chapter 02 – section 01 integers and the number line

Chapter 02 – Section 01

Integers and the Number Line

© William James Calhoun

To state the coordinates of a point on a number line, graph integers on a number line, and add integers by using a number line.

Initial terms:• number line - a line with equidistant markings representing numbers

Example of a number line:

We can put points on the number line to represent specific numbers.

If A = 5, we would plot A on the number line like this…

0 1 2 3 4 5 6 7 8 9 10Number Line


As the need arose in our societies and cultures, different number systems came into being.

The concept of “more than one” of something gave us as set of numbers known as the natural numbers.

• natural numbers - the numbers you learned from Sesame Street1, 2, 3, 4, 5, 6, 7, …

The concept of a symbol to represent the absence of value gave us the number “0”.

This changed our number system into a set known as the whole numbers.

• whole numbers - the natural numbers plus zero.0, 1, 2, 3, 4, 5, 6, 7, …


The need to describe numbers representing the same strength, and yet not acting on a body in the same direction presented a problem.

For example, if you are playing tug-of-war and one side pulls with a force of 100 Newtons (metric unit of force) and the other side also pulls with 100 Newtons of force, how can you describe the fact that neither side is winning the pull?

Both pull with equal strength, just in opposite directions.

To describe this “other direction”, the set of negative numbers were created.

• negative numbers - whole numbers less than zero and having a negative sign on them, used to refer to same strength but opposite direction

-1, -2, -3, -4, -5, -6, -7, ...


If you take all the whole numbers, zero, and the negative numbers, you get a new set called the integers.

• integers - the set of zero, whole positive and whole negative numbers

Notice we have only mentioned whole-number values, both positive and negative.

There will be another section on the infinite fractional numbers between these whole numbers.

There are also more numbers between the whole numbers which can not be written as fractions – again discussed in another section.

Beyond all that, there are numbers that do not even appear on the number line anywhere. For this, you really have to wait until Algebra 2.


A way to represent groups within groups and overlapping groups in mathematics is called a Venn diagram.

• Venn diagram - figure used to represent sets of numbersEach inner area is contained in the outer areas.

These are not specific to mathematics – you will see them across all subjects.

Here is a sample of a Venn diagram for our number sets.

Sets ExamplesNatural numbers 1, 2, 3, 4, 5, …Whole numbers 0, 1, 2, 3, 4, …Integers …, -2, -1, 0, 1, 2, …

NaturalNumbers

Whole Numbers

Integers

Venn Diagram


A couple more terms:

• graph - to draw, or plot, the points of a set on a number line

• coordinate - the number that corresponds to a point on a number line

We will have (x, y) coordinates and a Cartesian plane when we get to two-dimensional graphs later.

To put “…” before or after a comma means the pattern goes on forever in the direction of the “…”.

To show a pattern goes on forever on a number line, you draw the arrow on the end of the graph bigger.

You will see an example of a bigger arrow in Example 1.


Now, if we take our original, natural number line…add in zero…copy the number line…flip it over the number zero…rewrite and put negative signs on our numbers…we will get the number line for integers that we will be using.

-4 -3 -2 -1 0-5

1 2 3 4 50 1 2 3 4 50

12345 012345 0

-5 -4 -3 -2 -1


EXAMPLE 1α: Name the set of numbers graphed.a.

b.

EX1EX1β

-7 -6 -5 -4 -3 -2 -1 0 1 2

Notice the bigger arrow!

The set is {-5, -4, -3, -2, -1, 0, 1, 2, …}.

-4 -3 -2 -1 0 1 2 3 4 5

The set is {-4, -2, -1, 1, 3}.


EXAMPLE 1β: Name the set of numbers graphed.

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7


You can use number lines to add and subtract positive and negative numbers.

The hope is that you will get away from using number lines in the near future – if not already.

For example, to find the sum of -6 and -5:1) Draw an arrow from zero six places to the left.2) Draw a second arrow from the end-point of #1 five places to the left.3) Where you are on the number line is what -6 + (-5) equals.

-12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

-6 + (-5) = -11


EXAMPLE 2α: At 4:08 A.M., the temperature in Casper, Wyoming, was -10°F. By 1:30 P.M., the temperature had risen 17° to the daytime high. What was the high temperature?

EX2EX2β

The high temperature for the day was 7°F.

1) Draw and label a number line.2) Draw an arrow from 0 ten places to the left for -10.3) Draw an arrow from the end point in #2 seventeen place to the right for positive 17.4) This end-point is the sum -10 + 17.

0 2 4 6 8 10 12-12 -10 -8 -6 -4 -2

7


EXAMPLE 2β: The next day in Casper, Wyoming, the low temperature was -8°F. The high was 21° higher than the low temperature. What was the high temperature?


A special note here.

The book, in its instructions for adding integers in the sections says, “Find each sum. If necessary, use a number line.

You will need to get away from number lines quickly.

Adding, subtracting, multiplying, and dividing integers is a necessary component of Algebra.

You will need to do these operations quickly and effeciently, which omits the use of graphing to add integers.

Your calculator can be a resource here, but still using it to find -4 + 6 is a waste of batteries/solar power.


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chapter 02 – section 01 integers and the number line

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