TEXAS COLLEGE
2404 N GRAND AVENUE
TYLER, TEXAS 75702

DIVISION OF NATURAL & COMPUTATIONAL SCIENCES
MATHEMATICS DEPARTMENT
RESEARCH SEMINAR IN MATHEMATICS
MATH 4460 01

THE NUMBER LINE
BY
George L Williams III

Contents * THE NUMBER LINE

* Extended real number line

* Drawing the number line

* Line segmentation

* Comparing numbers

* Arithmetic Operations

* Arithmetic Operations (cont.)

* Algebraic properties

* Cartesian Plane/Cartesian Coordinate System

* An Overview

* My words

* Applications of the number line

* Resources

*

THE NUMBER LINE
Mathematics is one of the most useful and fascinating divisions of human knowledge. In mathematics, a real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers, such as the integer – 5 and the fraction 4/3, and all the irrational numbers such as positive square root of 2,√2. Real numbers can be thought of as points on an infinitely long number line. In basic mathematics, a number line is a picture of a straight line on which every point is assumed to correspond to a real number and every real number to a point.
Often the integers are shown as specially-marked points evenly spaced on the line. Although this image only shows the integers from −9 to 9, the line includes all real numbers, continuing forever in each direction, as shown by the arrows and also numbers not marked that are between the integers. It is often used as an aid in teaching simple addition and subtraction, especially involving negative numbers.
On the number line, numbers located to the right of a given number are greater than (>) the given number; numbers located to the left of a given number are less than (<) the given number. Thus, +2 is located to the right of +1, and +2> +1; -1 is located to the left of 0; and -1<0. The set of real numbers can be represented geometrically as the set of all points along a line called a number line or graph line. To help us visualize this concept, we first draw a horizontal straight line, select a point on this line and label it 0, then select another arbitrary point to the right of 0 and label it 1. The point that represents 0 is called the origin, and the line segment determined by the point representing 0 and the point representing 1 defines the scale unit.
By measuring successive scale units moving from left to right, starting at 0, we can associate the set of positive integers 1, 2, 3, 4 … with equal spaced points on the line. Moving from right to left, starting at 0, we can associate the set of negative integers -1, -2, -3, -4 … with equal spaced points on the number line. The remaining real numbers can be “located” or “plotted” on the real line by using decimal representations. The number corresponding to a point on the number line is called the coordinate of the point, and the point is called the graph of the number.
2
1
-4
-3
-2
-1
3
4

THE NUMBER LINE
Extended real number line
In mathematics, the affined extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ (read as positive infinity and negative infinity respectively). These new elements are real numbers. It is useful in describing various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The affined extended real number system is denoted Ṝ or (−∞, +∞). When the meaning is clear from context, the symbol +∞ (positive infinity) is often written simply as ∞ (infinity).

Negative Numbers (-) Positive Numbers (+) | |
8 is Greater than 5
-5 is greater than -8
Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are less. Thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left. Note that a negative number with greater magnitude is considered less. For example, even though (positive) 8 is greater than (positive) 5, written: 8>5
5 is less than 8
-8 is less than -5

Negative 8 is considered to be less than negative 5: -8 < -5
Because, for example, if you have (-8) you have less than if you have (-5).) Therefore, any negative number is less than any positive number, so
-8 < 5 and -5 < 8.
John Wallis, an English mathematician, is generally credited as the originator of the idea of the number line where numbers are represented geometrically in a line with the negative number represented by lengths opposite in direction to lengths of positive numbers. In mathematics, a negative number is a real number that is less than zero. Such numbers are often used to represent the amount of a loss or absence. The relationship between negative numbers, positive numbers, and zero is often expressed in the form of a number line as seen below.
THE NUMBER LINE
Drawing the number line
The number line is usually represented as being horizontal. Positive numbers always lie on the right side of zero, and negative numbers always lie on the left side of zero. An arrowhead on either end of the drawing is meant to suggest that the line continues indefinitely in the positive and negative real numbers, denoted by. The real numbers consist of irrational numbers and rational numbers, the latter of which include the integers, which in turn include the natural numbers (also called the counting numbers or whole numbers). It was not until the 19th century when British mathematicians like e Morgan, Peacock, and others, began to investigate the ‘laws of arithmetic’ in terms of logical definitions that the problem of negative numbers was finally sorted out. Outlined below are examples of such number lines.

THE NUMBER LINE
Line segmentation

Figure 1

Let L be a horizontal straight line as in Figure 1. Choose a point O (to be called the origin) on L. Choose a point L (to be called the unit point) to the right of O. We assign to the point O the number 0 and to the point I the number 1, and we call the line-segment OI = IP in the sense that the segment IP can be made to coincide with the segment OI by superposition. Actually, we think of taking the segment OI and placing it on the line L with its left-hand endpoint coinciding with I and its right-hand endpoint falling then on P. To the point P we assign the number 2. This geometric construction of laying off the unit segment twice to the right of O, starting with its left-hand endpoint at O, to find the position of the right-hand endpoint obviously corresponds to the arithmetic of forming the sum 1 + 1. Continuing the geometric operation of successive super-positions on L of the unit segment to the right of O, we assign to the right hand endpoint the number after three super-positions, the number 4 after four super-positions, and so on. Thus for the point Q with the number 3 assigned, we may write OI = IP = PQ to symbolize the three super-positions used to arrive at the endpoint Q. We may, in this fashion, find points to the right of O to which may be assigned each of the natural numbers.
If we choose Ī, (pronounce: horizontal bar I), to be the point on L to the left of O such that ĪO coincides with OI, then Ī can have assigned to it the number -1. Similarly, to P and Q in Figure 1 are assigned the numbers -2 (horizontal bar P) and -3(horizontal bar Q), respectively. In this manner we can assign to specified points to the left of O the negative integers. Again, we may briefly note the analogy between the geometric operation of super-positions of the unit segment to the right and left of O, with that of the arithmetic of the integers as follows: The point Ī to which is assigned the number -1 is the left-hand endpoint of the unit segment when the unit segment is superimposed on L with it right-hand endpoint at O; if we follow this geometric operation by superposition of the unit segment on L with its left-hand endpoint at Ī, the right-hand endpoint of the unit segment falls on O. This is in analogy with the arithmetic equality (-1) + 1 = 0.
When we assign the number x to a point X on the line L (on which an origin O and a unit point I have been chosen), we call x the coordinate of X. By assigning to each point of L a number, we construct a coordinate system on the line L. If this assignment can be properly carried out, the coordinates will reflect geometric properties of the line and its segments. It is for this reason that we have so carefully chosen points e.g., P and Q with coordinates 2 and 3, respectively, so that PQ should be congruent to the unit segment.
THE NUMBER LINE
Comparing numbers
If a particular number is farther to the right on the number line than is another number, then the first number is greater than the second (equivalently, the second is less than the first). The distance between them is the magnitude of their difference—that is, it measures the first number minus the second one, or equivalently the absolute value of the second number minus the first one. Taking this difference is the process of subtraction.

Thus, for example, the length of a line segment between 0 and some other number represents the magnitude of the latter number.

THE NUMBER LINE
Arithmetic Operations
Two numbers can be added by "picking up" the length from 0 to one of the numbers, and putting it down again with the end that was 0 placed on top of the other number.

THE NUMBER LINE
Arithmetic Operations
Two numbers can be multiplied as in this example: To multiply 4 × 3, note that this is the same as 4 + 4 + 4, so pick up the length from 0 to 4 and place it to the right of 4, and then pick up that length again and place it to the right of the previous result. This gives a result that is 3 combined lengths of 4 each; since the process ends at 12, we find that 4 × 3 = 12.

Division can be performed as in the following example: To divide 10 by 5—that is, to find out how many times 5 goes into 10—note that the length from 0 to 5 lies at the beginning of the length from 0 to 10; pick up the former length and put it down again to the right of its original position, with the end formerly at 0 now placed at 5, and then move the length to the right of its latest position again. This puts the right end of the length 5 at the right end of the length from 0 to 10. Since two lengths of 5 filled the length 10, 5 goes into 10 two times (that is, 10 ÷ 5 = 2).

The section of the number line between two numbers is called an interval. If the section includes both numbers it is said to be a closed interval, while if it excludes both numbers it is called an open interval. If it includes one of the numbers but not the other one, it is called a half-open interval; [1, 5], close interval; (1, 5), opened interval; [1, 5), and (1, 5] half-opened intervals. The arithmetic operations of R can be partially extended to R as follows:

THE NUMBER LINE
Arithmetic Operations

Here, "a + ∞" means both "a + (+∞)" and "a − (−∞)", while "a − ∞" means both "a − (+∞)" and "a + (−∞)".

THE NUMBER LINE
Algebraic properties
With these definitions R is not a field, nor a ring, and not even a group or semi group. However, it still has multiple convenient properties:
The algebraic properties listed apply given a, b, and c are real numbers. This is not an exhaustive list of algebraic properties.

In general, all laws of arithmetic are valid in R as long as all occurring expressions are defined.
THE NUMBER LINE
Cartesian Plane/Cartesian Coordinate System

Alternatively, one real number line can be drawn horizontally to denote potential values of one real number, commonly called x, and another real number line can be drawn vertically to denote potential values of another real number, commonly called y. Together these lines form what is known as the Cartesian coordinate system, and any point in the Cartesian plane denotes the values of a pair of real numbers. Rene’ Descartes’ influence in mathematics is the Cartesian coordinate system. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.
Choosing a Cartesian coordinate system for a one-dimensional space—that is, for a straight line—involves choosing a point O of the line (the origin), a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative; we then say that the line "is oriented" (or "points") from the negative half towards the positive half. Then each point P of the line can be specified by its distance from O, taken with a + or − sign depending on which half-line contains P.
A line with a chosen Cartesian system is called a number line. Every real number has a unique location on the line. Conversely, every point on the line can be interpreted as a number in an ordered continuum such as the real numbers.

(-3, 1)
(-1.5, -2.5)
(2, 3)
(0, 0)

Illustration of a Cartesian coordinate plane, four points are marked and labeled with their coordinates:
(2, 3) in green, (-3, 1) in red, (-1.5, -2.5) in blue, and the origin (0, 0) in purple.

The Number Line:
An Overview

One of the most overlooked tools of the elementary and middle school classroom is the number line. Typically displayed above the chalkboard right above the alphabet, the number line is often visible to children, though rarely used as effectively as it might be. When utilized in the elementary classroom, the number line has often used to help young children memorize and practice counting with ordinal numbers. Less often, perhaps, the number line is used like a ruler to illustrate the benchmark fractions like ½ or ¼. Beyond an illustration for these foundational representations of whole numbers and some fractions, however, the number line is underutilized as a mathematical model that could be instrumental in fostering number sense and operational proficiency among students.

Recently, however, there has been a growing body of research to suggest the importance of the number line as a tool for helping children develop greater flexibility in mental arithmetic as they actively construct mathematical meaning, number sense, and understandings of number relationships. Much of this emphasis has come as a result of rather alarming performance of young learners on arithmetic problems common to the upper elementary grades. For example, a study about a decade ago of elementary children in the Netherlands – a country with a rich mathematics education tradition – revealed that only about half of all students tested were able to solve the problem 64-28 correctly, and even fewer students were able to demonstrate flexibility in using arithmetic strategies. These results, and other research like them, prompted mathematics educators to question existing, traditional models used to promote basic number sense and computational fluency.

Counting from 28 64 is equal to 36 to the right to

The Number Line:
An Overview

Surprising to some, these research findings suggested that perhaps the manipulative and mathematical models typically used for teaching arithmetic relationships and operations may not be as helpful as once thought. Base-10 blocks, for example, were found to provide excellent conceptual understanding, but weak procedural representation of number operations. The hundreds chart was viewed as an improvement on arithmetic blocks, but it too was limited in that it was an overly complicated model for many struggling students to use effectively. On the other hand, the number line is an easy model to understand and has great advantages in helping students understand the relative magnitude and position of numbers, as well as to visualize operations. As a result, Dutch mathematicians in the 90ʼs were among the first in the world to return to the “empty number line,” giving this time-tested model a new identity as perhaps the most important construct within the realm of number and operation. Since that time, mathematics educators across the world have similarly turned to this excellent model with great results. The number line stands in contrast to other manipulative and mathematical models used with the number realm. Some of the reasons for developing the number line as a foundational tool are illustrated below as key ideas.

Key Concept #1: The linear character of the number line. The number line is well suited to support informal thinking strategies of students because of its inherent linearity. In contrast to blocks or counters with a “set-representation” orientation, young children naturally recognize marks on a number line as visual representations of the mental images that most people have when they learn to count and develop understanding of number relationships.

Key Concept #2: Promoting creative solution strategies and intuitive reasoning. A prevalent view in math education reforms is that students should be given freedom to develop their own solution strategies. But to be clear, this perspective does not mean that it is simply a matter of allowing students to solve a problem however they choose. Rather, the models being promoted by the teacher should themselves refine and push the student toward more elegant, sophisticated, and reliable strategies and procedures. This process of formalizing mathematics by having students recognize, discuss, and internalize their thinking is a key principle in math education reforms, and is one that can be viewed clearly through the use of a model like the number line – a tool that can be used both to model mathematical contexts, but also to represent methods, thinking progressions, and solution strategies as well. As opposed to blocks or number tables that are typically cut off or grouped at ten, the open number line suggests continuity and linearity -- a representation of the number system that is ongoing, natural, and intuitive to students. Because of this transparency and intuitive match with existing cognitive structures, the number line is well suited to model subtraction problems, for example, that otherwise would require regrouping strategies common to block and algorithmic procedures.

Key Concept #3: Cognitive engagement. Finally, research studies have shown that students using the empty number line tend to be more cognitively active than when they are using other models, such as blocks, which tend to rely on visualization of stationary groups of objects. The number line, in contrast, allows students to engage more consistently in the problem as they jump along the number line in ways that resonate with their intuitions. While they are jumping on the number line, they are able to better keep track of the steps they are taking, leading to a decrease in the memory load otherwise necessary to solve the problem.

The Number Line:
My words

The most important teaching point to convey regarding the number line is the notion that unlike a ruler, it is open and flexible. The number line models the natural ways in which we think about all number relationships and number operations. There has been much research done with the approach to teach mental math strategies in classrooms by way of the empty number line as it is able to create a mental image of the strategies that are being taught and it can also aide in the leap to more easily make mental calculations without paper. Using the empty number line also increases student’s confidence in their ability to use numbers flexibly which leads to further development in their understanding of number sense.
Within the school district I was brought through, the use of this tool that I have mentioned in my research has but a glimmer in my memory for we were taught to pass an exit exam versus the means to become self sufficient in this subject by uses of a number line or more complex algorithms. It pains me in a sense to know that I honestly feel that within these walls of Texas College, the professor allowed groomed me in the language of mathematics through basic concepts such as the number line, the Cartesian coordinate system, one-two-three dimension equations within the polar coordinate system.
This research has mention the foundation of the number line, its uses and how to use as well yet the empty number line is a powerful tool as well. In contrast to the number line there are neither scales nor any other pre-given objective landmark on the empty number line and in the case of such there is no rule which would require, for example, the same spatial distance between the marks which correspond to two pairs of numbers having an equal arithmetical distance. This particular number line therefore is a reproduction of the normal number line that is not faithful to the scale but which respects the order of numbers. I leave you with these words; the number line is an extremely powerful model.

The Number Line
Applications of the number line
1.

5≤x-1 :x⃓x≥6;6, ∞ x ≥6

2. A line drawn through the origin at right angles to the real number line can be used to represent the imaginary numbers. This line, called imaginary line, extends the number line to a complex number plane, with points representing complex numbers.

A complex number is of the form a + bi, where a ʌ b ε Ʀ.

Represent the complex number 4 + 2i.
Z-axis
X-axis
Y-axis
2
4 + 2i

3. https://www.youtube.com/watch?v=iI_2Piwn_og&spfreload=10 Resources

Klein, Anton S., Meindert Beishuizen, and Adri Treffers. "The empty number line in Dutch second grades: Realistic versus gradual program design." Journal for Research in Mathematics Education (1998): 443-464.

Radford, Luis, et al. "Historical formation and student understanding of mathematics." History in mathematics education. Springer Netherlands, 2002. 143-170.

World Book, Inc., Scott Fetzer company. The world book encyclopedia. Vol. 14. World Book, 2014.

Bates, Grace Elizabeth, and Fred Ludwig Kiokemeister. The real number system. Allyn and Bacon, 1960.

Tanton, James Stuart. "Encyclopedia of mathematics/James Tanton." (2005).

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[ 1 ]. There are three laws of arithmetic: The associative law, the commutative law, and the distributive law. These laws are:
Associative law: For addition: (a + b) + c = a + (b + c)/For multiplication: (ab)c = a(bc)
Commutative law: For addition: a + b = b + a/For multiplication: ab = ba
Distributive law: a(b + c) = ab + ac
Arithmetic is branch of mathematics in which numbers, relations among numbers, and observations on numbers are studied and used to solve problems.