Real number

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A real number ℝ is a rational or irrational number. Usually when people say "number" they mean "real number". There are also numbers which are not real. Those are called complex numbers.

Some real numbers are positive. A positive number is bigger than zero. If a positive number is added to another number, that number gets bigger. Zero is also a real number. If zero is added to a number, that number does not change.

There are also negative real numbers. They are smaller than zero. If a negative number is added to another number, that number gets smaller. This is called "subtraction".

The real numbers are infinite. That means that there is no first or last real number. No matter how many real numbers are counted, the last one will never be reached.

The real numbers are continuous. That means that if two different real numbers are taken, there will always be another one in between them, no matter how close together the two numbers are. This makes the real numbers special. There are more real numbers than there are numbers that are not continuous, even though the numbers that are not continuous might be infinite, too.

[edit] Different types of real numbers

There are different types of real numbers. Depending on the application, only a subset may be needed. These subsets have special names: Sometimes all the real numbers are not talked about at once. Sometimes only special, smaller groups of them are talked about. These groups have special names. They are:

Natural numbers: These are real numbers that are not fractions or decimals, and are bigger than zero. Given enough time, all natural numbers can be counted to.
Whole numbers: These are real numbers that are not fractions or decimals, and also zero.
Integers: These are real numbers that are not fractions or decimals.
Rational numbers: These are real numbers that can be written down as fractions.
Transcendental numbers can not be obtained with an equation with integer components. Very often, such numbers are also irrational.
Irrational numbers: These are real numbers that can not be written as a fraction.

The number 0 (zero) is special. Sometimes it is taken as part of the group to be considered, and at other times it is not. It is the Identity element for addition and subtraction. That means that adding or subtracting zero does not change the original number. For multipilcation and division, the identity element is 1.

One real number that is not rational is $\sqrt{2}$ . This number is irrational. If a square is drawn with sides that are one unit long, the length of the line between its opposite corners will be $\sqrt{2}$ .

[edit] Proof that the square root of 2 is not rational

The number $\sqrt{2}$ is not rational. Here is the proof.

Think that the number is rational. So there are some numbers $a, b$ such that $a/b=\sqrt{2}$ .
Either a or b is odd. If a and b were even, then the fraction can be shortened (for example, instead of writing $\frac{2}{4}$ , $\frac{1}{2}$ ) could be written instead.
If both sides are squared, then the answer is a² / b² = 2 and a² = 2 b².
The left side is $2 b 2$ . This number is even. So the left side must be even too. So $a 2$ is even. If an odd number is squared, then an odd number will be the result. And if an even number is squared, an even number would be the result too. So $a$ is even.
Because a is even, it can be written e: $a = 2 k$ .
The equation from the point 3 is used. into 6. We get 2b² = (2k)²
An exponentiation rule can be used (see the article) - the result is $2 b 2 = 4 k 2$ .
Both sides are divided by 2. So $b 2 = 2 k 2$ . This means that $b$ is even.
In 2, it is said that a is odd or b is odd. But in 4, it was said that a is even. In 7, it was said that b is even. This is impossible.