Chapter 9
Section 9.1: Perimeter
A Polygon is an enclosed geometric figure with three or more sides.
Its Perimeter is the sum of the length of all of its sides.
A Rectangle is a 4 sided enclosed figure. It has four 90° angles. A rectangle has both parallel sides equal in
length.
page 548 of Basic Mathematics
9th Edition by Marvin L. Bittinger
The perimeter of a rectangle
= l + w + l + w = 2 · l + 2 · w = 2 · ( l + w )
page 548 of Basic Mathematics
9th Edition by Marvin L. Bittinger
The perimeter of a square = s
+ s + s + s = 4 · s
Section 9.2: Area
Using the above figures for a
rectangle and square,
the area of a rectangle = l ·
w
and the area of a square = s
· s = s2
A parallelogram is an
enclosed geometric figure with two pairs of sides parallel like the figures
below:
page 553 of Basic Mathematics
9th Edition by Marvin L. Bittinger
To find the area of a
parallelogram, use the drawing below:
page 553 of Basic Mathematics
9th Edition by Marvin L. Bittinger
the area = b · h
A triangle figure is like
that of below:
page 554 of Basic Mathematics
9th Edition by Marvin L. Bittinger
The area of the triangle = 1/2
· b · h
And a trapezoid is like the
figure below:
page 556 of Basic Mathematics
9th Edition by Marvin L. Bittinger
and the area of the trapezoid
= 1/2 · h · (a + b)
Section 9.3: Circles
Using the figure below
page 563 of Basic Mathematics
9th Edition by Marvin L. Bittinger
where r is the radius and d
is the diameter.
d = 2 ∙ r and r = d/2
pi (pronounced pie) is
designated by the Greek character π = 3.14 or 22/7
The circumference of the
circle is basically its perimeter = π · d = 2 · r · π
The area of a circle = π
· r2 = π · (d/2)2 = π · (d2)
/ 4
Section 9.4: Volume
The volume of a rectangular
solid is the number of 1 X 1 X 1 cubes needed to fill the rectangular solid.
The volume of a rectangular
solid = length (l) ∙ width (w) ∙ height (h).
For further information see
page 572 of Basic Mathematics 9th Edition by Marvin L. Bittinger
The volume for a circular
cylinder = base (B) · height (h)
The base (B) is the area of a
circle = π ∙ r2
Therefore the volume of a
circular cylinder = π ∙ r2 · h
The volume of a sphere = 4/3
· π · r3
The volume for a circular
cone = 1/3 · base (B) · h
Using the same value for the
base (B) as above,
The volume for a circular
cone = 1/3 · π ∙ r2 · h
For further information on
these formulas see pages 572 through 575 in Basic Mathematics 9th
Edition by Marvin L. Bittinger.
Be advised that any
combinations of the above can be used.
For further information on
this see page 576 Example 6 in Basic Mathematics 9th Edition by
Marvin L. Bittinger.
Section 9.5 Angles and
Triangles
The most common way of
measuring an angle is with a protractor.
An angle is comprised of two
rays which meat at a common point called a vertex (see figure below)
page 582 of Basic Mathematics
9th Edition by Marvin L. Bittinger
The following are ways to
classify angles.
Right Angle : An angle that measures 90°
Straight Angle : An angle that measures 180°
Acute Angle : An angle that measures less than 90°
Obtuse Angle : An angle that measures less than 180° but
greater than
90°
Complementary
Angles : Two angles that together
measures 90°
Supplementary
Angles : Two angles that together
measures 180°
Triangles are made up of 3 line segments. Each line segment is called a side.
A triangle has 3 angles in
it. The sum of all 3 angles measure 180°
Triangles can be classified
as well
Equilateral
Triangle: A
triangle with all of the sides having equal length. such a triangle has all angles the same
measure = 60°
Isosceles
Triangle : A triangle with two of
the sides having equal
length.
Scalene
Triangle : A triangle with all of
the sides having different
lengths.
Right Triangle : A triangle with one angle a right angle
Obtuse Triangle : A triangle with one angle an obtuse
angle
Acute Triangle : A triangle with all three angles less
than 90°
Section 9.6 Square Roots and
Pythagorean Theroem
All numbers are the product
of 2 factors, for instance 4 = 2 · 2.
These
_
_
factors are called the square
root. √ denotes the square root of like √4 = 2.
_
We say the for numbers that
do not have even factors like √4 are irrational
_
like √3 because there
are an unknown number of digits after the decimal point and the fractional
value is not known. An example finding
the square root of 3, 1.72 = 2.89, 1.732 = 2.9929, 1.7322
= 2.999824 and so on implying one gets closer without obtaining 3. My calculator estimates it to be 1.7320508
but if one were to estimate it out, they would find that 1.73205082 ≠
3 either.
Refer to the figure
page 593 of Basic Mathematics
9th Edition by Marvin L. Bittinger
Then Pythagorean’s Theorem
states the measure of the hypotenuse2 = the measure of Leg A2
+ the measure of leg B2 or c2 = a2 + b2. Be advised that this works only on right
triangles. c2 = a2
+ b2 is known as Pythagorean’s equation. For further information on this see pages 593
to 595 in Basic Mathematics 9th Edition by Marvin L. Bittinger.
Chapter 10
Section 10.1 The Real Numbers
R indicates
the real number system. R is comprised of both positive and
negative numbers. Negative numbers are
all numbers less than 0. Remembering to
chapter 1, there was an additive identity 0.
0 was obtained by subtracting a number a by itself or a – a = 0.
Now suppose I were to tell
you all that there an opposite to 1 such that 1 + the opposite = 0. Using the a example, a – a = a + (-a) = 0
then the opposite of 1 is -1 The numbers consisting of all these opposite
numbers are called the negative numbers.
Showing this function I’ll introduce a number line:
-3
-2 -1 0
1 2 3
←├─├─├─┤─┤─┤─┤→
This number line also gives
us an indication of the ordering of these negative numbers. A corollary to this ordering is the larger
the number, the smaller will be its opposite.
A whole number is called an
integer. Remembering back to chapter 1,
there we dealt with positive whole numbers or positive integers. The mathematic notation for this is Z+ designating the positive
integers. Z denotes all integers both positive and negative. See pages 615 and 616 in Basic Mathematics 9th
Edition by Marvin L. Bittinger for uses of integers in the real world.
Another element of the real
numbers is the rational numbers. This is
usually designated by us math people as Q
for Quotient. The form for this is x/y
where x and y are integers. See pages 617
and 618 in Basic Mathematics 9th Edition by Marvin L. Bittinger for
further applications.
The last element of the real
numbers is the irrational numbers. This
does not have a letter denoting it so we just call it R or part of the real numbers.
_ _
_
Some examples of these
elements are π, √2 , √3,
√5, or any number that
does not have an exact square
root. See pages 618 and 619 in Basic
Mathematics 9th Edition by Marvin L. Bittinger for further
applications.
Now for the order. Here is where it becomes tricky. For the positive numbers, see section 1.4 pages
29 to 34 in Basic Mathematics 9th Edition by Marvin L. Bittinger for
reference. For negative numbers,
consider the opposite of the whole number.
As an example consider -5 and -8.
5 < 8 according to section 1.4 and the opposite of that indicates -5
> -8, which is the case. Now consider
-2 and 3. 3 > 2 but 2 > -2
therefore 3 > -2, which implies any positive number is greater than any
negative number which implies 1 > -2.
To further prove this point look at the number line
-3 -2 -1 0 1
2 3
←├─├─├─┤─┤─┤─┤→
the further right one goes,
the greater the value; and conversely the further left one goes, the lesser the
value.
Now putting all the
components together we have the Real Number System.
The last we see in this section
is an operation called the absolute value.
This is the distance on the number line between the number and 0. For further information see page 620 of Basic
Mathematics 9th Edition by Marvin L. Bittinger. Basically, if x is a number in R then |x| is the absolute value of x
which means x = x if x > 0, x = x if x = 0, and x = (-1) ∙ x (see
section 10.4 in Basic Mathematics 9th Edition by Marvin L. Bittinger
for operation)
Section 10.2 Addition of the
Real Numbers
To add 2 + 1 lets use the
number line
-3
-2 -1 0
1 2 3
←├─├─├─┤─┤─┤─┤→
├→┤
and as you see we went right
one space which by section 1.2 in Basic Mathematics 9th Edition by
Marvin L. Bittinger is correct 2 + 1 = 3.
Using this example, when adding positive number move right. On the other hand when adding a negative
number move left for instance (-2) + (-1)
-3
-2 -1 0
1 2 3
←├─├─├─┤─┤─┤─┤→
├←┤
Addition of two negative
numbers is equal to – ( | the first number | + | the second number| ) = - ( 2 +
1 ) = -3. Now looking at 2 + (-1) by the
number line,
-3 -2 -1 0 1
2 3
←├─├─├─┤─┤─┤─┤→
├←┤
and again since the line
moves to the left, 2 + (-1) is the same as 2 – 1 = 1. For further information see page 623 in Basic
Mathematics 9th Edition by Marvin L. Bittinger
In addition without a number
line see page 624 in Basic Mathematics 9th Edition by Marvin L.
Bittinger.
Be advised that opposites are
most commonly referred to as additive inverses, this is why 0 is the additive
identity. Also refer back to section 1.2
in Basic Mathematics 9th Edition by Marvin L. Bittinger to remember
that if x is a positive whole number, 0 + x = x, the same is true for the real
numbers. This make this the additive
identity property. Using x as a real
number and x < 0, then –x = |x|. If y
= the additive inverse of x, then
x + y = 0.
Section 10.3 Subtracting of
the Real Numbers
Theoretically, if x, y and z
are real numbers x - y = z implies x = y + z.
x – y = x + (-y) and they
work the same on the number line and as in the previous section.
Section 10.4 Multiplication
of the Real Numbers
In multiplying numbers,
multiply the absolute value of each number with the following consideration:
If both numbers are positive or negative then the answer
will be positive.
If one is positive, and one is negative then the answer
will be negative.
For examples see pages 635 to
636 in Basic Mathematics 9th Edition by Marvin L. Bittinger.
Section 10.5 Division of the
Real Numbers and Order of Operations
Using again x, y, and z are
real numbers; if x/y = z this implies x = y ∙ z.
In dividing numbers, divide
the absolute value of each number with the following consideration:
If both numbers are positive or negative then the answer
will be positive.
If one is positive, and one is negative then the answer
will be negative.
For examples see page 639 in
Basic Mathematics 9th Edition by Marvin L. Bittinger.
Remember from section 1.6 in
Basic Mathematics 9th Edition by Marvin L. Bittinger. Division by 0
is undefined and 0/x = 0 using the x from above.
For examples on reciprocals see
pages 640 to 642 in Basic Mathematics 9th Edition by Marvin L.
Bittinger. For examples of order of
operations see pages 643 to 644 and section 1.4 in Basic Mathematics 9th
Edition by Marvin L. Bittinger.