What is real number?

Answer 1

Any number that does not produce a math error on a calculator: any number that can be represented by a decimal place, no imaginary numbers such as a squareroot of -1 (or any negative number)

Answer 2

Don't add extra meaning to the name of the category. Real numbers are just a category of numbers. They include all rational numbers and irrational numbers, HOWEVER, that's only because rational and irrational numbers are defined as belonging to the set of reals. (A bit of circular logic, there).

Real numbers do not include 'complex numbers' (also known as imaginary numbers, but, once again, don't assign any special meaning to the name, 'imaginary' is just the name of the category).

By the way, the only way you could depend on the error message of a calculator to tell you whether or not a number is real or not is if you buy a really cheap calculator. Good calculators, such as a TI-85, TI-86, etc, can handle complex numbers as well as real numbers. In fact, you can even solve problems with complex numbers on a slide rule, if you have enough understanding of complex numbers.

For curious guy, couldn't you factor the square root of 4 a little further and break it down into: sqrt(2) * sqrt(2) * sqrt(-2) * sqrt(-2)?

Just being facetious. Sometimes, you have rules just so you can have some sort of organization that people might comprehend. Same goes for the rules about what constitutes a 'Real' number and an 'Imaginary' number. They're just names of categories that are organized in a way to help a person keep the rules straight.

Answer 3

The field of all rational and irrational numbers is called the real numbers, or simply the "reals," and denoted . The set of real numbers is also called the continuum, denoted . The set of reals is called Reals in Mathematica, and a number can be tested to see if it is a member of the reals using the command Element[x, Reals], and expressions that are real numbers have the Head of Real. The real numbers can be extended with the addition of the imaginary number i, equal to . Numbers of the form , where and are both real, are called complex numbers, which also form a field. Another extension which includes both the real numbers and the infinite ordinal numbers of Georg Cantor is the surreal numbers. most all real numbers are lexicons, meaning that they do not obey probability laws such as the law of large numbers

Answer 4

The collection of real numbers is obtained by 'completing' the collection of rational numbers (fractions). There are a couple of main ways of doing this technically: Dedekind cuts, and equivalence classes of Cauchy sequences. The easiest to understand is the Dedekind cuts method.

The basic idea of both methods is that the collection of rational numbers has a lot of 'holes' that can be filled in. They are filled in by real numbers. For example, there are rational numbers whose square is less than 2 and rational numbers whose square is more than 2, but none whose square is equal to 2. This means there is a 'hole' in the rational numbers. When the hole gets filled in, the real number there is called the square root of 2.

It turns out that every real number has a decimal expansion, but the part to the right of the decimal place can be infinite (the part to the left is always finite). The expression as a decimal expansion is not unique, however, since .9999... and 1.0000.. are the same real number (there are no rational numbers between them!).

The completeness property of the real numbers is the crucial one for many applications in calculus, analysis, topology, etc. Essentially, any subset of the real numbers which has an upper bound (i.e. some real number bigger than everything in the set) has a LEAST upper bound (an upper bound smaller than all other upper bounds). This allow proofs of existence results for solutions of equations, etc.

Answer 5

I think that this is the best definition of a real number but website below has other definitions of a real number:

Rational (fractions) and irrational (numbers with non-recurring decimal representation) numbers. The set of real numbers is denoted as ‘R’ for real. In computing, any number with a fractional (or decimal) part. Basically, real numbers are all numbers except imaginary numbers (such as the square root of -1). See Types of Numbers

Answer 6

The term "number" is undefined.
The term "point" is undefined.
The term "dimension" is undefined.
A point, however, is the graphical representation of a number.
A line is the set of all points in one dimension.
The *set of real numbers* can be assigned one-to-one correspondence with the points on a line. They are the numbers which create a continuum in one dimension.

A "real number" is an element of the set of real numbers.

I just want to state that my preference is not to call these "real numbers" but linear numbers for the reason that people assume that complex numbers [planar numbers] are indeed imaginary. They are not; they have practical application.


Bob G, you get i²*2*2 = -4.

Answer 7

Real numbers consist of rational numbers and irrational numbers.

You will find that all integers, fractions, mixed numbers and decimals (terminating and recurring) can be expressed in the form of a/b or -(a/b), where a and b are whole numbers and b is not equal to zero. These numbers are called rational numbers. Eg. 2, 0, -1, -1.25, -0.66

Numbers such as sqrt. 2 and sqrt. 3 are irrational numbers because they cannot be expressed in the form of a/b or -(a/b), where a and b are whole numbers and b is not equal to zero.

Here's a brief idea of real numbers:
Real numbers
1. rational numbers
1.1 non-integers
1.2 integers
1.2.1 negative integers
1.2.2 whole numbers
1.2.2.1 zero
1.2.2.2 natural numbers
2. irration numbers

Answer 8

A real number is any number that can be represented as a point on a number line.

For example, Pi is a real number, as it can be placed on a number line (at around 3.14). However, a number like the square root of -1 cannot be placed on a number line, and so is not a real number.

Answer 9

Numbers can be divided into REAL or IMAGINARY.

If you solve the equation:

x^2 = 4

the answers are +2 and -2, both are real.

X^2 = -4

the answers are +2i and -2i, both imaginary, where i is square root of -1.

Complex no. is formed by combination of real and imaginary:
Eg. 2+ 2i is complex.

So in general the numbers ( can include fraction, rational or irrational) which do not have "i" involved in them are REAL...


[ to Bob G below...

I think you can't do that because a quadratic (highest power 2) equation has only 2 solutions.... ]

Answer 10

a real number is a number that can be negative or positive, but it can not have a .09 or something like that, hope I helped...