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Summer 2016 - Session 2
Math 1300
FUNDAMENTALS OF MATH
Section #16535
Monday - Friday, 10am-12pm
Instructor: Dr. Angelynn Alvarez
[email protected]
06/05/2016
Section 1.1 - (Real) Numbers
Section 1.1 - (Real) Numbers
Before working with and computing numbers, it is beneficial to know
what kind of numbers there are and how to identify them.
Real numbers are the numbers you see in everyday life, in the real
world.
There are two major types of real numbers:
First we will discuss rational numbers and its subcategories.
Natural Numbers
-This is the smallest category of the real numbers.
The Natural Numbers are the “counting numbers”, which start with
the number 1:
{1, 2, 3, 4, 5, 6, 7,…,50, 51,…,100, 101, 102,...,etc.}
Examples:
• 17 is a natural number.
• 3,252,352 is a natural number.
• -1 is not a natural number because it is not positive, and you
cannot count “-1 objects”.
• 0 is not a natural number.
Whole Numbers
-This is the second smallest category of the real numbers.
Whole numbers are the natural numbers and the number “0”:
{0, 1, 2, 3, 4, 5, 6, 7,…,50, 51,…,100, 101,...,etc.}
Observe: From the diagram, one can see that every natural number
is a whole number.
Integers
The integers is the set of numbers which consists of
• The natural numbers (the “counting numbers”)
• The number 0
• The negatives of the natural numbers
{…-51,…, -4, -3, -2, -1, 0, 1, 2, 3, 4,…10,...,51,…}
Observe: From the diagram,
• Every natural number is a whole number.
• Every whole number is an integer.
• Every natural number is an integer.
WARNING:
• NOT every integer is a natural number.
• NOT every integer is a whole number.
• NOT every whole number is a natural number.
Examples:
• -3 is an integer but it is not a natural number.
• -10 is an integer, but it is not a whole number.
• 0 is a whole number, but it is not a natural number.
Rational Numbers
A rational number is a number that can be put in the form of
where 𝑝 and 𝑞 are integers and 𝑞≠0.
In other words, rational numbers are fractions.
We call 𝑝 the numerator, and we call 𝑞 the denominator.
!
!
,
Writing Fractions in Simplest Form
• Some rational numbers can be “simplified” if both the
numerator and denominator are divisible by the same integer.
• If the numerator and denominator are not divisible by a
common integer, then we say it is simplified.
• Sometimes it may take multiple steps of division to arrive at the
fraction’s simplified form.
Examples: Simplify the following fractions.
15
30
42
70
24
40
13
40
The following are also rational numbers:
Examples:
INTEGERS
(we can write every integer, say n,
!
as 𝑛 = .)
!
TERMINATING DECIMALS
- decimals that terminate to the
right of the decimal
REPEATING DECIMALS
- decimals that do not terminate,
but the numbers to the right of the
decimal follow a certain pattern
MIXED NUMBERS
- numbers that consist of an
integer and a fraction; they are of
!
the form 𝑚 !
Remark about mixed numbers: We always prefer mixed numbers
to be written as a single fraction, instead of a mix of an integer and a
fraction.
Converting mixed numbers to single fractions
!
Given a mixed number, say 𝑚 , we can write it in the form of a
single fraction
!
!
!
as follows:
𝑚
!
!
=
! × !!!
!
Examples:
!
Convert the following mixed numbers into the form .
!
2
1
3
7
1
8
!
−7 ! (For negative mixed numbers: temporarily ignore the “−“, then
include it in the conversion.)
Converting single fractions to mixed numbers:
Given a single fraction, say
!
!
, we can write it as a mixed number 𝑚
as follows:
1.
2.
Divide 𝑝 ÷ 𝑞. You then get a quotient and a remainder.
The mixed number is of the form:
𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟
𝑞𝑢𝑜𝑡𝑖𝑒𝑛𝑡
𝑞
Note: Sometimes
!"#$%&'"!
!
may need to be simplified.
Hint: Sometimes it is easier to simplify the single fraction before
converting it into a mixed fraction.
Examples:
!
Convert the following fractions into the form 𝑚 ! .
14
3
324
60
426
54
!
!
Irrational Numbers
Irrational numbers are numbers that cannot be written as fractions-!
-that is, they are numbers that cannot be written in the form ! .
NONTERMINATING, NONREPEATING DECIMALS are irrational
numbers. (These are decimals which go on forever, and the numbers
to the right of the decimal do not follow any pattern).
Examples:
2
3
𝜋
Real Numbers
A real number is a number that is either rational or irrational (but not
both).
Real numbers are the numbers we have been studying so far (and
will be working with throughout the course).
Summary
• Natural Numbers (counting numbers):
• {1, 2, 3, 4, 5,…}
• Whole Numbers (counting numbers + 0):
• {0, 1, 2, 3, 4, 5,…}
• Integers: (whole numbers + the negatives):
{…-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5,…}
• Rational numbers (fractions) - Numbers on the form
!
!
• Irrational numbers: Numbers that are not rational
(nonterminating, nonrepeating decimals)
• Real Numbers: Numbers that are either rational or irrational,
but NOT both.
Example: Classify the following numbers into the following
categories:
-10,
!!"
!
, 15, 1.2, 𝜋,
Natural:
Whole:
Integers:
Rational:
Irrational:
3, 1, 0,
!
!
!
, 19, 1.233333, 4 , 17.1
!
Dividing By Zero: The worst thing you can ever do
Notice in the definition:
A rational number is a number that can be put in the form
!
!
,
where p and q are integers and q≠0.
YOU CAN NEVER HAVE 0 IN THE DENOMINATOR.
Ordering and Comparing Real Numbers
There are 3 possible ways to order any 2 real numbers. Call the two
real numbers x and y.
(1)
x is greater than y
𝑥>𝑦
(2)
x is less than y
𝑥<𝑦
(3)
x is equal to y
𝑥=𝑦
The comparison of two real numbers depend on their relative location
on the real number line:
On a number line, we say…
• x is greater than y if it is to the right of y on the number line
• x is less than y if it is to the left of y on the number line
• x is equal to y if it in the same exact place as y on the number
line
Hint: If the symbols < and > represent the mouth of
Pac-Man, always remember that Pac-Man always wants
to face and eat the bigger number.
Example: Fill in the following blanks with < , >, 𝑜𝑟 =.
1.31 _____ 2.47
1 ____ 0.7
36 _____ 10
−0.2 _____ 2
1.8 ____ 1.8
−0.2 _____ − 0.9
Classifying Real Numbers: Positive vs. Negative
• A real number is positive if it is greater than zero.
If 𝑥 is a positive real number, we write: 𝑥 > 0.
• A real number is negative if it is less than zero.
If 𝑥 is a negative real number, we write: 𝑥 < 0.
Caution: The number 0 is neither positive nor negative! It is
considered neutral.