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

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”

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.

Natural exponents (powers)

Natural (see the next entry for explanation) exponent “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.

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.

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

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.

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.

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.

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 numbers, 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.

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