A Review of Our
Number Systems and
an Introduction to e
This PowerPoint contains notes, examples and practice problems for
you to complete. It is important that you slowly work through this
PowerPoint and not just move through it quickly. This is your chance
to practice this material, so you must work through it and learn it!
Remember that, as you work through this PowerPoint, do not go onto
the next part until you are certain that you understand what is being
presented.
Also, do not look at any answers until you have tried the problem
yourself.
If you have any questions, ask your teacher!
Part I: Reviewing Our Different Number
Systems
Can you write down two examples of the following types of numbers:
Real Numbers:
Irrational Numbers:
Rational Numbers:
Imaginary Numbers:
Integers:
Complex Numbers:
When you believe you have two examples for each
type, please check your answers.
Now, discuss and answer the following questions in your notebook:
1. How do we define a complex number or what is the
definition of a complex number?
2. How do we define a rational number or what is the definition of a
rational number?
3. How do we define an irrational number or what is the
definition of a irrational number?
When you are ready, look at the next slide for possible
answers. If you are unsure, ask!
Now, discuss and answer the following questions in your notebook:
1. How do we define a complex number or what is the
definition of a complex number?
A complex number is written in the form
number and bi is an imaginary number.
a  bi
where a is a real
2. How do we define a rational number or what is the definition of a
rational number?
A rational number is a real number which is written as a fraction
a
or in the form
where a and b are both integers.
b
3. How do we define an irrational number or what is the
definition of an irrational number?
An irrational number is a real number which cannot be written as
a
a rational number or it cannot be written in the form
where a
b
and b are both integers.
Now, write in the following types of numbers into the diagram below:
1. Real Numbers
2. Integers
3. Rational Numbers
4. Irrational Numbers
5. Imaginary Numbers
6. Complex Numbers
The entire grey
rectangle and
everything in it.
Both the grey and
blue rectangles
combined
Just the grey
part of the
rectangle.
Just the blue rectangle
Both the white
and blue parts.
Just the blue part.
When you are finished, the next slide has the answer.
Now, write in the following types of numbers into the diagram below:
1. Real Numbers
2. Integers
3. Rational Numbers
4. Irrational Numbers
5. Imaginary Numbers
6. Complex Numbers
The entire grey
rectangle and
everything in it.
Just the grey
part of the
rectangle.
Real Numbers
Irrational Numbers
Both the white
and blue parts.
Both the grey and
blue rectangles
combined
Complex Numbers
Just the blue rectangle
Imaginary Numbers
Rational Numbers
Just the blue part.
Integers
Part II: An Introduction to e
Using your calculator, complete the following table in your notebook for the
following exponential function:
f ( x)  (1  1x ) x
Give your answers to 5 decimal places!
x
f ( x)  (1  1x ) x
Answer
10
f (10)  (1  101 )10
2.59374
100
2.70481
1000
2.71692
10,000
2.71814
100,000
2.71827
1,000,000
2.71828
10,000,000
2.71828
Try the first row then
look at my answer to
see if you are correct.
If you are correct,
keep working and
check your
answers when you
are finished. If you
are incorrect, ask
your teacher.
Did you notice anything about your answer as you made x larger and
larger? Discuss this with your partner.
So, If we evaluate this function with larger and larger values of n, this
function eventually approaches a number.
This number that the function approaches is an irrational number.
Since it is an irrational number, it cannot be written as a fraction and is
found in a lot of different mathematical equations. Because it is used
so often, it is given a special letter, the letter e.
This is an important number to know and is used frequently in
higher level mathematics classes.
Here is an approximate value of e = 2.7182818284590452354
The number e is also in your calculator. It is found in two places. They are:
Here
Here
Let’s find the following values using our calculator. Remember to use
parentheses if they are needed!
For example, the value:
e
2
3
will be evaluated in our calculator as shown below:
So:
2
3
e  1.95
Notice the parentheses that are
used when typing the value into
your calculator. Without the
parentheses, your answer will be
wrong!
Find the following values using our calculator. Remember to use
parentheses if they are needed!
Find:
Answers:
Find:
Answers:
3
 20.1
 12
 0.607
e
e
3
4
e
e
2
e
 2.12
e 6
 0.00248
 23.1
ee
 15.2
 4.11
e
4 3
 0.268
Part III: An Example of How We Will Use
the Number e
Remember the basic formula for exponential Growth and Decay
r nt
n
A(t )  A(1  )
Let’s invest $1000 at a rate of 5% for 10 years. Use this
information to complete the following table:
Number of Times Per year
Amount after 10 years
Annually
$1628.89
Quarterly
$1643.62
Monthly
$1647.01
Daily (360 days per year)
$1648.66
Hourly (360 days per year)
$1648.72
Every Minute (360 days)
$1648.72
The values in this table are all values for discrete exponential growth.
These are problems where the growth or decay is evaluated at a
particular time.
As we increased the number of times we evaluated the exponent, the
values began to approach a number. This number is related to the
number e. When we evaluate exponential growth continuously, or,
instead of every minute, or every second, but all of the time, we use a
formula with e. Continuous exponential growth can be found using the
formula:
A(t )  A  e
rt
So, if a problem specifically states that it is found using
continuous growth or decay, then use this formula. If it does not
state continuous growth or decay, then use the discrete formulas
we have used earlier.
Here is an example, that we can solve together.
The number of bacteria cells in a culture grows continuously at a rate of
2.31% per day. If 25 cells are present now, approximately how many are
present after 12 days?
Notice that this problem states that it is growing continuously. If it did not
state that the growth is continuous, then we would use our basic
exponential growth or decay formula. So, find the following values:
A  25
So, we have
the formula:
A(t )  A  e
rt

A  25  e0.023112
r  0.0231
A  32.98575577
t  12
Answer: 33.0 cells
using our
calculator
Here is an example, that we can solve together.
The population of a country grows continuously at a rate of 2.12% per
year. If the country currently has 5.7 million people, how many people
will there be after 5 years?
Notice that this problem states that it is growing continuously. If it did not
state that the growth is continuous, then we would use our basic
exponential growth or decay formula. So, find the following values:
A  5.7
So, we have
the formula:
A(t )  A  e
rt
A  5.7  e0.02125
r  0.0212
A  6.337384696
t 5
Answer: 6.34 million
people
using our
calculator
Here is an example, that we can solve together.
If I invest $1000 in an account that pays 5% annual interest, how much
do I have in the account after 3 years?
Notice that this problem does NOT state that it is growing continuously.
So, we will use our basic exponential growth or decay formula with the
following values:
A  1000
r
t
 0.05
3
So, we have
the formula:
A(t )  A  (1  r )
3
A  1000(1  0.05)
t
A  1157.625
Answer: $1157.63
using our
calculator
The number of trees in a forest grows continuously at a rate
1.2% per year. If there are 2,350 trees in the forest now, how
many trees were there in the forest 12 years ago?
Answer: 2030 trees
The price of a computer increases at a rate of 7.3% annually. If
the price of the computer is $1600 now how much will it cost in
3 years and 4 months?
Answer: $2023.57
Notice that this question does NOT use the word continuous in it.
It is NOT continuous growth but discrete growth and so we use
the basic formula for discrete growth, or: A(t )  A(1  nr ) nt
where n = 1, or once a year.
The number of bacteria in a culture is decaying continuously at
a rate of 2.67% per day. If the culture currently has 350 cells,
how many cells were there in the culture 13 days ago.
Answer: 495 cells
A radioactive isotope is decaying continuously at a rate of
.0051% per year. If 2 kg is present now, how much is present
after 150 years?
Answer: 1.98 kg
The number of people who attend a university grows
continuously at a rate of 1.2% each year. If 5600 people
currently attend the university now, how many students
attended 5 years ago?
Answer: 5270 people
The price of a car decreases at a rate of 11% annually. If the
price of the car is $4500 now how much will it cost in 4 years?
Answer: $2823.40
Notice that this question does NOT use the word continuous in it.
It is NOT continuous growth but discrete growth and so we use
the basic formula for discrete growth.