Introductory Mathematics, MAT100a,d, Fall 2020

September 08

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 Organisationsal Issues
The course is based on a textbook; consult the syllabus and get yourself a copy
from the Bookstore, or an electronic one. Because of the hybrid system, it
would be crucial for you to be able to use the textbook.
You will be assigned homework electronically. You can do the HW with the
help of others, but make certain you understand the solutions yourself.
Quizzes, the Midterm Exam and the Final Exam are indeed means to check if
you did the HW in a responsible way.
I have an office hour (advertised in the syllabus). I plan to observe it; you are
welcome to visit and discuss things.

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 Student Assistants
Along with Mr. Plamen Ivanov, who will be giving you the recitation classes
that are part of this course, I have hired three student assistants; you can
contact them by e-mail and ask them to help you understand things in
personally arranged meetings, or group meetings (that get typically scheduled
before quizzes and exams).
These are:
Katrin Dyulgerova, knd180@aubg.edu
Vyara Pavlova, vkp190@aubg.edu
Gent Mulaku, gnm190@aubg.edu
My assistants are students like you, and have their own classes and exams.
Meetings can be arranged only if and when convenient for them too.

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 Quizzes and Exams
Very importantly, these will take place on the ground, except for students who
have chosen to be entirely online, that is, not come to Blagoevgrad at all.
However the tasks will be given electronically, and your answers will be graded
by the same system too. This means that you need to bring your laptop to the
classroom and use it for doing whatever the assessment exercise is just the way
you do for your homework. A smartphone might do as a backup; it is down to
you how comfortable that would be, if you have no laptop, or it happens to
malfunction.
Please read the syllabus carefully to learn about the course organisation, and
especially the grading system.

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 Course Material
Along with the textbook and the HWs, I shall e-mail to you these slides after
the lectures. I shall also circulate HW, quiz and exam papers from previous
years (when exams were on paper) for additional exercise.
The forms for that obsolete HWs have tables for grading on them; you can
ignore that. The relevant grading is to be of your electronically submitted
stuff, and will appear on Canvas shortly after each exercise.

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 Active Participation
When online, the closest we can achieve to appearing at the whiteboard for
active participation is sharing your handwritten notes.
To this end I want your cameras to be showing your notes, and not necessarily
your face.
The same applies to students whose turn is to be on the ground; this would
make their writings visible to the ones online.
To my knowledge tripods for positioning smartphones with the camera pointing
down towards the desk are available. (I am not a smartphone user myself.)

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 The real axis
In this course we are to only use numbers, which can be used to measure
lengths and determine the positions of points on a straight line.
A real axis is straight line with
a distinguished point O on it, called the origin,
a distinguished positive direction (one of the two possible ones),
a unit interval.
The position of an arbitrary point on a real axis is determined by the number
of applications of the unit interval as a yardstick, that are needed to reach that
point. That number is called the co-ordinate of the point.
Points which are in the positive (negative) direction relative to the origin O
have positive (negative) co-ordinates. The co-ordinate of the origin itself is 0.
Points which take a whole number of applications to reach, have integer
co-ordinates.
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The integer numbers are . . . − 2, −1, 0, 1, 2, 3, . . ..
The positive integer numbers 1, 2, 3, . . . are also known as the natural
numbers.
Sometimes 0 is regarded as a natural number too.
The non-negative integers are termed whole numbers.

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 The decimal representation of numbers
Most of the points on a real axis do not admit integer co-ordinates. A point
may require a certain whole number of applications of the unit interval to get
close to; closer than the length of another application. To cover the remaining
distance, an interval which is 1/10th of the size of the unit one is taken, and
the applications of this interval are counted. Then the co-ordinate of the point
assumes the form
1
n+ ×m
10
where m is the number of the applications of the 1/10th length unit interval.
Obviously m never needs to be more than 9, or else there would be enough
space to fit another complete application of the unit interval.

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 The decimal representation of numbers
If this is still not enough to reach the point precisely, a 1/100th unit long
interval is taken, and a co-ordingate of the form
1 1
n+ × m1 + m2 + . . .
10 100
Obviously m1 , m2 , etc., never need to be more than 9, or else there would be
enough space to fit another complete application of the unit interval (for m1 ),
or a complete application of the 1/10th sise unit interval for m2 .
The whole part of the co-ordinate can be written as

10k−1 nk + . . . + 102 n3 + 10n2 + n1

too, where n1 , n2 , . . . , can be chosen to be between 0 and 9 in a unique way.

Then

nk nk−1 . . . n1 .m1 m2 . . . ,

a sequence of digits, is the decimal representation of the point’s co-ordinate.

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 Rational Numbers and Irrational Numbers
Most of the points on the real axis require infinitely many mk s for their
decimal representation. E. g. , if we divide the unit interval into 3 equal parts,
the intermediate endpoints of the subintervals would have

0.333 . . . and 0.666 . . .

for their decimal representations because

1 1 1 1 2 1 1 1
=3 +3 +3 + . . . and = 6 + 6 +6 + ....
3 10 100 1000 3 10 100 1000
The infinite sequences m1 m2 . . . above are periodical, because the same (finite
sequence of) digits occurs again and again.

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 Rational Numbers and Irrational Numbers
A decimal d number whose representation has the form
0.m1 m2 . . . mk m1 m2 . . . mk . . . m1 m2 . . . mk . . .
can be written as
. . 0} 1 0| .{z
m1 m2 . . . mk ×0. 0| .{z . . 0} 1 . . . 0| .{z
. . 0} 1 . . .
| {z }
a natural number k−1 times k−1 times k−1 times

Now, since
1
0.0 . . . 010 . . . 01 . . . 0 . . . 01 . . . = ,
9| .{z
. . 9}
k times
m1 m2 . . . mk
we get 0.m1 m2 . . . mk m1 m2 . . . mk . . . m1 m2 . . . mk . . . = ,a
9| .{z
. . 9}
k times
proper fraction.
359 1 359 1
E.g. 0.359359359... = 999 because 999 = 0.001001 . . . , and 999 = 359 999 .
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This means that periodical decimal fractions can be written as the ratios of
integer numbers. Conversely, performing the division on two integer numbers
always leads to a periodical decimal representation.

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 Rational Numbers and Irrational Numbers
Numbers which can be written as ratio m n where m and n are both integer,
and n is not 0, for the expression to be meaningful, are called rational.
Some points do not admit a periodical decimal representation of their
co-ordinates on the real axis. The numbers, which are required to supply such
points with co-ordinates, are irrational.
The real numbers include both the rational and the irrational ones; they are all
the numbers which may happen to represent the positions of points on the real
axis.

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 Arithmetical operations on the real numbers and algebraic
expressions
In this course we are to use the basic arithmetical operations +, −, ×, /,
raising to a power and radicals. Later on we shall study exponentiation,
logarithms and the trigonometric functions too.
+, −, × and / each have a geometric interpretation on the real axis. E.g.,
addition can be seen as obtaining the length of segments which combine
segments of the lengths of the given operand numbers.
The operands of +, −, ×, / have specific names: summands, factors,
numerators and denominators, etc. To reason about unspecified quantities, we
are going to denote the respective numbers by letters, e. g., assuming that a
and b are numbers, we can write
a + 5, a − b/2, etc.
to express the sum of a and 5, the difference of a and b divided by 2, etc.,
whatever number a might be. The above a
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 Arithmetical operations on the real numbers and algebraic
expressions
To make statements about unspecified numbers we link expressions by = or <,
to indicate that, for the hypothetical numbers written with letters involved,
two expressions evaluate to the same number, or to numbers which are ordered
in some specified way, e.g.,

a − 5 = 2b, 3 − b < a, etc.

mean that subtracting 5 from a and multiplying b by two produce the same
number; subtracting b from 3 is a smaller number than a, etc. (A number x is
smaller than another one y , written x < y if y is the co-ordinate of a point
which is in the positive direction from the point whose co-ordinate is x.)
To be able to reason about such statements, we need to know some properties
of the arithmetical operations. These properties are expressed by equalities and
inequalities written in terms of arbitrary real numbers as above too.

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 Algebraic properties of + and ×
Multiplication × is commonly written as a dot (.), or altogether omitted:
a × b = a.b = ab.

property addition multiplication

commutativity a+b=b+a ab = ba
associativity (a + b) + c = a + (b + c) (ab)c = a(bc)
neutral element a+0=a a.1 = a
1
inverses a + (−a) = 0 a. = 1 for a 6= 0
a
inverses the negative of a the reciprocal of a
distributivity a(b + c) = ab + ac
no divisors of zero a.b = 0 only if a = 0 or b = 0

The validity of properties such as the above can be checked by analysing the
way the operations are performed on the decimal representation of numbers.

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 Derived properties
Properties which look similarly fundamental, can sometimes be derived from
the ones already formulated. E.g., here follows a formal proof of a.0 = 0:

1 a.(0 + 0) = a.0 + a.0 distributivity

2 0+0=0 direct check
3 a.0 = a.0 + a.0 substitution of 0 for 0 + 0 in 1
4 a.0 + (−a.0) = (a.0 + a.0) + (−a.0) the same number is added on both sides of 3
5 0 = a.0 + (a.0 + (−a.0)) inverse on the LHS, associativity on the RHS
6 0 = a.0 + 0 inverse on the RHS
7 0 = a.0 neutral element

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 Derived arithmetical operations: − and /
The above properties refer explicitly to + and × only; subtraction and division
can be defined in terms of + and ×, using negatives and reciprocals,
respectively:

a − b=
ˆ a + (−b) the negative of b is added to achieve the effect of subtracting b
a
ˆ a × 1b the reciprocal of b is multiplied to achieve the effect of dividing by b
=
b
The properties of − and / are derived from these definitions, e.g.
a+b a b
distributivity = + for c 6= 0
c c c
is derived by rewriting the equality as
1 1 1
(a + b) × =a× +b×
c c c
Equalities such as the above, which hold for the involved letters denoting any
numbers, may be subject to restrictions such as c 6= 0, are called identities.

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 More Derived properties
1
involutivity −(−a) = a 1 =a
a

−a a a
(−a)b = a(−b) = −(ab) = =−
b −b b

a c ab b
= iff ad = cb =
b d ac c

a c ad + bc ac ac
+ = =
b d bd bd bd

a
b ad
c =
d bc

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 Using the Properties to Manipulate Proper Fractions
a
Ratios with integer a, b are called proper fractions (as opposed to decimal
b
ones).
1 1 7 5 1 12
+ = + = (7 + 5) =
5 7 7.5 5.7 35 35
17 15 2.17 3.15 1 11
− = − = (2.17 − 3.15) = −
3 2 2.3 3.2 6 6
Exercise
1 3 11 13
Write + as a single proper fraction. Write − as a single proper
50 20 63 35
fraction.

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 Absolute value
If a 6= 0, then either a is positive, and −a is negative, or a is negative, and −a
is positive.
The absolute value of a, written |a| is the non-negative one, of the numbers a
and −a.
|.| ”erases” the sign of a: |5| = | − 5| = 5
Obviously |a| ≥ 0, and |a| = 0 only if a = 0. Furthermore, ||a|| = |a|.
Given two numbers a and b, the distance between their corresponding points
on a real axis is |b − a| units (measured using the unit interval of that axis).
Thanks to the |.| in |b − a|, we have a single expression for all the possible
orderings of a and b on the axis: |b − a| = |a − b|.

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 Exponents
ax - ”a raised to the power x”. a is called the base; x is the exponent.
For a natural n, an =
ˆ a × . . . × a:
| {z }
n times

33 = 3.3.3 = 27, 210 = 2.2.2.2.2.2.2.2.2.2 = 1024

For n = 0, a0 = 1, only for a 6= 0. 00 is undefined! In the sequel we assume
non-zero bases in exponents.

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 Exponents with negative bases
Examples: (−1)2 = (−1).(−1) = 1; (−1)2 = (−1).(−1).(−1) = −1,
(−1/2)3 = −1/8, . . .
If a < 0, then a = −|a| = (−1) × |a|, and

an = a × . . . × a = (−1) × |a| × . . . × (−1) × |a| =

| {z } | {z }
n times n times
|a| × . . . × |a| × (−1) × . . . × (−1) = |a|n × (−1)n
| {z } | {z }
n times n times

Now

 1 if n is even
(−1)n =
 −1 if n is odd

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Hence, for a < 0,

 |a|n if n is even
an =
 −|a|n if n is odd

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 Properties of exponentiation
am an = a × · · · × a × a × · · · × a = am+n
| {z } | {z }
n times m times
am m −n m−n
= a × a = a
an
(ab)n = an bn
³ a ´n an
= n.
b b
(an )m = a × · · · × a × · · · × a × · · · × a = anm
| {z } | {z }
n times n times
| {z }
m times

Negative and fractional exponents

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If n < 0, then −n, the negative of n, is a positive number and an =
ˆ .
a−n
1 1 1 1
E.g, 10−5 = 5
= , (−11) −2
= 2
= − .
10 10000 −11 121

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The End

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