Lecture Notes: Number Systems  

The Natural Numbers, N:  The set 0, 1, 2, 3, …

Observations:

1. If we add two natural numbers together, we get a natural number Þ closure property.  We say, “The natural numbers are closed under addition.”
2. If a, b Î N (i.e. a and b are natural numbers) then a + b = b + a Þ commutative property. We say, “Addition is commutative on the natural numbers.”
3. If a, b, c Î N, then a + (b + c) = (a + b) + c Þ associative property.  We say, “Addition is associative on the natural numbers.”
4. For all natural numbers, a + 0 = 0 + a = a    Þ  0 is the additive identity for the natural numbers, (i.e. adding 0 to anything doesn’t change it).

We can define the operation “subtraction” as follows:  a – b = x   if and only if  a = x + b.

Observations:

1. if a, b Î N,   a – b may not be a natural number.  Example:  5 – 7 = -2,   -2 is not a natural number.   Thus, subtraction is not closed on the natural numbers.
2. if a, b Î N   a – b may not equal b – a.  Example:  5 – 7 = -2  but 7 – 5 = 2.  Thus, subtraction is not commutative on the natural numbers.
3. if a, b, c Î N, (a – b) – c may not equal a – (b – c).  Example:  (4 – 2) –3 = -1, but 4 – (2 – 3) = 5.  Thus subtraction is not associative on the natural number.

If we wanted to subtraction to be closed, what additional numbers would we need to add to the set of natural numbers?  Answer:  …, -3, -2, -1

The set of numbers: …, -3, -2, -1, 0, 1, 2, 3, … are called the Integers, and are denoted Z.

Observations:

1. Z is closed under addition and subtraction.  In fact, we can define subtraction in a new way:  If a, b Î Z, then a – b = a + (-b).  This wouldn’t make sense for N since –b is not an element of N. 
2. If a + b equals the additive identity, we say b is the “additive inverse” of a, and a is the “additive inverse” of b.  For integers, the additive identity is 0.  Thus, if a + b = 0, they are additive inverses of each other.  For integers, this means b = -a.
3. All integers have additive inverses.
4. The only natural number with an additive inverse is 0 (since 0 + 0 = 0, thus 0 is its own additive inverse).
5. Addition is associative and commutative on the integers.

We can define the operation “multiplication” as follows:

            If a is not the additive identity, (i.e. for N and Z, if a ¹ 0), a x b = b + b + …+ b  (add b to itself a times). 

If a is the additive identity, then a x b = a (i.e. for N and Z, if a = 0, then a x b = 0).

If a < 0 then a x b = (-b) + (-b) + (-b) + … + (-b)

Observations:

1. The integers are closed under multiplication
2. Multiplication is commutative on the integers
3. Multiplication is associative on the integers
4. 1 is the multiplicative identity for the integers (i.e. 1 x a = a x 1 = a for all integers a.  Multiplying an integer by 1 doesn’t change the value of the integer).
5. Not every integer has a multiplicative inverse.  We say b is the multiplicative inverse of a if  a x b = b x a = the multiplicative identity.  Example, (1/5) x (5) = 1.  Thus 1/5 is the multiplicative inverse of 5, and vice versa.  1 and –1 are the only integers with multiplicative inverses.
6. We also note that multiplication “distributes” over addition, i.e. a x (b + c) = a x b + b x c.  This is called the Distributive property

Questions to think about:

Is N closed under multiplication?  Is multiplication associative, commutative on N?  What is the multiplicative identity for N?  Do all natural numbers have a multiplicative inverse?  If not, which do? 

Rational Numbers, Q: The set of numbers a/b where a and b are both integers.

Observations:

1. Q is closed under multiplication and addition
2. Addition and multiplication are both commutative and associative on Q.
3. Every rational number has a multiplicative inverse and an additive inverse.
4. 1 is the multiplicative identity, and 0 is the additive identity for the rational numbers.

Summary:

• In this section we covered three important number systems:  the natural numbers (N), the integers (Z), and the rational numbers (Q).

• We learned about properties which describe the actions of an operation on a number system.  Examples: the closure property, the commutative property, and the associative property.

• We also learned about the distributive property, which relates the interaction of two operations on a given number system.

• We learned about additive and multiplicative identities and inverses.

Examples:

Are the odd numbers closed under addition?  Þ No, the sum of two odd numbers is even.  Example: 3 + 3 = 6 which is not an odd number

Are the odd numbers closed under multiplication? Þ Yes.  Example: 3 x 3 = 9 which is an odd number.  The product of any two odd numbers is odd.

In the English language, the collection of words into phrases is often not associative.  

Example: High School Student:  (High School) Student is a student of a High School, but High (School Student) is a student who is high.

The following phrases are all non-associative.  Can you tell why?

1. Man Eating Tiger

2. Brown Smoking Suit

3. Quit Smoking Cold Turkey

4. Three Year Old Teacher

*	1	-1	i	-i
1	1	-1	i	-i
-1	-1	1	-i	i
i	i	-i	-1	1
-i	-i	i	1	-1

Given the following table, answer the following questions:

1.      Is the set {1, -1, i, -i} closed under *?    

Yes, all entries of the table are in {1, -1, i, -i}.  Thus, if you take * of any two elements of the set {1, -1, i, -i } you’ll get an element of the set.

2.      Is * associative?  

Try it out (1 * i) * -i = i * -i = 1   and 1 * (i * -i) = 1 * 1 = 1.  If you try this with all possible combinations, you’ll find that * is associative on the set {1, -1, i, -i }

3.      Is * commutative? 

Yes.  An easy way to check this, is to see that the ij^th entry of the table equals the ji^th entry.  If this is always true, then the operation is commutative.

4.      Is there a * identity?  

Yes.  We see that 1 * any element of the set equals that element of the set.  Thus, 1 is the * identity.

5.      Do all elements have * inverses?  Yes.  -1 * -1 = 1, 1 * 1 = 1, and –i * i = i * -i = 1.

 

·	a	b	c
a	b	c	a
b	c	a	b
c	a	b	c

Given the following table, answer the following questions:

6.      Is the set {a, b, c} closed under ·?    

Yes, all entries of the table are in {a, b, c}.  Thus, if you take · of any two elements of the set {a, b, c} you’ll get an element of the set.

7.      Is · associative?  

Try it out (a · b) · c = c · c = c   and a · (b · c) = a · b = c.  If you try this with all possible combinations, you’ll find that · is associative on the set {a, b, c}

8.      Is · commutative? 

Yes.  From the table we can see that a · b = b · a,  a · c = c · a, and b · c = c · b.  An easy way to check this, is to see that the ij^th entry of the table equals the ji^th entry.  If this is always true, then the operation is commutative.

9.      Is there a · identity?  

Yes.  We see that c · any element of the set equals that element of the set.  Thus, c is the · identity.

10.  Do all elements have · inverses?  

Yes.  b · a = a · b = c the · identity.  Thus, a and b are · inverses.  c · c = c.  Thus c is its own · inverse.

   

Q	a	b	c
a	c	a	b
b	a	b	e
c	b	c	c

Given the following table, answer the following questions:

11.  Is the set {a, b, c} closed under Q?    

No.  b Q c = e which is not an element of the set {a, b, c}

12.  Is Q associative?  

No. (a Q c) Q c = b Q c = e  but aQ(cQc) = a Q c = b.

13.  Is Q commutative? 

No. c Q b = c, but b Q c = e

14.  Is there a Q identity?  

No. 

15.  Do all elements have Q inverses?  

Question doesn’t make sense without a Q identity.

 

Let the operation ¯ mean “chose the smaller of the two” and ® mean “chose the one on the right”.  Thus, a®b = b, and a¯b = b if b< a and a¯b = a if a< b.  Does ¯ distribute over ® for the natural numbers?

Let’s look at some examples:

Does 5¯(4®7) = (5¯4)®(5¯7)

5¯(4®7) = 5¯7 = 5

(5¯4)®(5¯7) = 4®5 = 5

 

5¯(7®4) = 5¯4 = 4

(5¯7)®(5¯4) = 5®4 = 4

 

6¯(3®4) = 6¯4 = 4

(6¯3)®(6¯4) = 3®4 = 4

 

2¯(7®4) = 2¯4 = 2

(2¯7)®(2¯4) = 2®2 = 2

  It certainly appears that ¯ distributes over ®

 

Vanishing Leprechaun Problem:  The following two pictures are formed by putting together a three-piece puzzle in two different ways.  How many leprechauns are in each picture?  What does this suggest about the operation of putting this puzzle together?

 

The first picture has 14 leprechauns, while the second has 15.  This implies that the position of the top two pieces is not commutative.  Can you figure out where the missing leprechaun went?

In-class Quiz:

1. For each of the following numbers, list all number systems to which it belongs (i.e. is it a natural number, an integer, a rational, or none of these).

a.  0

b. 5

c. –213

d. 5/6

e. 1

 2. Is subtraction closed on the natural numbers?  Give an example.

3. Is subtraction closed on the integers?  Give an example.

 4. What is the additive identity for the rational numbers?

 5. Which integers have additive inverses?

 6. Which integers have multiplicative inverses?

 7. For each of the following, state if it is an example of the closure, the associative, the distributive, or the commutative property.

            a. 5 + 7 = 7 + 5

            b. 6 + (9 + 5) = (6 + 9) + 5

            c. 6(7 + 3) = 6*7 + 6*3

            d. odd number x odd number = odd number