Download Chapter 02 – Section 01

Chapter 02 – Section 01
Integers and the Number Line
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:
0
1
2
3
4
5
6
7
8
9
10
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…
© William James Calhoun
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 Street
1, 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, …
© William James Calhoun
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, ...
© William James Calhoun
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.
© William James Calhoun
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 numbers
Each 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.
Whole Numbers
Integers
Natural
Numbers
Sets
Natural numbers
Whole numbers
Integers
Examples
1, 2, 3, 4, 5, …
0, 1, 2, 3, 4, …
…, -2, -1, 0, 1, 2, …
Venn Diagram
© William James Calhoun
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.
© William James Calhoun
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.
-5
-4
-3
-2
-1
0
-5
5
-4
4
-3
3
-2
2
-1
1
0
1
2
3
4
5
© William James Calhoun
EX1β
EXAMPLE 1α: Name the set of numbers graphed.
a.
-7
-6
-5
-4
-3
-2
-1
Notice the bigger arrow!
0
1
2
The set is {-5, -4, -3, -2, -1, 0, 1, 2, …}.
b.
-4
-3
-2
-1
0
1
2
3
4
5
The set is {-4, -2, -1, 1, 3}.
© William James Calhoun
EXAMPLE 1β: Name the set of numbers graphed.
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
© William James Calhoun
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
© William James Calhoun
EX2β
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?
1)
2)
3)
4)
Draw and label a number line.
Draw an arrow from 0 ten places to the left for -10.
Draw an arrow from the end point in #2 seventeen place to the right for positive 17.
This end-point is the sum -10 + 17.
7
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
The high temperature for the day was 7°F.
© William James Calhoun
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?
© William James Calhoun
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.
© William James Calhoun
PAGE 75
#17 – 39 odd
plus #41
© William James Calhoun