1.1: Apply Properties of Real #s
Objectives:
1. To describe real
numbers and their
subsets
2. To approximate
square roots to the
nearest tenth without
a calculator
Assignment:
• Make your own
visual/graphic
representation of real #'s
with multiple examples in
each set: Real,
Irrational, Rational,
Integer, Whole, Natural
• Working with Irrational
Numbers, 2, 4, 8, 12, 16,
20, 22
Vocabulary
Real numbers
Rational
numbers
Irrational
numbers
Integers
Natural numbers
Whole numbers
OBJECTIVE 1
You will be able to describe
real numbers and their subsets
Building the Real Number Line
We group different
kinds of numbers
into sets.
The first set of
numbers is the natural
numbers. These are
the counting numbers:
{1, 2, 3, 4, …}
Building the Real Number Line
These numbers
can be arranged
by size on a ray,
where the size
increases as you
move to the
right.
The first set of
numbers is the natural
numbers. These are
the counting numbers:
{1, 2, 3, 4, …}
1
2
3
4
5
Building the Real Number Line
The set of natural
numbers is
represented by
the capital letter
N or a
blackboard bold
ℕ.
The first set of
numbers is the natural
numbers. These are
the counting numbers:
{1, 2, 3, 4, …}
1
2
3
4
5
Building the Real Number Line
Some authors do not
make a distinction
between whole and
natural numbers,
but others do.
Next, we add zero
to the natural
numbers to get the
whole numbers:
{0, 1, 2, 3, 4, …}
0
1
2
3
4
5
Building the Real Number Line
All the natural numbers are positive. If
we add the negative numbers to the
whole numbers, we get the integers:
{…, -3, -2, -1, 0, 1, 2, 3, …}
-5
-4
-3
-2
-1
0
1
2
3
4
5
Building the Real Number Line
Now we have a number line that extends
infinitely in two directions, and the numbers
decrease in size as you move to the left.
The set of integers is represented by Z or ℤ
for the German word zahl.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Building the Real Number Line
However, we still do not have the real
number line, since there are an infinite
number of gaps.
For example, how many numbers exist
between 0 and 1? Between any two
integers?
-5
-4
-3
-2
-1
0
1
2
3
4
5
Building the Real Number Line
Next, we have all
the numbers that
can be written as
fractions, called
rational numbers.
-5
-4
-3
-2
-1
Rational numbers
𝑎
are of the form
𝑏
where 𝑎 and 𝑏 are
integers and 𝑏 ≠ 0.
0
1
2
3
4
5
Building the Real Number Line
The set of rational
numbers is
represented by Q
or ℚ for quotient.
-5
-4
-3
-2
-1
Rational numbers
𝑎
are of the form
𝑏
where 𝑎 and 𝑏 are
integers and 𝑏 ≠ 0.
0
1
2
3
4
5
Building the Real Number Line
The rational numbers
can also be written
as:
Rational numbers
𝑎
are of the form
𝑏
where 𝑎 and 𝑏 are
integers and 𝑏 ≠ 0.
– Fractions: -¾ or 10/3
– Terminating or
repeating decimals:
-0.75 or 3.33333…
-5
-4
-3
-2
-1
-3/4
0
1
2
3
10/3
4
5
Building the Real Number Line
However, we still have
an infinite number of
gaps, since not all
numbers can be
written as a ratio of
integers.
-5
-4
-3
-2
-1
So we add in the
irrational numbers,
which are simply the
opposite of rational
numbers: 2, 𝑒, and
𝜋.
0
1
2
2
3
𝑒 𝜋
4
5
Building the Real Number Line
Finally we have a real number line, where
a real number is defined as either
rational or irrational.
What about the integers, whole, and natural
numbers?
-5
-4
-3
-2
-1
0
1
2
3
4
5
Building the Real Number Line
Finally we have a real number line, where
a real number is defined as either
rational or irrational.
The set of real numbers is represented by R or ℝ. All
of the previous sets are subsets of the real numbers
(sets within a set).
-5
-4
-3
-2
-1
0
1
2
3
4
5
Exercise 1
Draw a Venn Diagram to represent the set of
real numbers.
Exercise 2
1. How many Real numbers are there?
2. How many Natural numbers are there?
3. How many even numbers are there?
Exercise 3
1. If a number is rational, can it also be
irrational?
2. If a number is irrational, can it also be an
integer?
3. If a number is an integer, must it also be
natural?
4. If a number is an integer, must it also be
rational?
Objective 2
You will be able to approximate square roots
to the nearest tenth without a calculator.
Exact vs. Approximate
Since irrational numbers have never-ending
and never repeating decimal expansions, we
can write them in exact or approximate
form:
Exact:
2
Approximate:
2 ≈ 1.4142 …
You have to round the decimal somewhere
Approximating Square Roots
So how do we approximate a square root
like 7?
Click to make a conjecture:
Approximating Square Roots
So how do we approximate a square root
like 7?
1.
2.
First find the 2
perfect squares
which 7 lies
between
Find the fraction of
“units”
5 “units”
4=2<
7 < 9=3
3 “units”
7 ≈ 2 35 = 2.6
Exercise 4
Without the use of a calculator, approximate
2, 8, and 13 to the nearest tenth.
Exercise 5
Which list shows the numbers in increasing
order?
Exercise 6: SAT
If X represents the sum
of the 10 greatest
negative integers and
Y represents the sum
of the 10 least
positive integers,
which of the following
must be true?
I.
II.
III.
X+Y<0
Y – X = 2Y
X2 = Y2
Exercise 7
Are all numbers real?
Solve the equation x2 + 1 = 0 for x.
When trying to solve the above equation,
you would have to take the square root of
a negative number, which is imaginary.
Exercise 7
Are all numbers real?
Complex Plane
The set of complex
numbers, C or ℂ,
includes both the
imaginary numbers
and the real numbers.
More on this later…
1.1: Apply Properties of Real #s
Objectives:
1. To describe real
numbers and their
subsets
2. To approximate
square roots to the
nearest tenth without
a calculator
Assignment
• Make your own
visual/graphic
representation of real
#'s with multiple
examples in each
set: Real, Irrational,
Rational, Integer,
Whole, Natural
• Working with Irrational
Numbers, 2, 4, 8, 12,
16, 20, 22
“I <3 math, for reals!”