Set Notation and Number Sets
As you start getting further along in algebra you will start to see a number of new symbols and terms
that might make you feel like learning math is more like learning a new language. This is actually very
true in a number of ways. Writing equations is very much like writing a sentence in any language. There
are rules about the order you write things, where you put your punctuation, and the symbols you use to
“spell” words and phrases.
The transition into algebra is where math stops becoming about arithmetic and calculations and more
about patterns and relationships. Hence the need for new symbols, new ways to write equations, and a
new way to think about numbers.
We will begin with the first and one of the most fundamental concepts in algebra: the set.
A set is a collection of objects. They could be numbers, shapes, equations, matrices, coordinates, or any
number of things. A set could have 1 object or 3 million objects or an infinite number of objects. There is
no limit to size.
A good way to think about a set is to think of a bag. Picture a backpack, a purse, or Santa Claus’ gift
sack….anything that can hold items. This is a set. We denote a set by using a pair of curled brackets { }.
So if you saw A={1, 2, 3} this would be a set containing three items: 1, 2, and 3. This set has the name
“A”
The things that you put inside a set are called elements. An element is a member of a given set. We use
the symbol  to mean “is an element of”. The symbol  means “is not an element of”.
Using the previous example, if A  {1, 2,3} then 1  A and 8  A (1 is in the set, 8 is not).
Here’s another example without numbers: Consider the set of all the different kinds of fruits in the
world. We will call it set F. (Imagine a giant bag filled with a bunch of fruit…this is our set. There could
even be a giant “F” printed on the bag if that helps you visualize it.)
What would be an element of this set?
An apple would be an element of this set. We could write this as: apple  F
A banana would be an element of this set. We could write this as: banana  F
A t-bone steak would not be an element of this set. We could write this as: tbone  F
You get the idea. We actually use sets all the time. Any time you categorize something or group items
together or talk about an item belonging to a certain collection you are talking about sets and elements.
That being said, let’s look at some common sets that you will come across.
There are a number of sets that you will need to remember as they will become very important as you
move through algebra. Let’s start with the most basic set: an empty one.
Imagine a bag with nothing in it. We call this an empty set (also sometimes called a null set) and it has a
special symbol:  . (The symbol represents the name of the set, just like we had sets A and F above)
Since the set has nothing in it, you would write it as a pair of empty brackets:   { } .
Take a look at the following statements. Can you tell which ones are true and which are false?
3 
7 
0 
Well, what numbers are in the empty set { } ? Is there a 3 in there? Is there a -7 in there? Is 0 in there?
No, no, and no. There is nothing in the empty set so any number you think of will not be in there.
ALL THREE OF THESE ARE FALSE STATEMENTS. The reason to mention this is that many people get
mixed up thinking that zero is an element of the empty set. This is not true.
The empty set has zero elements. It does not have zero as an element. Can you see the difference?
Alright, let’s move on to some more interesting sets. There are five fundamental sets that we are going
to examine: Natural Numbers, Integers, Rational Numbers, Real Numbers, and Complex Numbers.
Each of them has a special name:
, ,
,
, and
respectively and each has some unique
properties.
We begin with the first set: a bag filled with natural numbers called
.
Imagine for a moment that you step into a time machine and are able to go back and watch the whole
history and development of math and numbers. What do you think would be the first thing you saw?
Square roots? Calculus? Graphs? Pi?
Probably not. The first numbers were the simplest. The positive whole numbers that were used to count
items. 1, 2, 3, 4, and so on. These most fundamental of numbers we call the natural numbers.
Definition: A natural number is any of the positive whole numbers 1, 2, 3, and so on.
Let’s take all these numbers and put them in a bag (an infinitely large bag since this set is infinite) and
label it .
 {1, 2,3...}
This is our first set. You might be wondering why zero is not included here since it could be considered a
“whole number” or a “counting number”. This is an excellent question and something you should ask
your teacher about.
There is no real agreement among mathematicians whether 0 should or should not be included in the
natural numbers. The general philosophy is to use it when it’s convenient and to leave it out if it creates
problems. Ask your teacher to see how he or she wants you to define natural numbers.
So now that we have developed our counting numbers (and 0), what comes next? What about the other
side of the number line? Let’s throw the negative numbers into our set.
The nice thing about all these basic sets is that they continue to build on each other. So let’s take what
we already have {1, 2, 3,…} and add in zero and all the negative counting numbers to get the set
{…,-3, -2, -1, 0, 1, 2, 3,…}. You might recognize this set from earlier. We call these values integers.
Definition: An integer is any of the counting numbers 0, 1, 2, 3, .... and their opposites -1, -2, -3, ...
So if we put all this together into a new bag we now have the set of all integers, called
the letter Z was chosen to represent the German word Zahlen meaning “numbers”.
. It is said that
 {..., 3, 2, 1, 0, 1, 2, 3, ...}
If you haven’t noticed already, the three little dots . . . means that the pattern continues on forever.
This set, like the natural numbers, is an infinitely large set.
This is a good point to pause and mention a new symbol and something called a “subset”. Notice how all
the values of
are inside of ? Whenever one set is completely inside of another set, we call this a
subset.
We can say that
In this case

is a subset of
. We have a symbol for this ”  ” meaning “is a subset of”.
since every element of
is also an element of
.
There is a lot more to this, but we will save that for another tutorial. It’s just important now to notice
that one set is contained in the other.
So we have our bag of natural numbers labeled
and our bag of integers labeled . What else is
missing on the number line? Well how about all the values in between the whole numbers? What kinds
of numbers are between 1 and 2?
Fractions! This is our next set. Let’s take all the positive counting numbers, all the negative counting
numbers, zero, and all the fractions and put them into one bag.
We call this collection the set of rational numbers.
p
Definition: An rational number is any number that can be written as a fraction
where
q
p, q  and q  0
Did that make sense? Remember what p, q 
That says that p and q are elements of
means?
. Or in other words, p and q are integers.
So a rational number is basically any fraction that you can come up with involving two integers.
4
6
365
2
9
1
3827
545
34
228
0
5
These are all examples of rational numbers. Notice how they include positives, negatives, whole
numbers and zero?
Notice also that q  0 . This is because you can’t have a fraction with zero on the bottom. Remember
that dividing by zero is undefined.
So what do we call this new set of rational numbers? We give it the name
. It is said that the letter
was chosen to represent the word quotient (the dividing of two numbers).
p
  | p, q 
q

and q  0 

Did you catch all that? Time for a new symbol, a vertical line “|” meaning “such that”.
Here’s how you read the previous statement:
is the set of all fractions
p
such that p and q are integers and q cannot equal zero.
q
If you are feeling confused about notation right now, just focus on the big picture of the sets and what is
in them. We have 3 so far: The natural numbers , the integers , and the rational numbers .
(Notice also that
is a subset of
and that
is a subset of
. We can write this as


.)
OK, on to the next set. We started with the most basic of numbers: 1, 2, 3, … then added in all the
negative numbers and zero. We then started filling in all the gaps between these numbers by adding in
all the possible rational fractions that could ever be created.
So what’s left? Doesn’t that just about cover it? What other numbers are there?
Well, believe it or not, there are some numbers that cannot be created by dividing two integers.
There are certain characteristics that all rational numbers have in common. If you take a rational
fraction and write it out in its decimal form it will do one of two things:
1. It will terminate. In other words, it has a finite number of decimal values. Examples: 0.5, 7.125,
-9.236892, etc.
2. It will repeat a sequence of numbers forever. Examples: 0.333, 4.828282,  2.18666, etc.
This is true for every single rational number! So what happens if you have a decimal that is nonterminating and non-repeating. In other words, it goes on forever without repeating itself over and over.
For example: 1.121122111222111122221111122222…. or -5.010010001000010000010000001…
See how these keep expanding without repetition? These numbers cannot be written as fractions! So
this means we need a new name for these kinds of numbers. If they are not rational, then what should
we call them?
How about irrational! And this is in fact what we call them, irrational numbers.
Definition: An irrational number is any number that cannot be written as the quotient of two integers.
Irrational numbers are non-repeating and non-terminating.
Some more examples of irrational numbers are  , e ,
2 , 5 , and 92837.262
In fact, if you take the square root of any non-perfect square, you will get an irrational number. This isn’t
the only way to get an irrational number though. There are infinitely many irrational numbers and an
infinite number of ways to get them.
If you think of all the possible ways to create an endless string of decimal numbers, you can imagine
how many different irrational numbers there are.
So let’s go back to our basic sets. We have
,
, and
. Now we have filled in all the gaps in
to
create a new set. This set has all the natural numbers, all the integers, all the rational numbers, and all
the irrational numbers. This creates all the numbers on the number line and we can categorize all these
numbers under a single name: real numbers.
Definition: The set of real numbers
irrational numbers.
is the set of all rational numbers combined with the set of all
So basically, any number you can think of (positive, negative, zero, fraction, decimal) is a real number.
As you saw in the definition, we give the set of real numbers the symbol
.
This is going to be one of the sets you talk about most since it contains all the numbers that we are used
to seeing. Now when you hear the phrase “real numbers” or “the real number line” you have a better
idea of what’s going on. Real numbers are all the fractions, decimals, positives, and negatives that we
use in the majority of mathematics.
So that’s it right? We have covered every number on the number line so what else could possibly be out
there?
Well hold on tight because there is one more left. But first, it’s story time.
Remember that time machine we mentioned earlier? Jump back in and let’s look at how these number
systems have developed over time.
Natural numbers obviously came first from the earliest carvings on cave walls and the first written
communication of numbers. Many cultures lived their entire existence knowing only the natural
numbers. Ancient civilizations in the Americas, Middle East and Asia eventually came across an
understanding and way to communicate the concept of zero, but nobody was really able to grasp the
idea of a negative number until the Chinese around 100BC.
In fact, many ancient cultures, including the Egyptians, had intricate systems for working with fractions
way before they understood the mathematics behind negatives. Would you believe that modern Europe
was still arguing about the validity of negative numbers up until the 17th century?
It was common throughout history to question the creation and usage of new number systems since a
lot of times these numbers did not apply to the kind of math that was being done at the time. For
example, in Europe they would often just ignore negative results because they believed negatives had
no real life relevance to what mathematicians were attempting to do.
When the ancient Greeks started creating definitions for irrational numbers, nobody believed them! In
fact, legend has it that the man who first discovered irrational numbers was killed because his discovery
was considered to be religious heresy! They believed that all numbers could be written as beautiful,
perfect ratios. Besides, fractions could take care of all their measurements and calculating needs…why
should they have to believe in these strange irrational numbers that may or may not actually exist?
Well, here is one more case of a new number system that was hard for people to grasp. The real
numbers meet all of our day to day calculating needs. What else could we possibly need?
It’s time to open your mind and accept the existence of something else, even if it seems strange and is
not something that you will need to pay your bills or balance your checkbook.
This new number system is the set of complex numbers.
You have, hopefully, by this point been introduced to the idea of a square root. For example, consider
the square root of 16. Square roots are a way to “unsquare” a number. If you take 4 2 you get 16. So if I
give you 16 can you tell me what number was squared to get 16?
Well, there are two options that could have given us 16. 42  4  4  16 or (4)2  (4)  (4)  16 .
What if we started with -16 instead? What number could you multiply by itself to get -16?
We just saw that 4 2  16 and (4)2  16 but what about -16?
Uh oh…we’ve hit a point where we need to come up with a new type of number. If we try to find the
value of 16 there is no real number that can help us here. We don’t want to just say that 16
doesn’t exist since there are obviously equations that require us to do these kind of calculations. For
example, how would you solve x 2  16 ?
To deal with this problem, mathematicians developed a new definition that could help them push
through this wall. They created what are called complex numbers.
It all starts by defining a new number called i . This number is equal to
1 .
Definition: i  1
Using this new value, we can find the answer to problems such as
writing
16 . This would be the same as
16  1 or 4  1 or 4i .
So
16  4i .
Using this value of i we can create whole new number lines and whole new sets of equations using
complex numbers.
Definition: A complex number is any number in the form a  bi where a, b 
and i  1 .
Here are some examples of complex numbers:
35i
7  19i
4 2
 i
5 3
6  4i
2 i
21
72.3  2.49i
8i
7
0
So our final set contains the following things:
-
All natural numbers (1, 2, 3, …)
-
All negative counting numbers and 0
-
All rational numbers
-
All irrational numbers
-
All complex numbers
We call this set
for complex (which is a very fitting name since this concept can be pretty complex)
 { a  bi | a, b 
and i  1 }
So let’s recap what we’ve learned about sets.

Sets can be any size and can contain all kinds of different objects.

Sets are denoted by using curled brackets { }.

The things inside sets are called elements. We use the symbols  and  to show that an item
3
5
is or is not in a particular set. (i.e.  
)

When one set is completely contained within another we call this a subset. We use the symbol
 to show this. (i.e. natural numbers are a subset of real numbers or
 )

The empty set (or null set) is a set with no elements.   { }

Natural numbers are positive whole numbers.

Integers are positive and negative whole numbers along with zero.

Rational numbers are any numbers that can be found by dividing two integers.

Irrational numbers are numbers that cannot be defined by the division of two integers.

Real numbers are the combination of all the previous numbers (natural, integer, rational, and
irrational). They make up every value on the number line.

Complex numbers involve the combination of real numbers and values containing i . ( i  1 )
That’s as far as we will go with sets and set notation in this tutorial. This is perhaps more than what will
be covered in your average middle school/high school algebra course but you will see this all again in
college algebra. Besides, it really helps you understand numbers much better when you have the big
picture in mind from the beginning.
Pretty much everything you deal with all the way through high school will be in the real number set.
Depending on the courses you take, you will probably see a little bit about complex numbers, but
probably won’t get too deep into that until college.
For those looking for more information on sets such as closure, identity elements, groups, fields, etc.
check out the “Upper Level” section of the tutorials.
That’s all for sets. We will leave you with a handy diagram to hopefully make sense of all that we talked
about here. Enjoy!
Note: 1. Irrational numbers are those real numbers that do not fall within the rational number section.
2. Zero is being included as an integer and not a natural number in this diagram
Complex Numbers
23.83  89.52i
 i
Real Numbers
.125
Integers
-25
Natural Numbers
-4647
36
1043
-12
7
15
98
-32
4i
6
83
2.121122111222...
4
 3 i
7
3
4
0
1
7  2i

Rational Numbers
17
-41
2.3684
-19275
-5
7
5
8
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