|
Number Theory Lesson
Primes
vs. Composites
Classes of Use:
This lesson can be taught on any level
of education. In the secondary classroom (grades 9-12), however, the
lesson can be taught more in dept. Students at this level already have a
basic understanding of infinity. But this lesson can show them how it
was discovered, and that is has more than just one level. Where as
students in a lower level would be learning the basics of sets. As well
as prime and composite numbers. So you see, this lesson can be aimed at
any age level.
It is just
the way you teach it and discuss it that changes the lesson.
History of Number Theory:
It is rather difficult to track the
ideas of number theory. Every great mathematician was using number
theory in mostly everything they did. The theory of numbers is the study
how they behave and why. Fermat was using number theory to explain his
last theorem.
a^n + b^n = c^n only holds for n less then or
equal to 2
Pythagorus used it when explaining triangle
theorems. Every study on numbers could be considered number theory.
Because that is what it is, a study of numbers.
This lesson deals with one of the most basic
ideas of numbers. In order to understand numbers, you must define them.
This lesson introduces the definition of prime and composite numbers.
Defining numbers as prime or composite was first documented in history
as early as 300 B.C. A
mathematician by the name of Euclid documented his definitions in his
book of elements. Books VII, VIII, and XV of Euclid’s Elements
go into great detail about the properties of prime and composite
numbers.
Proposition VII - 31: Any composite number is measured by some prime
numbers, and
Proposition VII - 32: Any number either is prime or is measured by
some prime number.
In the next propositions, Euclid goes in to great
detail about the behavior of primes and composites. This lesson only
introduces primes and composites, we focus more on the origin of the
concepts.
The second part of the lesson introduces the
concept of sets and calculating their cardinality. Because we are
looking at the sets of all positive integers, prime numbers, and
composite numbers, we are dealing with infinite sets. The concept of
infinity using sets was studied by great mathematicians such as Galileo
and Cantar.
Galileo studied infinite sets. He compared
sets to one another. He discovered the theory that if the sets can be
placed in a one to one correspondence, then the sets have the same
number of elements. This led to the work of George Cantar in the 1800s.
He used Galileo’s theory to derive relationships between sets.
- if card A = 4, and card B = 4, then card (A u B)
= 8
He then looked at the relationship between the set
of all integers, the set of all evens, and the set of all odds. He
discovered that they are all equal. The symbol he used for their
cardinality was ‘aleph nought’.
In this lesson, the students are learning about
concepts that were, at one time, thought to be unsolvable. Yet, these
are concepts that are critical for I higher level of mathematical
education.
Teacher Instructions:
It is
important in this lesson that you read through the worksheet with the
students.
- Make
sure that they have an understanding of the definitions of prime and
composite numbers.
- It is
important that you write the definitions on the board, or say them aloud
for the students to write down the correct definitions.
- When
determining if a number is prime of composite, help them along with
their hints. Write some numbers on the board as examples of all the
steps you take when determining prime or composite.
- When
going over the definitions of sets, read them aloud as a class.
-The
definitions are important in the last section.
-
Explain the concept of one-to-one relationship.
- Discuss
the concept of infinity with the class. Not all students grasp the
concept the same. Talk with each one of them about their views of this
new concept of infinity.
Primes vs. Composites
Who Will Win?
Definitions:
- A prime number
is defined as any integer greater than 1 that has no factors besides 1
and itself.
- A composite
number is any integer that is non prime.
Define a
composite number.
- What are the factors of 1?
- Because 1
only has a factor of one, which is itself, does that make it a prime?
Why or why not?
- Are there
any even primes? If so how many are their and why?
- All ______ numbers greater than
are
composite.
Determining Prime or
Composite:
We know
the difference between even and odd numbers. It is easy to determine if
a large number is even. If the last digit is even, we know it’s even.
This is one easy to tell if a large number is composite or prime. But
what about large numbers that’s last digit is odd?
- What are the properties of multiples of 5?
(Hint: What are their last digits?)
- Any number who’s last digit is or
is a multiple of 5 , and therefore
because .
- What are the properties of multiples of 9?
(Hint: Remember your times tables.)
That was a quick reminder for determining
multiples. Yes this works on determining if a number is a multiple of 9
to determine composite or prime, but it can be used for a broader scale.
-
Because this works for multiples of 9, can it work for factors of 9?
- If you sum the
digits of a multiple of their sum will be a multiple
of .
There are
three quick tools for determining if an integer is prime or composite.
What if all those fail. Well that’s it, no more quick tools. After that
it is guess and check. But there is a quick tool to make that process
not so rigorous.
-
By taking the square root of your number, you can lesson this process.
If the square root equals an integer, your number is
. If not, then you only need to check
less than the square root by
.
If you ever receive an integer as an answer, then your number is
.
If not, then your number is .
Now
to the question: who will win? Primes or Composites?
-Are
there more primes than composites, or more composites then primes?
We start by defining a
set of numbers. A set is defined by a group of one or more values.
We define
set A as having the variables a, b, c, and d.
A = {a, b,
c, d}
We call the
cardinality of a set the number of values within the set.
- The cardinality of set
A = .
If two sets
have the same cardinality, then those sets are equal. Their members can
be placed in a 1 - 1 (one to one) correspondence. That means that
each member of one set has a corresponding member in the other set.
We define
set B as having members: x, y, z, q
B = {
, , , }
- The cardinality of B
= .
- card B = (short
hand way of writing.)
- So card =
card
- Now to determine the
cardinality of the set of all primes, and all composites.
We first
define the set of all positive integers
Z = {1, 2,
3, 4, 5, 6, .... }
card Z =
- Create set P
consisting of all the primes you know.
- Create set C
consisting of all the composites you know, or until you are tired of
writing.
- Can each member of set
P be mapped to a member of set C?
- Do they appear to
have the same cardinality?
- Now lets look back at
the set of all positive integers. Remember its cardinality is infinity.
When you compare set Z to set P, what do you notice?
- What happens when you
compare set Z to set C?
- What does this mean
about the cardinality of set P and set C?
SO EVERYONE WINS!!!!
(or losses, depending on your feelings about Infinity)
Primes vs. Composites
Who Will Win?
Definitions:
- A
prime number is defined as any integer greater than 1 that has no
factors besides 1 and itself.
- A
composite number is any integer that is non prime.
Define a composite number.
Any integer greater than 1,
that has factors other than 1 and itself. Or, any integer greater than 1
that can be evenly divided by more than just one and itself.
- What are the factors of 1?
only 1
- Because 1 only has a
factor of one, which is itself, does that make it a prime? Why or why
not?
No. 1 is not
a prime number because a prime is defined by any integer greater than 1.
- Are there any even primes? If so how many are their and why?
Yes. 2 is the only even prime.
It is larger than 1 and only has 1 and itself as factors, therefore,
prime. Every even number greater than 2 can be divided by 2.
-
All even
numbers greater
than 2
are
composite.
Determining Prime or Composite:
We know the difference between
even and odd numbers. It is easy to determine if a large number is even.
If the last digit is even, we know it’s even. This is one easy to tell
if a large number is composite or prime. But what about large numbers
that’s last digit is odd?
-
What are the properties of multiples of 5?
(Hint: What are their last digits?)
A multiple of 5 will
always end in 0 or 5.
- Any number who’s
last digit is 0
or
5
is a multiple of 5 , and
therefore composite
because 5
is a factor .
- What are the
properties of multiples of 9?
(Hint: Remember your times tables.)
The digits in a multiple
of 9 add up to a multiple of 9. An easy tip from the 9 times table.
That was a quick reminder for determining
multiples. Yes this works on determining if a number is a multiple of 9
to determine composite or prime, but it can be used for a broader scale.
-
Because this works for multiples of 9, can it work for factors of 9?
Yes. This process works
for 3.
- If you sum the
digits of a multiple of 3
their
sum will be a multiple of 3
.
There are
three quick tools for determining if an integer is prime or composite.
What if all those fail. Well that’s it, no more quick tools. After that
it is guess and check. But there is a quick tool to make that process
not so rigorous.
-
By taking the square root of your number, you can lesson this process.
If the square root equals an integer, your number is composite
. If not, then you only need to
check all the prime numbers
less than the square root by dividing
the number by all of them.
If you ever receive an integer as an answer, then your number is composite
. If not, then your number is prime
.
Now
to the question: who will win? Primes or Composites?
-Are
there more primes than composites, or more composites then primes?
We
start by defining a set of numbers. A set is defined by a group of one
or more values.
We define
set A as having the variables a, b, c, and d.
A = {a, b,
c, d}
We call the
cardinality of a set the number of values within the set.
- The cardinality of set
A = 4
.
If two sets
have the same cardinality, then those sets are equal. Their members can
be placed in a 1 - 1 (one to one) correspondence. That means that
each member of one set has a corresponding member in the other set.
We define
set B as having members: x, y, z, q
B = {x,
y, z, q}
- The cardinality of B
= 4
.
- card B = 4
(short
hand way of writing.)
- So card A
=
card B
- Now to determine the
cardinality of the set of all primes, and all composites.
We first
define the set of all positive integers
Z = {1, 2,
3, 4, 5, 6, .... }
card Z =
infinity
- Create set P
consisting of all the primes you know.
P = {2, 3, 4, 5, 7, 11, 13, 17, 19, 23, 29, 31,
37, 41, 43, 47, ....}
- Create set C
consisting of all the composites you know, or until you are tired of
writing.
C = {6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21,
22, 24, 25, ....}
- Can each member of
set P be mapped to a member of set C?
No
- Do they appear to have
the same cardinality?
No
- Now lets look back at
the set of all positive integers. Remember its cardinality is infinity.
When you compare set Z to set P, what do you notice?
Every member of set P can
be mapped to a member in set Z.
- What happens when you
compare set Z to set C?
Every member of set C can
be mapped to a member in set Z.
- What does this mean
about the cardinality of set P and set C?
card P = card Z = infinity
card C = card Z = infinity
SO
EVERYONE WINS!!!!
(or losses, depending on
your feelings about Infinity)
Documentation
-http://www.lessonplanspage.com/MathPrimeVsCompositeNumbers7HS.htm
- A History of Mathematics, Katz, Victor J. 2004 (pages 57-58,
447-448) |
