Interval notation and the Number Line
The real numbers can be represented on a number line, a line theoretically extending infinitely in two opposite directions as shown here:
The arrowheads at the opposite ends of the drawing of the number line mean that line in concept extends infinitely in those directions, even though the drawing of the line cannot be extended forever in those directions. Note that the right side of the number line stretches to positive infinity and the left side stretches to negative infinity. Numbers in a set can be shown as dots on (or near) a number line. For example, the above set of natural numbers from 1 to 8 would be shown as follows:
Often, a series of numbers will go on infinitely in one or both directions. For example, the set of natural numbers, consisting of numbers one naturally counts with, starts with 1, 2, 3, 4, and goes on to infinity. The indefinite continuation of an infinite set of numbers (or similar elements) can be written as several dots (called an ellipsis) after some numbers or elements listed showing the initial trend. Thus, the set of natural numbers can be represented as follows:
where the three dots represent the continuing trend of an infinite set of elements The set of integers can be represented as follows:
There is no particular requirement that the listing of an infinite set of elements stop at 10 or 8 or any particular number, as long as a clear, understandable trend is given. Intuitively we can the set of natural numbers is a subset of the set of integers when we find the number 1 in the integers and see that the set continues to count upward forever. So far, we have discussed discrete numbers. Discrete means consisting of one or more isolated, individual numbers or points; or not having a continuous range (interval) of numbers or points.
Between every two integers, there are an infinite number of fractional, or rational numbers. Furthermore, between any two fractional numbers, there are an infinite number of other fractional numbers, and so on. This characteristic is sometimes referred to as continuity. Such a continuous set of numbers is represented as a bold line segment on (or near) the number line, similar to the way a continuous set of points is represented by a line segment in geometry. A continuous set of numbers which includes all the numbers between two given numbers is often called an interval. The two numbers that the continuous set of numbers are between are the endpoints of the line segment. One, both, or neither of the numbers at the endpoints of the interval may be included with the set of numbers in the interval. If the number at the endpoint is included, that endpoint is a closed endpoint and is represented by a solid dot. If the number at the endpoint is not included, that endpoint is a open endpoint and is represented by a hollow dot (a tiny hollow circle). As an example, shown below on a number line is the interval between 1 and 8 which includes 1 (is closed at 1) but does not include 8 (is open at 8):
To save ourselves time describing these intervals we often represent them with two types of parentheses, [ ] and ( ). The square brackets, [ ], mean an inclusive interval- that is, the numbers inside the brackets are included, much like a solid point on our number line interval. The rounded brackets, ( ), mean an exclusive interval- that is, the numbers inside the brackets are excluded from the interval, much like the hollow point on the number line. We can also use the symbol to show that a statement is inclusive on the left and the symbol to show it is inclusive on the left. The symbols and show that the interval is exclusive. Consider these examples where we let X represent any number on the interval:
or all numbers from 4 to 9.
or all numbers between 4 and 9.
or all numbers between 4 and 9 including 4.
As with the third example, we can use combinations of the two types of parentheses to display any interval on the real numbers. A set of continuous numbers can also be defined which starts (or ends) at one number and extends infinitely in either the positive direction or the negative direction. Geometrically, such a set is represented by a ray on the number line, where the continuous set of numbers is shown as a bolder part of the line. If the endpoint is included in the set, the endpoint is closed and represented by a solid dot. If the endpoint is not included in the set, the endpoint is open and represented by a hollow dot. As an example, a set of numbers greater than or equal to 1, is shown on a number line below:
This is equivalent to the interval . In another example, a set of numbers less than 8, , is shown on a number line below:
A set which contains all the solutions to an algebraic equation is called that equation's solution set, i.e. all the numbers that if substituted for an "unknown" variable in that equation would make it true. A formula is a math "process" that finds an answer to different unknown variables by using other variables and numbers. An example of a formula is Einstein's formula: ; if you know the mass of an object, M, and you multiply it by the speed of light squared (), you get its energy, E. Formulae like these can be rearranged to find the values of different variables, too.