Real Natural
How many different kinds of numbers are there and what is the distinction, i.e. whole numbers, real, natural ?
There are Natural , Real, Whole and is there anything else and what is the distinction between the different types of numbers and what do they mean ? Thanks
There are as many different kinds of numbers as you care to describe. Here are some of the most important kinds:
The first two sets you work with in arithmetic are:
{1, 2, 3, ... } and
{ 0, 1, 2, 3, ... }
Either of these sets might be called the "Natural Numbers," the "Counting Numbers," or the "Whole Numbers." Consult definitions in your text, because they differ from one text to another. To avoid ambiguity you can call the first set the "Positive Integers," and the second set the "Nonnegative Integers."
Integers are the Natural Numbers together with 0 and all of the negatives of zero: ..., -3, -2, -1, 0, 1, 2, 3, ... . In contrast to Natural, Counting, and Whole Numbers, use of "Integers" is standard.
The Rational Numbers are numbers which can be expressed as fractions with an integer numerator and integer denominator. In decimal form, these are numbers which can be expressed as terminating or repeating decimals. The integers are a subset of the rationals.
The Real Numbers are numbers which can be expressed as decimals, including those which neither terminate nor repeat like π or √2. The rationals are a subset of the reals.
(There are a couple of extensions of the real numbers which are sometimes used in analysis: the Extended Real numbers, which consists of the real numbers together with positive and negative infinity; and the Projective Real numbers, which consists of the real numbers together with a single point at infinity. In Topology, the Projective Real number line is called the "one-point compactification" of the reals.)
The Irrational Numbers are those Real Numbers which are not Rational Numbers.
The Complex Numbers are those numbers which can be written in the form a+bi, where a and b are real numbers and i = √(-1). The reals are a subset of the rationals. For historical reasons, the number "i" is sometimes called the "imaginary" number, and the "bi" part of a+bi is sometimes called the "imaginary" part. Really, the number i is no more imaginary than, say, -1.
Those sets will get you quite a ways, though other sets of numbers are also interesting.
The Algebraic Numbers are numbers which are the roots of polynomials with rational coefficients. This includes all rationals and some but not all of the reals and complex numbers.
The Transcendental Numbers are those real and complex numbers which are not algebraic.
The Infinitesimal Numbers are numbers which are defined to be greater than zero but smaller than any positive real number. These numbers are studied in Nonstandard Analysis.
The Cardinal Numbers are numbers which are used to describe the size of sets. The natural numbers (in the sense of "nonnegative integers") are a subset of the Cardinal Numbers. But sets can also have infinitely many members -- and it turns out that there are different sizes of infinity! The Cardinal Numbers include numbers designating different sizes of infinity.
The Transfinite Numbers are cardinal numbers larger than any natural number.
The Ordinal Numbers are also infinite numbers, but are used to designate order properties of sets. The natural numbers are a subset of the ordinals, but not the transfinite numbers: every transfinite number corresponds to infinitely many ordinal numbers.
The Ring of Integers modulo n is the set of n integers {0, 1, 2, ..., n-1}. You perform addition and multiplication on such a ring by adding and multiplying in the usual way, then finding the remainder upon division by n.
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