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MATH PAK IV
All material is (C) 1993 protected by Dan Dalal
All Rights Reserved WorldWide.
No Duplication and/or Modification allowed to this file.
Welcome to MATH 101, of MATH PAK IV. If you feel that this file is not
in its entirety or there has been tampering done to the file, then you
can send a blank, formatted disk(either 5.25 or 3.5") to :
Dan Dalal
Dept: MPK4
374 Don Basillo Way
San Jose, CA 95123
A new, untampered copy of the MATH101.DAT file will be sent to you.
There is no cost for this service, but you MUST include return postage,
or SASE, otherwise, it will not be sent to you.
<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>
MATH 101
This file is intended for users who want a basic "re-fresh", or re-grasp of
their original mathematical knowledge, held once upon a time...
Questions are asked along the way, which you are encouraged to try. If you
cannot solve a particular question, then go back and review the topic again
and consult a more comprehensive math text, available at your school library
or local public library. Or, talk with your math teacher or professor.
You can exit from this by entering Alt-X. To make the view window larger
or smaller, use the F5 key(if so equipped).
==============================================================================
Some basic math symbols and terminology :
Symbol Meaning
+ Addition process
- Subtraction process
x or * Multiplication process
/ or ÷ Division process
< Less than
> Greater than
= Equals
≥ Greater than or equal to
≤ Less than or equal to
≈ Approximately
√ Square root
∞ Infinity
π Pi ≈ 3.1415
Σ Summation process
± Plus or Minus
What is an integer ?
An integer, is a number that has no decimal parts to it. For example,
an integer would be the number '2'. The numbers 3,222,-84, 325
and 45 are all examples of positive and negative integer numbers.
What is a real number ?
A real number, is any number, regardless of it having a decimal(fraction)
part or not. For example, a real number would be the number '3.43'. Notice
that the number has a fractional part to it : .43 . Some real numbers may
not always have a fractional part, like the number 68. This would imply
that all INTEGER numbers can be considered a part of the real number world !
The numbers -4.55, 34, -32.2, 3233.31 and 43.4 are all examples of positive
and negative real numbers.
The number line :
You probably remember your math teacher in the second or third grade talking
about this one ! A number line represents all real numbers, from negative
infinity to positive infinity. Numbers to the left of the number 0, are
called "negative numbers and have a negative sign(-) in front of the number,
whereas numbers on the right side of the number 0, are called "positive"
numbers and have a positive sign(+) in front of the number. When a sign is
not present in front of a number, it can be assumed that the number is a
positive(+) number.
Here is an example of a number line :
|-----|-----|-----|-----|-----|-----|-----|-----|
... -4 -3 -2 -1 0 1 2 3 4 ...
negative positive
Note that the point zero(0), is called the ORIGIN of the number line.
Do you know what a RATIONAL number is ?
A RATIONAL number, is any number which can be expressed as the quotient
(the result of a division process) of two integers. For example, the
fraction 4 / 3 is a rational number, as is 16 / 5.
A IRRATIONAL number, is any number which cannot be express as the
quotient of two integers.
Give three examples of each of the following :
a. integer
b. real number
c. rational number
d. irrational number
What does the symbol "<" mean ? Can you give an example ?
Fractions.
A fraction can be thought of as a way to do division. For example, given
the fraction 3/4(3 over 4), aren't we actually saying we want to divide
the number 3 by the number 4 ? By the way, the number "3" is on top of the
fraction and the top number is called the NUMERATOR. The bottom number, "4"
is called the DENOMINATOR, or :
NUMERATOR
__________
DENOMINATOR
Can you give three examples of a fraction ?
Given the fraction 4/5, which number is the number is the numerator ?
The denominator ?
Square roots.
The square root of a number, denoted by the symbol "√", means that
we are trying to find a number, such that when that number is multiplied
by itself, will equal the number we started with.
When we see for example the expression √2, this tells us that we want
to find a number, either positive or negative(because when a negative(-)
and negative(-) are multiplied together, they equal a positive) that will
give us a result of 2, which is under the √ symbol/sign.
In this case, the √2(read "the square root of two"), would be ≈
± 1.414, or 1.414, -1.414( as -1.414 * -1.414 ≈ 2).
What would the √1 = ?
RULE : You cannot have(or calculate) the square root(√) of a NEGATIVE(-)
number. Why this is, will be explained later...
Absolute value.
The absolute value of a number, denoted by two vertical marks(│) on
the left and right side of a number, indicate the DISTANCE from that
number to the ORIGIN, or point zero(0).
For example, given │23│, this would indicate the distance(always a
positive value) from the point "23" to the point(origin) 0, or 23.
This also holds true for negative numbers. For example, given │-56│ ,
this would indicate the distance from point "-56" on the number line
to the point(origin) 0, or 56, so │-56│ = 56.
What is │-3│ ?
Basic Algebra.
Algebra, in its simplest form, is the study of numbers and how they work
with each other.
We've so far talked about integers, real numbers, rational and irrational
numbers, absolute value and fractions...All a part of Algebra !
There are several properties that we can discuss here :
1. When we say "a < b", we mean that the value of "a" is less than
the value of "b".
2. When we say "a > b", we mean that the value of "a " is greater(more)
than the value of "b".
3. Given if a - b > 0, then a > b or b < a.
4. Given :
a > b , this means that the point "a" is to the right of point "b"
on the number line.
a = b, this means that point "a" and point "b" are the same point
on the number line.
a < b, this means that the point "a" is to the left of point "b"
on the number line.
5. If a,b and c are real numbers, then if
a > b and b > c, then a > c
if a < b and b < c, then a < c
if a > b then a + c > b + c
if a < b then a + c < b + c
6. Given " a ≥ b", then this means that point "a" is greater than or
equal to the point "b" on the number line.
7. Given "a ≤ b", then this means that point "a" is less than or
equal to the point "b" on the number line.
8. An open interval on a number line, can be denoted as (a,b), or
we can say a < x < b, where x is the collection of all points between
points "a" and "b". The points "a" and "b" are called "end points."
9. A closed interval on a number line, can be denoted as [a,b], or
we can say a ≤ x ≤ b, where x is the collection of all points between
points "a" and "b".
Graphing.
Mathematicians use what is called the Cartesian Coordinate System to
do their graphing with. In the Cartesian Coordinate System, two
mutually perpendicular number lines are used. The up-down(vertical)
number line, is called the "y-axis" and the left-right(horizontal)
number line, is called the "x-axis". We can show a small portion of it
here :
I
I 4
I
I 3
Quadrant II I Quadrant I
I 2
I
I 1
_____________________________x-axis
-3 -2 -1 0 1 2 3 4
I -1
I
I -2
Quadrant III I Quadrant IV
I -3
I
I y-axis
Now, the point at which both lines meet, is called the ORIGIN and we
can assign an ordered pair((a,b)) to the point as (0,0), where the
first number is the "x" value and the second number, is the "y" value.
Where the ordered pair lie on the plane, is called a "point".
The first number, is also referred to as the "abscissa" of the point
and the second number is referred to as the "ordinate" of the point.
Some rules to remember :
1. If the ordered pair of numbers are both positive(+,+), then
the point is in Quadrant I.
2. If the ordered pair of numbers are (-,+), then the point is
in Quadrant II.
3. If the ordered pair of numbers are (-,-), then the point is
in Quadrant III.
4. If the ordered pair of numbers are (+,-), then the point is
in Quadrant IV.
In what quadrant would the point (-45,56) be in ?
What about the point (5,-2.3) ?
Quadratic Equations.
Quadratic equations can be given in the general form :
ax² + bx + c = 0
The Quadratic formula, can be given as :
-b ± √(b² - 4ac)
x = ___________________
2a
The expression b² - 4ac, is called the "discriminant" and can determine
the outcome of the roots of the quadratic function.
Rules:
1. If the discriminant is > 0, then there are two real roots and
the graph(a parabola) will intersect the x-axis at two points.
2. If the discriminant is = 0, then there is one real root and one
root which is = (b) / (2a). The graph of the function will
intersect the x-axis at only one point.
3. If the discriminant is < 0, then there are two imaginary roots
and the graph of the function does not intersect the x-axis.
Using the quadratic formula given above, can you solve for the following
function : 4x² - 3x + 2 = 0 ? What are the two roots ? What does the
discriminant tell you about the graph of this function ?
Linear equations.
Linear equations are equations that deal with first degree(x) polynomials
and have as their graphs, straight lines on the Cartesian Coordinate System.
The graph of a linear function, is a straight line.
The "slope" of a line, can be given as the (change in y) / (change in x).
The slope of a line is usually represented by the letter "m".
m = (rise) / (run)
or m = (y2 - y1) / (x2 - x1)
For example, given two points on a line as (0,0) and (3,4), then :
x1 = 0, x2 = 3, y1 = 0 and y2 = 4.
Using the above formula, we get (4 - 0) / (3-0) or 4/3 = 1.33 as the
slope of the line.
Rules for slope :
1. if m > 0, the line slants upward from left to right.
2. if m = 0, then the line is parallel to the x-axis and the
slope is undefined.
3. if m < 0, then the line slants downward from left to right.
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