Fundamentals of Mathematics
1.1 Real Numbers
Ricky Ng
Lecture 1
August 26, 2013
Ricky Ng
Fundamentals of Mathematics
Course Info
Math 1300 Section 15446
Instructor: Wai Hin (Ricky) Ng
Office Hours: TBA
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Fundamentals of Mathematics
Grading
The course grade will be determined as follows:
Daily Poppers
Homework
Weekly Quizzes
Test 1-4
Final Exam
10%
10%
20%
40%
20%
Details in the syllabus.
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Fundamentals of Mathematics
Announcements
Course webpage:
http://www.math.uh.edu/⇠rickula/teaching.html
Read the course syllabus and course policies.
CourseWare webpage:
https://www.casa.uh.edu
Purchase an access code ASAP at the University
Bookstore and create a student CASA account. DO NOT
BUY ANY TEXTBOOK!
Purchase Popper Bubbling Forms (Math 1300, section
15446) at the University Bookstore.
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Fundamentals of Mathematics
Real numbers
In this course, we are dealing with numbers in real life, the real
numbers. There are two big categories of real numbers:
rational and irrational.
We first discuss the set of rational numbers, which includes
natural numbers, whole numbers, and integers.
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1.1 Real Numbers
Natural numbers
The natural numbers are simply the counting numbers we
see in daily life:
{1, 2, 3, 4, 5, . . . }
These numbers are the “building blocks” of the real number
system.
Definition (Primes and Composite)
A natural number > 1 is called prime if its only factors are 1
and itself.
A natural number > 1 is called composite if it is not a prime.
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1.1 Real Numbers
Natural numbers
Every natural number other than 1 can be classified as either
a prime or composite. The number 1 is neither a prime nor
composite.
Question
What is the best way to check whether a natural number is
prime or not?
Find its factors.
Example
Is 18 a prime or composite? What about 19? 24?
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1.1 Real Numbers
18
19
24
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1.1 Real Numbers
Whole numbers
The whole numbers are just the natural numbers together
with 0:
{0, 1, 2, 3, 4, . . . }
Thus, every natural number is a whole number.
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1.1 Real Numbers
Integers
The set of integers basically includes the negative whole
numbers:
{. . . , 3, 2, 1, 0, 1, 2, 3 . . . }
So, every natural number is an integer, BUT not the other way
around.
For example, -12 does not belong to {1, 2, 3, . . . }.
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1.1 Real Numbers
Every integer can be classified as even or odd.
Definition
An integer is even if it is divisible by 2.
An integer is odd if it is not divisible by 2.
Example
51 is odd because 2 does not divide it.
-20 is even, since 20 = 10 ⇥ 2.
What is the easiest way to check? Look at the last digit...
15446 is even and 15445 is odd.
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1.1 Real Numbers
Rational numbers
Rational numbers are probably the number we deal with the
most in this course... Yes, they are the fractions!
Definition (Rational number)
A rational number r is a real number that can be written as
r=
p
,
q
where p and q are integers, and q 6= 0.
Since every integer p and can be written as p = p1 , all
integers are rational.
Division by zero is not allowed!
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p
0
is not defined.
1.1 Real Numbers
Other than integers, the following are rational numbers as well:
1
Mixed numbers
2
Terminating decimals
3
Repeating decimals
Definition (Mixed number)
A mixed number is of the form m ab , where m is a whole
number and ab is a rational number.
Recall that this notion means
m
a
a
b a
m⇥b+a
= m+ = m ⇥ + =
.
b
b
b b
b
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1.1 Real Numbers
Other than integers, the following are rational numbers as well:
1
Mixed numbers
2
Terminating decimals
3
Repeating decimals
Definition (Mixed number)
A mixed number is of the form m ab , where m is a whole
number and ab is a rational number.
Recall that this notion means
m
a
a
b a
m⇥b+a
= m+ = m ⇥ + =
.
b
b
b b
b
Hence, a mixed number is also rational (improper).
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1.1 Real Numbers
Example
Write the mixed number 2 23 as an improper fraction.
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1.1 Real Numbers
Decimals
Terminating decimals are rational. How do we convert it into a
rational number?
Rule you should know
Count the number of digits after decimal, say n, and write that
part after decimal as a fraction over 10n .
Example
Write the following terminating decimal numbers as rational
numbers: 2.6 and 3.103
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1.1 Real Numbers
2.6 =
3.103 =
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1.1 Real Numbers
What about repeating decimals?
Rule you should know
Count the number of digits after decimal, say n, and write that
part after decimal as a fraction over 10n 1, or 9| .{z
. . 9}.
n
Example
Write the following repeating decimals as rational numbers: 0.4̄
and 2.43
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1.1 Real Numbers
0.4̄ =
2.43 =
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1.1 Real Numbers
Definition (Irrational numbers)
A number is called irrational if it is not rational.
p
p
Examples: 2, ⇡, 5 + 3
Rule you should know
If a decimal number is non-terminating and non-repeating,
then it is irrational.
Rule you should know
p
If p is a prime number, then p is irrational.
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1.1 Real Numbers
Real numbers
Definition (Real number)
A real number is a number that is either rational or
irrational, but not both.
Every real number can be written as a decimal number.
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1.1 Real Numbers
Caution
There are two special real numbers, 0 and 1, in the
multiplication ⇥. For any real number x 6= 0,
x⇥0=0
This explains why
are equivalent...
x
1
x ⇥ 1 = x.
= x, because division and multiplication
Rule you should know
c
a ⇥ b = c () a = ,
b
for c 6= 0.
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1.1 Real Numbers
This also explains why division by zero is undefined.
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1.1 Real Numbers
This also explains why division by zero is undefined.
For instance, if a = 100
0 made any sense, then it is equivalent to
say
a ⇥ 0 = 100,
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1.1 Real Numbers
This also explains why division by zero is undefined.
For instance, if a = 100
0 made any sense, then it is equivalent to
say
a ⇥ 0 = 100,
However, a ⇥ 0 can only be ZERO.
So don’t ever divide by zero unless you believe 100 = 0.
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1.1 Real Numbers
Exercise
Classify the following numbers into the following categories:
p
p
4 25
2.1, 7, 6, ⇡, 13, 1.232323..., ,
, 0.24
9 3
Natural:
Integers:
Rational:
Irrational:
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1.1 Real Numbers