what is a real number? what are some examples? whats isn't a real number? Some examples?

Answer 1

All numbers, including rational and irrational numbers, with no imaginary component are real.

Answer 2

In math it is common to form sets of numbers into what are known as fields. For any two elements in these fields, you can add them, subtract them and multiply them. For any element in a field and another element that isn't 0 (for that field) you can divide the first by the second. Division by 0 is not defined, thus can not be done.

A basic construction of the Real numbers is this:

Start with your basic counting numbers 1, 2, 3, 4, 5, . . . and so on (maybe 0 is included). You can add any of these numbers together and they are still in the set (for example 3+5=8 is in the set). But what happens when you subtract. If a
This is why we introduce negative numbers {-1,-2, -3, -4, . . .}. Now we can add and subtract and we still have all the numbers (make sure you add 0 in too, if you didn't before). We can also multiply any two numbers and stay in the set (3*4=12). But what happens when we divide by a number? 4/1=1, 15/(-3)=-5. What about 5/2? 2<5/2<3 so 5/2 is not in the set.

Now we add all ratios (except for dividing by 0). Thus all numbers a/b where b≠0. Now we can add, subtract, multiply, and divide. Actually, we now have a field. We call this field the Rational numbers. But is this field all of the numbers?

It may seem like it is all of them, but really it isn't. It can be shown that for any rational number q q^2≠2 (I'm not going to show that, so please believe me). Since q^2≠2, either q^2<2 or q^2>2. Consider 1 (it is obviously rational): 1^2=1<2. Therefore there is a rational number that has a square less than 2. Likewise 2^2=4>2, and there is a rational number that has a square greater than 2. Why is this important? Because we know that there are numbers in each of these sets:
A= the set of all rational numbers whose square is less than 2
B= the set of all rational numbers whose square is greater than 2

Now, I'm already being long winded, so I'll just say this next part:
Since we know that there is a number in each of the sets there has to be numbers between the sets. But the number between the sets is not rational, so we know that we don't have all the numbers with only the rational numbers. So, the Real numbers are all the numbers that can't be written as rational numbers, but are still between two rational numbers.

But now consider solving x^2=-1. Let's do what we did before: We can easily show that there are no rational numbers r such that r^2=-1. What are all the rational numbers r such that r^2>-1. Well, r^2≥0>-1, so all of them. That means that there aren't any rational numbers r such that r^2<-1. Thus our solution x is not between any two rational numbers, and is not real. But we want a solution, so we make a number i such that i^2=-1. We call any number that can be written as a+bi (where a and b are both real numbers) a complex number. This is actually as far as we can go in this direction.

Examples:

Integers: 1, 6, 13, -27, 100006643;
Rational numbers: 1, 3, 3/8, 6/27, 0.11111...;
Real numbers: 3/8, 1, √2, π, e:
Complex numbers: 1, -16, i, 38i, -2+46i, 2-3i

Answer 3

Real numbers include the Integers (..., -3, -2, -1, 0, 1, 2, 3 ...) the Rationals which are numbers that can be represented as A/B where A and B are Integers. And also the irrational numbers like pi which cannot be represented by whole fractions.

Some examples of real numbers: -1, 599, pi, e, 1.523

The Complex numbers, Quaternions, and Octonians are examples of numbers that are not real. An example of a Complex number is: i
which is the sqrt(-1)

Answer 4

pi, minus 1, a million are all real numbers. The square root of minus one is different; 1 times 1 is 1, -1 times -1 is 1. So we call the square root of minus 1 an imaginary number. But this doesn't mean it doesn't exist or that we can't do useful things with it. A diagram of real numbers is called a number line. Draw a straight line. Mark zero on it. 1 is to the right of zero, 2 is to the right of 1, 1 1/2 is half way between 1 and 2, minus 1 is to the left of zero. Now draw a line through zero at right angles to your number line. The square root is one up from zero. Let's call it i. 2i is another one up from i. minus i is one down from zero. If you combine real and imaginary numbers, like 3 + 5i they're called complex numbers.

Answer 5

it's numbers like 3, -980, even fractions like 9/19 or -9/19. it's also irrational numbers like sqaure root of 2 and pi [3.14......] and things like that . It's anything on the 'number line'.
The opposite of a real number is an imaginary number, like square root of -1.
Wikipedia describes it really well, because when I was studying this I went there and I finally understood it haha

Answer 6

a real number are like the numbers 1, 2 or 3 ...

some numbers are not defined, like the square root of a negative number, and for that we have 'imaginary numbers' ... those are not real numbers :)

take care :)

Answer 7

I like to say real numbers are any number that can be plotted on the number line.

Answer 8

In mathematics, the real numbers may be described informally in several different ways. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as pi and the square root of 2; a real number can be given by an infinite decimal representation, such as 2.4871773339...., where the digits continue in some way; the real numbers may be thought of as points on an infinitely long number line.

These descriptions of the real numbers, while intuitively accessible, are not sufficiently rigorous for the purposes of pure mathematics. The discovery of a suitably rigorous definition of the real numbers — indeed, the realisation that a better definition was needed — was one of the most important developments of 19th century mathematics. Popular 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.


Basic properties of the real numbers

Real numbers may be rational or irrational; algebraic or transcendental; and 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, no matter how many more might be added at the end.

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.
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Uses of the real numbers

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.

Computers can only approximate most real numbers. Most commonly, they can represent a certain subset of the rationals exactly, via either floating point numbers or fixed-point numbers, and these rationals are used as an approximation for other nearby real values. Arbitrary-precision arithmetic is a method to represent arbitrary rational numbers, limited only by available memory, but more commonly one uses a fixed number of bits of precision determined by the size of the processor registers. In addition to these rational values, computer algebra systems are able to treat many (countable) irrational numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their rational approximation.

Mathematicians use the symbol R (or alternatively, \Bbb{R}, the letter "R" in blackboard bold) to represent the set of all real numbers. The notation Rn refers to an n-dimensional space of real numbers; for example, a value from R3 consists of three real numbers and specifies a location in 3-dimensional space.

In mathematics, real is used as an adjective, meaning that the underlying field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra.

Answer 9

A real number is a complex number z = x+i*y where the imaginary part y is exactly zero. 5, Pi, sin(1), etc. are real. i=sqrt(-1) is not real, because it has an imaginary part.

Answer 10

a real number is any number that can be expressed whether it is positive or negative, it also has a real square root like 4,16, there are also imaginary numbers which are the expressions of negative square roots like 4i which is the square root of -16