Chapter 02 Section 01

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Chapter 02 Section 01

• Integers and the Number Line

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2-1 INTEGERS AND THE NUMBER LINE
OBJECTIVES
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

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2-1 INTEGERS AND THE NUMBER 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 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,

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2-1 INTEGERS AND THE NUMBER LINE

• 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, ...

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2-1 INTEGERS AND THE NUMBER LINE

• 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.

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2-1 INTEGERS AND THE NUMBER LINE

• 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.

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2-1 INTEGERS AND THE NUMBER LINE

• 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.

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2-1 INTEGERS AND THE NUMBER LINE
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.
1
2
3
4
5
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2-1 INTEGERS AND THE NUMBER LINE
EX1ß
EXAMPLE 1a Name the set of numbers
graphed. a. b.
Notice the bigger arrow!
2
-7
-6
-5
-4
-3
-2
-1
0
1
The set is -5, -4, -3, -2, -1, 0, 1, 2, .
5
-4
-3
-2
-1
0
1
2
3
4
The set is -4, -2, -1, 1, 3.
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2-1 INTEGERS AND THE NUMBER LINE
EXAMPLE 1ß Name the set of numbers graphed.
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
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2-1 INTEGERS AND THE NUMBER LINE
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
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2-1 INTEGERS AND THE NUMBER LINE
EX2ß
EXAMPLE 2a At 408 A.M., the temperature in
Casper, Wyoming, was -10F. By 130 P.M., the
temperature had risen 17 to the daytime high.
What was the high temperature?
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.
7
0
2
4
6
8
10
12
-12
-10
-8
-6
-4
-2
The high temperature for the day was 7F.
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2-1 INTEGERS AND THE NUMBER LINE
EXAMPLE 2ß The next day in Casper, Wyoming, the
low temperature was -8F. The high was 21
higher than the low temperature. What was the
high temperature?
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2-1 INTEGERS AND THE NUMBER LINE
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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2-1 INTEGERS AND THE NUMBER LINE
HOMEWORK
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