Real numbers are ones that can be expressed as decimals, regardless of whether the number of decimal digits is finite or infinite.
For example, both and are real numbers.
We can arrange all real numbers on the real line, according to their relative sizes, stretching from to .
Note that the infinities themselves cannot be written as decimals and are therefore not real numbers. There are, however, no largest or smallest real numbers.
Furthermore, there are no gaps in the real line, and so between any two distinct real numbers we can always find more of them.
The symbol is often used to refer to the set of all real numbers, which we know to be infinitely many.
When squaring a real number, the product is always positive, so what then is the square root of a negative number?
It turns out that there exist other types of numbers than real ones, with the defining property:
, and any scalar multiple thereof, is called an imaginary number.
As an example,
is an imaginary number.
Complex numbers are composed by adding a real number and an imaginary number . The result is , which is a point in the complex coordinate system, with the real part on the -axis, and the imaginary on the -axis.
As seen in the figure, can also be expressed in terms of the length and angle . This is called the polar form, and is written .
A third alternative is to write the complex number in exponential form as , where is Euler's number.
To compactly express:
then we can use the following notation:
Here, the hard bracket means that can take on the value and the bracket means that cannot take on the value , but it can come arbitrarily close.
The picture below show how we represent open and closed sets on the real number line
Combining and intersecting interval's
If we want to combine two intervals then we use the symbol,
we call this the union between two sets. For example if we want all in and all between then we could write this as
If we instead want all that are in two intervals. For example all in that is also in . That is, the interval . Then we could write this as
We call this the intersection between the two sets.
Inequalities and absolute values
The absolute value of a number is defined by the formula
Equations with absolute value
Take a look at the following equation:
This equation reads like a question:
- What number should be such that the distance between and is exactly 2?
The answer is of course and
To solve this algebraically, we use the definition of the absolute value:
The first equation yields the solution and the second .
Inequalities with absolute value
The following inequality:
also reads like a question:
- What number should be such that the distance between and is greater than or equal to 2?
The answer is of course and .
We can solve this inequality algebraically by using the definition of the absolute value:
Solving the first inequality yields the solution and the second .
Remember the following when multiplying and dividing with negative numbers in inequalities: