Algebra/Sets and the Number Line

Sets and the Number Line

In mathematics, a set is a group of things. The things do not have to be tangible objects; they can be abstract things. Each of the things in a set is called an element of the set. In algebra, the elements of a set are often numbers; in geometry they are often points, which are infinitesimally small locations in space. The number of elements in a set could be countable or could be infinite, as long as the elements are quantifiable, definable, or determinable in some way either now or in the future. Infinity is not a number, it is a concept - it means that however big of a number you can think of, infinity is bigger. The objective of algebra is often to take the information available in a situation or problem, and to define or determine a set of elements such as numbers as simply and precisely as possible. There may be no elements in a set; such a set is called an empty set or a null set. If all the elements of one set are also elements of a second set, then the first set is a subset of the second set. In algebra, letters symbolizing sets are commonly capital (upper case) letters, whereas variables standing for numbers are often symbolized by small (lower case) letters. Elements of a series of closely related variables are sometimes symbolized by a letter followed by a subscript number (integer) such as x₁, x₂, etc. The elements in such a set can be more generally symbolized by the letter followed by a subscript lower case letter such as i, j, k, etc. standing for the subscript numbers; for example, x_i where i could stand for 1, 2, 3, etc.

A set can be symbolized with braces around a list of symbols representing the elements of the set, with each element being separated by a comma. For example, a set S containing natural or whole numbers from 1 to 8, inclusive, could be shown as follows:

$\{1,2,3,4,5,6,7,8\}$

An empty set is symbolized as follows: { } or the symbol ∅. 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:

$\{1,2,3,4,5,6,7,8,9,10...\}$

where the three dots represent the continuing trend of an infinite set of elements The set of integers can be represented as follows:

$\{...,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,...\}$

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. The set of natural numbers and the set of whole numbers are both subsets of the set of integers. 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 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 exluded from the interval, much like the hollow point on the number line. Consider these examples:

$[4,9]=4,5,6,7,8,9$ or all integers from 4 to 9.

$(4,9)=5,6,7,8$ or all integers from 5 to 8.

$[4,9)=4,5,6,7,8$ or all integers from 4 to 8.

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 $[1,\mathrm{\infty }]$. In another example, a set of numbers less than 8, $[-\mathrm{\infty },8)$, 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: $E=M{c}^2$; if you know the mass of an object, M, and you multiply it by the speed of light squared ($c^2$), you get its energy, E. Formulae like these can be rearranged to find the values of different variables, too.

Practice Problems

Name whether each expression would be represented by a number line, line segment, or ray. If the answer is a line segment or ray, identify each end of the segment and whether the non-infinite ends would be represented by a filled or hollow dot.

1. >= 8

2. < 5 >

3. < -4

4. >= 2 < 7

Identify a probable next value in the set, given the values shown.

1. {..., 1, 2, 3, ...}

2. {3/4, 4/4, 5/4, ...}

3. {..., -2, 0, 2}

How would you represent the following number sets, assuming that sides of the number set which aren't explicitly closed continue forever?

1. {..., 9, 10, 11}

2. {9, 10, 11, ...}

3. {8, 9, 10, 11, 12, 13}

Answers

1. Ray - The ray's lower bound is equal to 8. The dot would be filled.

2. Line

3. Ray - The ray begins at less than 4, and the dot would not be filled.

4. Line segment - The lower bound of the segment begins at greater or equal to 2, and would be filled. The upper bound of the line segment is located at any value less than 7, and would be hollow.

1. {..., 0, 1, 2, 3, 4, ...}

2. {3/4, 4/4, 5/4, 6/4, ...}

3. {..., -4, -2, 0, 2}

1. <= 11; A ray's arrow points facing the negative end of the number line, who's upper bound starts at == 11 (filled dot).

2. >= 9; A ray's arrow points facing the positive end of the number line, who's lower bound starts at == 9 (filled dot).

3. >= 8 <= 13; A line segment's lower bound == 8, and upper bound == 13. Both would be filled dots. Template:Subject