Download Numbers Natural 0, 1, 2, 3, 4, or 1, 2, 3, 4

Numbers
Natural
Integers
Rational
Real
Complex
0, 1, 2, 3, 4, ... or 1, 2, 3, 4, ...
..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...
a
⁄b where a and b are integers and b is not zero
The limit of a convergent sequence of rational numbers
a + bi or a + ib where a and b are real numbers and i is the square root of −1
Special Numbers
Phi (Golden Ratio): where the Greek letter phi,
value is approximately equal to:
, represents the golden ratio. Its decimal
π
: Where the Greek letter
is a mathematical constant that is the ratio of a circle's
circumference to its diameter. Its decimal value is approximately equal to 3.14159.
Peano axioms
The Peano axioms give a formal theory of the natural numbers. The axioms are:
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There is a natural number 0.
Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a)
is a + 1.
There is no natural number whose successor is 0.
S is injective, i.e. distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠
S(b).
If a property is possessed by 0 and also by the successor of every natural number which
possesses it, then it is possessed by all natural numbers. (This postulate ensures that the
proof technique of mathematical induction is valid.)
It should be noted that the "0" in the above definition need not correspond to the number zero.
"0" simply means some object that when combined with an appropriate successor function,
satisfies the Peano axioms. All systems that satisfy these axioms are isomorphic, the name "0" is
used here for the first element (the term "zeroth element" has been suggested to leave "first
element" to "1", "second element" to "2", etc.), which is the only element that is not a successor.
For example, the natural numbers starting with one also satisfy the axioms, if the symbol 0 is
interpreted as the natural number 1, the symbol S(0) as the number 2, etc. In fact, in Peano's
original formulation, the first natural number was 1.
Constructions based on set theory
A standard construction in set theory, a special case of the von Neumann ordinal construction, is
to define the natural numbers as follows:
Set 0 := { }, the empty set, and define S(a) = a {a} for every set a. S(a) is the
successor of a, and S is called the successor function.
By the axiom of infinity, the set of all natural numbers exists and is the intersection of all
sets containing 0 which are closed under this successor function.
This then satisfies the Peano axioms. Each natural number is then equal to the set of all
natural numbers less than it, so that
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0={}
1 = {0} = {{ }}
2 = {0, 1} = {0, {0}} = {{ }, {{ }}}
3 = {0, 1, 2} = {0, {0}, {0, {0}}} ={{ }, {{ }}, {{ }, {{ }}}}
n = {0, 1, 2, ..., n−2, n−1} = {0, 1, 2, ..., n−2,} {n−1} = {n−1}
S(n−1)
(n−1) =
and so on.
When a natural number is used as a set, this is typically what is meant. Under this definition,
there are exactly n elements (in the naïve sense) in the set n and n ≤ m (in the naïve sense) if and
only if n is a subset of m. Also, with this definition, different possible interpretations of
notations like Rn (n-tuples versus mappings of n into R) coincide. Even if the axiom of infinity
fails and the set of all natural numbers does not exist, it is possible to define what it means to be
one of these sets. A set n is a natural number means that it is either 0 (empty) or a successor, and
each of its elements is either 0 or the successor of another of its elements.
Russell’s Paradox
R is the set of all sets that are not members of themselves. This creates the famous paradox: