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:

Rational Numbers:

Integers:

Irrational Numbers:

Imaginary Numbers:

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?

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 an irrational number?

A complex number is written in the form where a is a real number and bi is an imaginary number.

a bi

A rational number is a real number which is written as a fraction

or in the form where a and b are both integers.a

b

An irrational number is a real number which cannot be written as

a rational number or it cannot be written in the form where a

and b are both integers.

a

b

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

Both the grey and blue rectangles combined

The entire grey rectangle and everything in it.

Just the grey part of the rectangle.

When you are finished, the next slide has the answer.

Both the white and blue parts.

Just the blue part.

Just the blue rectangle

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

Both the grey and blue rectangles combined

The entire grey rectangle and everything in it.

Just the grey part of the rectangle.

Both the white and blue parts.

Just the blue part.

Real Numbers Complex Numbers

Imaginary Numbers

Integers

Rational Numbers

Irrational Numbers Just the blue rectangle

Part II: An Introduction to e

Using your calculator, complete the following table in your notebook for the

following exponential function: 1( ) (1 )xxf x Give your answers to 5 decimal places!

x Answer

10

100

1000

10,000

100,000

1,000,000

10,000,000

1( ) (1 )xxf x

2.59374

2.70481

2.71692

2.71814

2.71827

2.71828

10110(10) (1 )f

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.

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

Did you notice anything about your answer as you made x larger and larger? Discuss this with your partner.

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:

23e

will be evaluated in our calculator as shown below:

So:23 1.95e

Notice the parentheses that are used when typing the value into your calculator. Without the parentheses, your answer will be wrong!

2e

Find: Answers:

4.11

Find the following values using our calculator. Remember to use parentheses if they are needed!

3e

34e

e

20.1

2.12

23.1

4 3e

Find: Answers:

0.268

12e

6e

ee

0.607

0.00248

15.2

Part III: An Example of How We Will Use the Number e

Remember the basic formula for exponential Growth and Decay

( ) (1 )n trnA t A

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

Quarterly

Monthly

Daily (360 days per year)

Hourly (360 days per year)

Every Minute (360 days)

$1628.89

$1643.62

$1647.01

$1648.66

$1648.72

$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:

( ) rtA t A e

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?

Answer: 33.0 cells

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

r

t

25

0.0231

12

So, we have the formula:

( ) rtA t A e 0.02311225A e

32.98575577A

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?

Answer: 6.34 million people

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

r

t

5.7

0.0212

5

So, we have the formula:

( ) rtA t A e 0.0212 55.7A e

6.337384696A

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?

Answer: $1157.63

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

r

t

1000

0.05

3

So, we have the formula:

( ) (1 )tA t A r 31000(1 0.05)A

1157.625A

using our calculator

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:

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

( ) (1 )n trnA t A

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 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.

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