigma (Σ) is a shorthand for summation and x_i denote indexed variables.

The consequence of commutative and associative properties of addition is that we can add numbers in any order we want and the result is always the same.
For example: 2 + 3 + 4 + 5 = 2 + 5 + 4 + 3 = 5 + 4 + 3 + 2 etc.

Commutative and associative properties of multiplication produce a similar result. In this case we change the order of multiplications.
For example: 2 ∙ 3 ∙ 4 ∙ 5 = 2 ∙ 5 ∙ 4 ∙ 3 = 5 ∙ 4 ∙ 3 ∙ 2 etc.

Bear in mind that when performing the operations described above, we use the PEMDAS convention.

Pythagorean theorem

For a triangle constructed with segments: a, b and c units long where the segments: a and b are at the right angle, the relation between lengths: a, b and c is given by the formula: a²+b²=c²

Rational numbers ℚ

Rational numbers are defined as a ratio of two whole numbers: a/b, where b cannot be equal to 0.

The set of all rational numbers is denoted with the symbol: ℚ.

Radians

Radian is a unit used to measure an angle of inclination expressed as a ratio of the length L of the arc cut from a circle by two lines intersecting at the circle’s centre. One radian corresponds to the situation L=R. On flat planes the ratio L/R is the same no matter how big is the circle.

Rational exponent (power)

Rational exponent (or power) of number x is defined as:

x^a/b:=(x^1/b)^a

where “a” and “b” are whole numbers, x^1/b is a b-th root of x and superscript ^a on the right indicates that x^1/b is raised to power “a” that is positive or negative.

Real numbers ℝ

All rational and irrational numbers together constitute a set of so called real numbers denoted with symbol ℝ.

Reductio ad absurdum

Reasoning technique relying on the equivalence:

~(A $\boldsymbol{{\Rightarrow}}$ B) $\boldsymbol{\Leftrightarrow}$ (A $\boldsymbol{\wedge}$ ~B)

meaning: negation (~) of the implication A $\boldsymbol{{\Rightarrow}}$ B is equivalent ( $\boldsymbol{\Leftrightarrow}$ ) to the predicate A and ( $\boldsymbol{\wedge}$ ) negation (~) of the predicate B.

Example:

A := “all balls in the bowl are black”

B := “a ball randomly chosen from the bowl is black”

Negation of B states:

~B = “a ball randomly chosen from the bowl is not black”

A and ~B cannot be true at the same time so (A $\boldsymbol{\wedge}$ ~B) is false. From the equivalence discussed above it follows that ~(A $\boldsymbol{{\Rightarrow}}$ B) must be also false, so the implication A $\boldsymbol{{\Rightarrow}}$ B has to be true.

In plain English the implication A $\boldsymbol{{\Rightarrow}}$ B means: if “all balls in the bowl are black” then “a ball randomly chosen from the bowl must be black”.

Remainder of division

If one natural number is greater than the other, we can ask: how many times one can fit into the other. For example 2 fits into 7 three times, however, we get a leftover equal to 1. This leftover is called the remainder of the division 7/2.

Right angle

Imagine segment AB and some line h intersecting AB in point C. If the inclination between line h and segment AC is the same as inclination between line h and segment BC then the angle between AB and h is right. Right angles are usually denoted with a small square in the point of intersection.

Roots

n-th root of number x is denoted as x^1/n or $\sqrt[n]{x}$ . It is such a number that raising it to power n produces x:

( x^1/n )ⁿ = x

For example:

Second root of 4 better known as square root of 4 i.e. 4^1/2 is equal to 2 because 2²=4

Third root of 27 better known as cube root of 27 i.e. 27^1/3 is equal to 3 because 3³=27

Segment

a collection of points along the shortest route between two locations. It has a length yet doesn’t have any width and height.

The segment linking points A and B is denoted with AB or BA, and its length is denoted with |AB| or |BA|.

Sigma (Σ) as a summation shorthand

Summation shorthand is useful for indexed variables e.g. x_i where index “i” is a natural number. If there is a need to add all x_i with the index “i” running from “n” to “m”, n<m, through all the consecutive natural indices, the following shorthand is used to describe the resulting sum:

$\sum_{i=n}^{m}x_i$

Example:

$\sum_{i=0}^{4}x_i=x_0+x_1+x_2+x_3+x_4$

Simulacra Mundi

The name of this website (simulacra mundi) is modelled upon the term axis mundi introduced by Mircea Eliade in “Images and Symbols”, a book about religious symbolism.

Latin word simulacrum means likeness, imitation, representation or model, like we would say today. Simulacra is simply plural of simulacrum.

Mundus among other things means cosmos and mundi is a genitive of mundus thus:

simulacra mundi = simulacra of mundus.

Thales’ theorem

Thales’ theorem expresses the relationships between the lengths of segments cut out of intersecting two lines by a pair of other two lines parallel to each other:

Segment AC is parallel to BD. The relations between the lengths of segments are given by the ratios:

|OC| / |OD| = |OA| / |OB|

|OC| / |CA| = |OD| / |DB|

|OC| / |OA| = |CD| / |AB|

Translation

Translation moves objects in space without changing their orientation.

2d translation
3d translation

Variable

A symbol representing a number but not a concrete one. A variable can be equal to any number unless some restrictions are imposed.

For example:

The equation:
a=5
is true only when “a” is equal 5.

Thanks to equations we can create dependencies between variables:
a+b=5
The above statement is true for many different values of “a” and “b” as long as their sum is 5.

Whole numbers ℤ

The set of whole numbers denoted with the symbol ℤ consists of all natural numbers, zero, and a number negative to every natural one.

A number negative to a positive “x” is denoted with “-x” meaning:
-x:=0-x

In other words, it is such a number that:
x+(-x)=-x+x=0