Learning component: Special sets of numbers
Learning group: Basic mathematical concepts
Table of Contents
Learning material notes........................................................................................................................2
Introduction..........................................................................................................................................3
Sets of numbers....................................................................................................................................3
Set of Natural Numbers - ℕ........................................................................................................3
Set of Primes - ℙ.........................................................................................................................5
Set of Integers - ℤ.......................................................................................................................6
Set of Rational Numbers - ℚ.....................................................................................................7
Set of Real Numbers - ℝ............................................................................................................8
Set of Complex Numbers - ℂ....................................................................................................8
Exercises...............................................................................................................................................9
Where to learn more...........................................................................................................................10
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Learning component: Special sets of numbers
Learning group: Basic mathematical concepts
Learning material notes
(From the learning plan)
Some sets of numbers are used frequently in mathematics and have achieved special significance.
These sets are given specific names and used in a variety of mathematical topics, e.g. set of natural
numbers labelled N. This component reviews these special sets.
This is a basic mathematical component and assumes as little knowledge as possible.
(Additional comments)
This module has also expanded slightly to include an introduction to the different sequences and not
just their special notations.
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Learning component: Special sets of numbers
Learning group: Basic mathematical concepts
Introduction
Consider the following sequence of numbers: 1, 2, 3, 4, 5. These numbers should be familiar and
will be used to represent many things in your life: the channels on the television, the number of
places at the dinner table or the books on loan from the library. There is nothing particular special
about this short sequence of numbers. It is simply the sequence you will use to count a number of
items. They are example of the ordinary counting numbers, called the natural numbers.
Of course, the set of numbers is not always sufficient. You may need negative numbers to count the
amount of money in your bank account or require real numbers to represent the length of an object
as accurately as possible. Some particular sequences of numbers, such as the natural numbers, are
used throughout mathematics. You will encounter these sets of numbers both in real-life and in a
variety of mathematical problems.
In fact, some of these sets of numbers are used with such regularity that they have been given
special names and symbols. We're going to look at a selection of these special numbers of numbers.
Sets of numbers
There are a few different special sets of numbers. We're going to look at each one in turn and find
out what they represent. Each set has a special symbol used to represent that set of numbers.
Mathematicians will use this symbol to represent the set.
Set of Natural Numbers - ℕ
{ 1, 2, 3, 4, 5, 6, 7, 8, … }
Background
The natural numbers are the counting numbers, used to count everything from the number of books
on the shelf to the number of elements in a set. In addition to the term 'natural numbers', these may
also be referred to as the counting numbers or the whole numbers.
There are two main uses for the natural numbers:
•
•
counting – counting the number of items, e.g. there are five books.
ordering – ordering a number of items, e.g. the fifth book I've read this month.
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Learning component: Special sets of numbers
Learning group: Basic mathematical concepts
The set of numbers usually starts at the number 1 but can sometimes include zero. If the set begins
at the number 1, the set can be thought of as the 'positive integers'.
Notation
The set of natural numbers is represented using the symbol ℕ. It is also acceptable, and common,
to give this as simply the letter N, usually written in a bold font.
ℕ = {1, 2, 3, 4, …}
Due to the ambiguity of whether the set should include zero, some mathematicians may explicitly
state whether it should be included. Sometimes this is included in the set notation itself. Adding a
subscript zero to the ℕ symbol indicates that zero is included in the set:
ℕ0 = {0, 1, 2, 3, 4, …}
Equally so, adding a subscript explicitly states that the set does not include the number zero:
ℕ1 = {1, 2, 3, 4, …}
Properties
The natural numbers has several interesting properties. These properties are statements of fact
about the numbers and the operations we can perform on them.
Choose any two numbers from the set of natural numbers. Now try adding these numbers together.
Your answer is another number also in the set of natural numbers. This property is called closure.
The natural numbers also demonstrate associativity and commutativity because the order of
certain operations are unimportant. For example:
a+b=b+c
a*b=b*a
The natural numbers also contain identity elements. These are numbers which, when combined
with another number, will leave the number unchanged. In the natural of the natural numbers,
adding zero to any number will result in the same number. Also, multiplying any number by 1, will
result in the same number.
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Learning component: Special sets of numbers
Learning group: Basic mathematical concepts
The natural numbers also demonstrate distributivity for multiplication over addition. For example:
5 * ( 4 + 3 ) = (5 * 4) + (5 * 3)
Formal definitions
Information about peano axioms definition of natural numbers to be added here.
Set of Primes - ℙ
{ 2, 3, 5, 7, 11, 13, 17, 19, … }
Background
Prime numbers are numbers greater than 1 that cannot be divided except themselves and 1. A prime
number has exactly two distinct natural number divisors. The first four prime numbers 2, 3, 5 and 7
can all only be divided by themselves and the number 1. However, the numbers 4 and 6 are not
prime numbers because 4 can be divided by both 2 and 4, and the number 6 can be divided by both
3 and 6. It should be noted that the number 1 is not a prime number. As there is only one distinct
divisor of the number 1.
Euclid's Theorem, developed in 300 BC, shows that that an infinite number of prime numbers exist.
Every non-zero natural number can be factorized into prime numbers. The fundamental theory of
arithmetic states that any non-zero integer has a unique product of one or more prime numbers. For
example, the number 27 can be written as the product of prime numbers:
27 = 3 * 3 * 3
Prime numbers are used widely in mathematics, perhaps most notably in public-key cryptography.
This form of cryptography takes advantage of the difficulty of factorization.
Notation
The set of prime numbers is represented using the symbol ℙ. It is also acceptable, and common, to
give this as simply the letter P, usually written in a bold font.
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Learning component: Special sets of numbers
Learning group: Basic mathematical concepts
ℙ = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37…}
Specific prime numbers can be identified with a subscript number before the letter P. For example,
the second prime number is 3 and this can be identified as P2 = 3.
Sieve of Eratosthenes
The sieve of Eratosthenes is an ancient algorithm used to identify prime numbers. It was created by
the Greek mathematician over 2000 years ago.
I plan to write an interactive example to help demonstrate prime numbers and the sieve.
Set of Integers - ℤ
{…, -3, -2, -1, 0, 1, 2, 3, ...}
Background
The integer numbers are the combination of the natural numbers and the negative of the non-zero
natural numbers. The set of integers contains all the whole numbers that descending from zero and
ascend from zero.
Notation
The set of integers is represented using the symbol ℤ. It is also acceptable, and common, to give
this as simply the letter Z, usually written in a bold font. The set of integers is written as follows:
ℤ = {…, -3, -2, -1, 0, 1, 2, 3, ...}
The set of integers can also be subdivided to separate positive and negative integers. An additional
plus or minus superscript can be used to indicate the particular set of integers. For example, ℤrepresents the set of negative integers, ℤ+ represents the positive integers and ℤ* is used to represent
the non-negative integers.
ℤ- = {…, -3, -2, -1}
ℤ+ = {1, 2, 3, …}
ℤ* = {0, 1, 2, 3, ...}
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Learning component: Special sets of numbers
Learning group: Basic mathematical concepts
Properties
The set of integers has several properties similar to the properties of the natural numbers. These
properties are statements of fact about the integers and the operations we can perform on them.
Choose any two numbers from the set of integers. Now try adding these numbers together. The
answer is another number also in the set of integers. Multiplying two integers also produces
another number found in the set of integers. This property is called closure.
The integers also demonstrate associativity and commutativity because the order of certain
operations are unimportant. For example:
a+b=b+c
a*b=b*a
The integers also contain identity elements. These are numbers which, when combined with
another number, will leave the number unchanged. For the integers, adding zero to any number will
result in the same number. Also, multiplying any number by 1, will result in the same number.
The integers also demonstrate distributivity for multiplication over addition. For example:
5 * ( 4 + 3 ) = (5 * 4) + (5 * 3)
Set of Rational Numbers - ℚ
Background
The rational numbers are the set of numbers that can be expressed as the fraction of two integers.
Rational numbers are given in the form:
a/b
where b is a non-zero integer, which may be equal to one. As a consequence of this, every integer
can be considered a rational number.
A real number that is not rational is called an irrational number.
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Learning component: Special sets of numbers
Learning group: Basic mathematical concepts
Notation
The set of rational numbers is represented using the symbol ℚ. It is also acceptable, and common,
to give this as simply the letter Q, usually written in a bold font.
Set of Real Numbers - ℝ
Background
A real number is the first of the sets we have encountered that is not expressed with only whole
numbers. A real number is a value on a continuous scale – all the numbers on the number line. So
positive numbers, negative numbers, integers, rational numbers etc. are all real numbers.
Real numbers are used extensively in science to represent the value of physical phenomena.
Physical variables such as mass, velocity and position are all likely to be represented using real
numbers.
Notation
The set of integers is represented using the symbol ℝ. It is also acceptable, and common, to give
this as simply the letter Z, usually written in a bold font.
Set of Complex Numbers - ℂ
Background
The set of complex numbers is the set denoting all complex numbers. Complex numbers consist of
a real and imaginary component of the form a + bi, where both a and b are real numbers and i2 = -1.
Complex numbers are found in a variety of engineering and mathematical fields where signals are
being processed or differential equations are being solved.
Notation
The set of integers is represented using the symbol ℂ. It is also acceptable, and common, to give
this as simply the letter C, usually written in a bold font.
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Learning component: Special sets of numbers
Learning group: Basic mathematical concepts
Exercises
Try the following sample exercises. The questions begin with simple questions and increase in
difficulty.
1. Add the next five terms to the following sets of numbers,
i. {1, 2, 3, 4, …}
ii. {2, 3, 5, 7, … }
iii. {-3, -2, -1, 0, … }
2. Identify the following sets of numbers and give the first five elements.
i. ℤ
ii. ℙ
iii. ℕ
3. Describe the rational numbers. Give some examples of rational numbers.
4. For the natural numbers, give examples of the following properties:
i. Distributivity
ii. Associativity
iii. Closure
5. Explain why the set of integers can be considered a subset of the set of rational numbers.
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Learning component: Special sets of numbers
Learning group: Basic mathematical concepts
Where to learn more
If you want to learn more about the different sets of numbers there a number of different places you
can look.
Many mathematical textbooks will have an introduction to these sets of numbers, usually at the start
of the book. Look in the contents page or the textbooks for the key phrases highlighted in bold
throughout this material.
The Internet contains a vast number of mathematical resources. You can try using your favourite
search engine to find material using the highlighted phrases, or trying following some of the links to
websites below.
Wolfram has an extensive mathematical website called 'MathWorld'. You can visit the site at
http://mathworld.wolfram.com/. Some of the material can be very advanced but you may find it
helpful as a resource. The following articles relate to the material covered in this learning unit:
http://mathworld.wolfram.com/PrimeNumber.html
http://mathworld.wolfram.com/NaturalNumber.html
http://mathworld.wolfram.com/RealNumber.html
http://mathworld.wolfram.com/ComplexNumber.html
http://mathworld.wolfram.com/RationalNumber.html
http://mathworld.wolfram.com/Integer.html
MathForum contains many mathematical questions and answers. Visit the MathForum at
http://www.mathforum.org. Specific questions and answers you may be interested in are as follows:
http://mathforum.org/dr.math/faq/faq.prime.num.html
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