what are real numbers?

Answer 1

Either Dedekind cuts of the set of rational numbers, OR equivalence classes of Cauchy sequences of rational number.

Answer 2

In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. A real number may be thought of as any point on an infinitely long number line.

The discovery of more rigorous definitions of the real numbers was one of the most important developments of 19th century mathematics. Definitions in use today include equivalence classes of Cauchy sequences of rational numbers; Dedekind cuts; a more sophisticated version of "decimal representation"; and an axiomatic definition of the real numbers as the unique complete Archimedean ordered field. These definitions are all described in detail below.
The term "real number" is a retronym coined in response to "imaginary number".

A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero.

Real numbers measure continuous quantities. They may in theory be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823211247…. The three dots indicate that there would still be more digits to come.

Uses
Measurements in the physical sciences are almost always conceived of as approximations to real numbers. While the numbers used for this purpose are generally decimal fractions representing rational numbers, writing them in decimal terms suggests they are an approximation to a theoretical underlying real number.

A real number is said to be computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable.

The real numbers can be constructed as a completion of the rational numbers in such a way that a sequence defined by a decimal or binary expansion like {3, 3.1, 3.14, 3.141, 3.1415,…} converges to a unique real number.
Axiomatic approach
Let R denote the set of all real numbers. Then:

The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
if x ≥ y then x + z ≥ y + z;
if x ≥ 0 and y ≥ 0 then xy ≥ 0.
The order is Dedekind-complete, i.e., every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R.
The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational.

Answer 3

any number between negative infinity (-∞) and positive infinity (+∞) is a real number as long as it is not something like i = √ (-1).

"A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero.

Real numbers measure continuous quantities. They may in theory be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823211247…. The three dots indicate that there would still be more digits to come.

More formally, real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. These two together define the real numbers completely, and allow its other properties to be deduced. For instance, we can prove from these properties that every polynomial of odd degree with real coefficients has a real root, and that if you add the square root of minus one to the real numbers, obtaining the complex numbers, the result is algebraically closed."

Answer 4

Real number



One of the infinitely divisible range of values
between positive and negative infinity, used to represent
continuous physical quantities such as distance, time and
temperature.

Between any two real numbers there are infinitely many more
real numbers. The integers ("counting numbers") are real
numbers with no fractional part and real numbers ("measuring
numbers") are complex numbers with no imaginary part. Real
numbers can be divided into rational numbers and irrational
numbers.

Real numbers are usually represented (approximately) by
computers as floating point numbers.

Strictly, real numbers are the equivalence classes of the
Cauchy sequences of rationals under the equivalence
relation "~", where a ~ b if and only if a-b is Cauchy with
limit 0.

The real numbers are the minimal topologically closed
field containing the rational field.

A sequence, r, of rationals (i.e. a function, r, from the
natural numbers to the rationals) is said to be Cauchy
precisely if, for any tolerance delta there is a size, N,
beyond which: for any n, m exceeding N,

| r[n] - r[m] | < delta

A Cauchy sequence, r, has limit x precisely if, for any
tolerance delta there is a size, N, beyond which: for any n
exceeding N,

| r[n] - x | < delta

(i.e. r would remain Cauchy if any of its elements, no matter
how late, were replaced by x).

It is possible to perform addition on the reals, because the
equivalence class of a sum of two sequences can be shown to be
the equivalence class of the sum of any two sequences
equivalent to the given originals: ie, a~b and c~d implies
a+c~b+d; likewise a.c~b.d so we can perform multiplication.
Indeed, there is a natural embedding of the rationals in the
reals (via, for any rational, the sequence which takes no
other value than that rational) which suffices, when extended
via continuity, to import most of the algebraic properties of
the rationals to the reals.

Answer 5

complex numbers have a real and an imaginary part (5, 3i)
i is used to represent the square root of negative one
(-1)^(1/2)
real numbers include all rational and irrational numbers but do not include i.

Answer 6

is a number that can be graph on a number line. Ther is two types of real numbers: Irrational and Rational. Irrational means it can't be a non repeting number and rational number are repeating numbers.

Answer 7

all numbers are real numbers, or we are all dreaming.

Answer 8

as opposed to imaginary numbers like the root of -1
squareroot of -1 = i (i for imaginary)

Answer 9

the ones that are not fake.

Answer 10

I failed math.