real numbers?

Question

what are real numbers and what makes them so real? 10 points for best answer!

Answer 1

Real numbers are numbers that, when squared, are positive.

The distinction "real number" was not used until the other numbers (imaginary numbers) began to be used. Imaginary numbers, when squared, result in a negative number. This doesn't "make sense" in terms of positive/negative values, and so these numbers were considered different or "imaginary."

The imaginary unit is called i. It is the square root of -1, meaning i² = -1. Notice that i cannot be positive OR negative, it's something else entirely, because anything positive or negative squared would be positive. A complex number is a number a+bi, where a and b are real numbers. This means that (a+bi)² might be positive, negative, or just another complex number.

It is important to note that the best distinction to make is that the real numbers have this property about squares being positive - this is what separates them from imaginary numbers. Like I said, i isn't positive OR negative - it's neither. It is an important property of the real numbers, because the real numbers are ORDERED. This is the important distinction. You can take any two real numbers and say x
Complex numbers (imaginary/real numbers all together) cannot be ordered! It is impossible. There is no way to establish such a "less than" relationship. I mean, what do you think, if I were to ask you which is bigger, i+2 or 2i+1? You could pick one arbitrarily, but you can't establish a rule that orders them.

The thing that sets apart complex numbers is that complex numbers can express the roots of ANY polynomial. When somebody says "find the real roots of _____" or "there is no solution to this polynomial equation" or whatever, they're talking about the limitations of real numbers. Sometimes a 4th degree equation has 4 roots. Sometimes it has none.

Complex numbers are great. How many complex roots does a 4th degree equation have? 4 roots. 10th degree? 10 roots. 7364th degree? 7364 roots. This is called algebraic closure, and it's pretty amazing that if you use complex numbers of the form a+bi, you get all the roots, ever.

Summary:
Real numbers - positive, negative, or zero. always square to be positive (or zero). have an order.
Complex numbers - no ordering. squaring could give anything. solve roots of any polynomial.

Answer 2

It is obvious that numbers have differences among themselves.
For example is easier to see that 3 is a different kind of number than 0.5 or even 6.6666666......
For this reason (and to study them better), numbers were grouped is sets
The most basic one is the Natural numbers (called N). They are integer numbers that go from 0 to +infinity (0, 1, 3, 4,......)

Then came the idea of Integer numbers (Z). They are similar to the natural numbers, but they are also negative. Their range is from -infinity to +infinity ( -inf...... -2, -1, 0, 1, 2, 3....)

Next, we have the rational numbers (Q). Here you have the fractions, and any number whose divition would give an quotient. Some rationals follow a pattern when written as decimals.
1/2, 0.45, 700/5, 3.333333333333.........
6.546546546546546.........

But what about the numbers that do not come from division? Number like sqrt(2)=1.4142..... or Pi or e?
They are called Irrational numbers(I). Irrational numbers are decimals that do not follow a pattern and go on forever, therefore they cannot be written as fractions.

N is a subset of Z
Z is a subset of Q
when you group Q and I together, you get the set of the real numbers (R).
So basically thats what it means for a number to be real.

Answer 3

Real number refers to the numbers you deal with every day. For instance, pi is a real number, the integers are real, fractions of real numbers are real, negative numbers are real.

Imaginary numbers, by contrast, are based on teh idea the sqrt(-1) is a number. This is often denoted as " i " or "j" in electrical engineering. So, while the sqrt(-4) is not real, it is imaginary and can be written as
sqrt(-4) = sqrt(-1)*sqrt(4) = i *2 =2 i

The " i" is a value so 2 + 2i is NOT equal to 4, 2 + 2i = 2 + 2i, you cna't reduce it further (ok, you can factor a 2 but then you have 2*(1 + i)).

You can multiply and divde reals and imaginaries, for instance 2i/2 = i and 2*2i = 4i. Also note that, by definition of i:
i * i = -1

Answer 4

organic numbers: a million,2,3,... entire numbers: organic numbers such as 0 integers: 0, a million, -a million ,2, -2,... (entire numbers and their negatives) rational numbers: if a and b are integers (b isn't 0) then a/b is a rational numbers actual numbers: the ending touch of the rational numbers (somewhat greater state-of-the-paintings to describe completely) irrational numbers: actual numbers that are no longer additionally rational numbers, as an occasion ?2

Answer 5

A real number is any number that can be shown on a number line. Real numbers include rational numbers, irrational numbers, and integers.
The only kind of numbers that are not real, are imaginary numbers, which include i.

Answer 6

Real numbers cover integers, rational numbers and irrational numbers (such as pi etc.)

They are called real numbers to distinguish them from complex or imaginary numbers.

Answer 7

Real numbers are those numbers which can be expressed as a fraction consisting of one interger divided by another, whether or not either of the two intergers has infinite digits. They are "real" in that they express definite tangible values such as "-12.3", "5/4", "0.3333...", "pi", and the square root of two.

The set of real numbers can be divided into two sets, the rational numbers which can be expressed as one interger with a finite number of digits divided by another, and the irrational numbers which can only be expressed as a fraction using an infinite number of digits.

Answer 8

http://en.wikipedia.org/wiki/Real_number