Concepts in Mathematics
By David Alderoty © 2015
over 2,650 words
To go to the previous chapter
left click on one of the following links:
www.TechForText.com/Ma/chapter-4
www.TechForText.com/Ma/chapter-4/PDF.pdf
To contact the author left click for a
website communication form, or use:
If you want to go to the table of contents
of this chapter left click on these words
To Access Additional Information with Hyperlinks
After I complete a writing task, I select a number of websites from other authors, and link to them, to provide additional information and alternative perspectives for the reader. The links are the blue underlined words, and they can be seen throughout this e‑book. The in‑line links, such as the link on these words, are primarily to support the material I wrote, or to provide additional details. The links presented at the end of some of the paragraphs, subsections, and sections are primarily for websites with additional information, or alternative points of view, or to support the material I wrote. The websites contain articles, videos, and other useful material.
The brown text that look like these fonts, represent quotes in this book. You can access the original source, by clicking on a link presented just before a quote.
If a link fails, use the blue underlined words as a search phrase, with www.Google.com. If the failed link is for a video use www.google.com/videohp. The search will usually bring up the original website, or one or more good alternatives.
Basic Numbers Sets, and Related Concepts
The set of natural numbers are represented by N={1,2, 3, 4, 5…}. These are the numbers used for counting. Some sources also include zero as a natural number, thus {0, 1, 2, 3, 4, 5…}. The set of natural numbers do not include negative numbers, fractions, decimals, irrational numbers, and imaginary numbers.
The set of integers, are represented by the following:
Z={…‑5, ‑4, ‑3, ‑2, ‑1, 0, +1, +2, +3, +4, +5…}
The set of integers includes negative numbers, zero, and the natural numbers. The integers do not include fractions, decimals, irrational numbers, and imaginary numbers.
The
set of rational numbers, usually denoted by Q, are numbers that can be represented by a quotient of
two integers, such as 
when A and B,
are integers. The set of natural numbers and integers are part of the
set of rational numbers. This is because they can be expressed in the form of , as shown below:
.
Numbers
with decimals are rational, if they can be
represented as a fraction consisting of two integers. Examples are: 
, 
, 
Even when the decimal of a number repeats infinitely, it is a rational number, if it can be expressed as a fraction, consisting of two integers. See the following examples:
When rational numbers are in the decimal format, they often have decimals that repeat in a pattern. This should be obvious from the examples presented above.
Some
examples of numbers that CANNOT be
represented by a fraction with two integers are: 
. Numbers in this category are
irrational and they are discussed under the following subheading.
The irrational numbers are NOT a set of sequential numbers, such as the previous examples, of natural numbers, and integers. Irrational numbers CANNOT be represented by a fraction with two integers. These numbers have decimals that DO NOT END, even if they are calculated to 100 decimal places. The digits in the decimals of irrational numbers do NOT repeat in a PATTERN, and they look like a set of random digits. The following are examples of irrational numbers:
The Sets of Real, Imaginary, & Complex Numbers
The set of real numbers, represented by R, includes natural numbers, integers, as well as rational and irrational numbers. Real numbers exclude imaginary and complex numbers, which are discussed below.
Imaginary
numbers can be thought of as a set of real numbers multiplied by the square
root of negative one 
).
For example, 
(-4), (-3), (1), 
, 
, 
, 
, 
. However, with imaginary numbers,
the square root of negative one is represented by i.
That is 
. Thus,
the examples presented above, can be represented as: ‑4i, ‑3i, ‑i, 
, 
, 2i, ei, and 
.
When the square root of negative one 
is
squared, the result is negative one. This
is because i
as a result, when i is squared, the result is -1. In terms of mathematical notation, this means 
This is a very important
relationship for calculations that involve imaginary numbers.
Imaginary numbers, can be added, subtracted, multiplied, and divided, and used in algebraic expressions. In this regard, you should keep in mind all of the following relationships:
Below there are some general examples involving imaginary numbers. See if you can calculate the indicated results.
(10i)(3i)=-30,
(10i)(3)=30i,
(-10i)(3i)=30,
10i+3i=13i
10i-3i=7i
10i-10i-3i=‑3i
(10i)(-10i-3i)=130
Complex numbers are a combination of both real and imaginary numbers, such as the examples at the end of this paragraph. With these examples, I placed the imaginary numbers on the left, and the real numbers on the right. The variables without the i represent real numbers, such as X+Y. The variables with the i represent imaginary numbers, such as iZ+iY.
i+10
i10+100
Complex Numbers, and Quadratic Equations
Complex
numbers are often encountered when solving quadratic equations. This happens when the discriminant (
) is negative. The reason for this is
obvious, if you examine the quadratic formula, which is presented below:
The following quadratic equation results in a mixed number, which can be seen in the step-by-step solution, presented below:
Thus a result is:
*Mixed number, with i
Numbers with decimals are rounded down.
The above cannot be simplified any further, because you cannot combine a real number with an imaginary number, by addition or subtraction.
*Note, some sources present this type of relationship as:
and 
This can be confusing, because it
looks like 
and 
The format that I used eliminates
this potential difficulty.
The Sets of Odd, and Even Numbers, and Related Concepts
Even
numbers are integers that can be divided by two, without a remainder, which
means no decimals. For example, three is NOT an even number, because 
and the remainder is
. (With decimals, the result is 1.5.)
However, eight is an even number, because 
with no remainder.
The
set of all even integers can be represented by 2Z, if Z represents any
integer. (The proof for this is obvious, which is: 
.)
Thus, in terms of a set, this can be presented as follows: 2Z={…‑6, ‑4, ‑2, 0, +2, +3, +4 +6…}
This can be symbolized in terms of a formula, as follows 2Z=Even_Number. With this formula, Z represents
any integer. This means that any integer that is entered into the formula will
result in an even number. See the following examples:
When Z=-3, 2Z=-6
When Z=-2, 2Z=-4
When Z=-1, 2Z=-2
When Z=0, 2Z=0 (Zero is even number)
When Z=1, 2Z=2
When Z=2, 2Z=4
When Z=3, 2Z=6
Integers that have a decimal of 0.5 when divided by two are odd numbers. Odd numbers can also be defined as an even number plus one, or an even number minus one. See the following examples:
1 is odd, because 
3 is odd, because 
4 is NOT odd because 
2
5 is odd because 
=2.5
In the previous subsection, I presented a formula for all even numbers, 2Z=Even_Number, (when Z=any integer). As stated above, odd numbers can be defined as an even number plus one, or an even number minus one. If 2Z represents an even number, then we can create a formula for odd numbers, by adding one to 2Z. This results in the following formula:
2Z+1= Odd_Number (when Z=any integer).
An alternative formula is an even number minus one, such as
2Z-1= Odd_Number.
Any integer that is entered into these formulas will always result in an odd number. See the following examples:
2Z+1= Odd_Number
If Z=-4 then 2Z+1=-7
2Z+1= Odd_Number
If Z=7 then 2Z+1=15
2Z+1= Odd_Number
If Z=5 then 2Z+1=11
2Z-1= Odd_Number
If Z=11 then 2Z-1=21
2Z-1= Odd_Number
If Z=0 then 2Z-1=-1
2Z-1= Odd_Number
If Z=10 then 2Z-1=19
The Set of Prime and Composite Numbers
A prime number is a natural number that CANNOT be factored into two or more natural numbers. A prime number can also be defined as a natural number that cannot be divided by any other natural number, without a remainder, or a decimal*. For example, 5, 7, 11, are prime numbers, and if they are divided by other natural numbers there will be a remainder.
*Of course, if a prime number is divided by
itself, there is no remainder, such as 
In addition, if a prime number is
divided by 1 there is no remainder, such as 
All prime numbers are odd, except for 2, and they can be thought of as the fundamental building blocks of other numbers. Probably the best way to understand prime numbers is to examine composite numbers, which is explained below.
Composite
numbers can be factored into two or more prime numbers. This is the same as
saying that composite numbers are comprised of two or more prime numbers
multiplied together. All composite numbers can be divided by one or more prime
numbers without a remainder. For example, 15 is a composite number, and it is
comprised of 3(5), and it can be divided by 3 or 5 without a remainder, such as
Below there are examples of composite numbers, which I factored in terms of two or more prime numbers. The prime numbers are highlighted in yellow.
10=2(5)
If you examine the yellow highlighted results presented above, you will see a general relationship, which can be expressed with the following formula:
With
the above formula,
represent the set of all prime numbers.
The lowercase letters a, b, c, and d, are exponents. The exponents relate to
the number of times a specific prime number appears as a factor, in relation to
a specific composite number. For example, 
With this example, a=2 and b=4, 
, 
. Based on the formula presented
above, and this example, all of the remaining numbers, in the set of prime
numbers, have an exponent of zero. Any number with an exponent of zero is
equal to one.
Devising a General Formula, for Odd Numbers That Are NOT Prime Numbers
When
two positive odd numbers are multiplied together, the result will be an odd
number that is NOT a prime number. This
can be restated as, when any two odd numbers a multiplied together, the
result will be an odd composite number.
We can devise an interesting formula with the idea presented above. Let
us assume that N and n represent any natural number that is equal to or
greater than one. That is N and n can equal any number in the set
{1, 2, 3, 4, 5…}.
We can represent all even numbers in this set by multiplying the above by two, as such 2N and 2n. Now, we can represent all odd numbers, in this set by adding one to the above as follows: 2N+1 and 2n+1. Thus, both of these expressions represent odd numbers, that are equal to or greater than plus one. As stated above, if two odd natural numbers are multiplied together, the result will be, a composite number that is odd. Thus, (2N+1)(2n+1)= Odd_Composite Any natural number that is entered into this formula will result in an odd composite number. See the following examples:
(2N+1)(2n+1)= Odd_Composite
If N=1, and n=1
(2(1)+1)(2(1)+1)= (3)(3)=9
(2N+1)(2n+1)= Odd_Composite
If N=2, and n=1
(2(2)+1)(2(1)+1)=(5)(3)=15
(2N+1)(2n+1)= Odd_Composite
If N=3, and n=1
(2(3)+1)(2(1)+1)=(7)(3)=21
(2N+1)(2n+1)= Odd_Composite
If N=4, and n=1
(2(4)+1)(2(1)+1)=(9)(3)=27
(2N+1)(2n+1)= Odd_Composite
If N=5, and n=1
(2(5)+1)(2(1)+1)=(11)(3)=33
(2N+1)(2n+1)= Odd_Composite
If N=6, and n=1
(2(6)+1)(2(1)+1)=(13)(3)=33
The Fundamental Theorem of Arithmetic, Extending the Theorem, and Related Concepts
What is The Fundamental Theorem of Arithmetic?
The concepts and related formulas presented in the previous section are ultimately based on the fundamental theorem of arithmetic. This theorem is clearly illustrated, on the website link to these words, as follows: “The fundamental theorem of arithmetic: every whole number larger than one is either a prime number or can be expressed as a unique product of prime numbers.” The product of prime numbers, results in a composite number, which is represented by the following formula:
With
this formula,
… represent the set of all prime
numbers. The lowercase letters a, b, c, and d, are exponents, and they relate
to the number of times a specific prime number appears as a factor, of a
composite number.
The fundamental theorem of arithmetic, applies to positive numbers only, which is indicated by the words: “…every whole number larger than one…” This exclusion appears to be the result of convention, and the concept is extended to negative numbers in the following subsection.
The Fundamental Theorem of Arithmetic, Modified for Negative Numbers
The
fundamental theorem of arithmetic, and related formulas can be used with
negative numbers, if we use 
as
a prime number. When this is done, the theorem applies to
all integers, positive and negative. The modified concept can be symbolized
by the following formula:
The
value of X in the above equation, determines whether the compound number will
be negative or positive. When X=0, the result is 
. When
X=1, the result is 
. (Note
this is based on conventional concepts that relate to exponents.) An example
based on the above equation is:
With the ideas presented above the Fundamental Theorem of Arithmetic: can be modified as follows: Every integer is either a prime number, or product of prime numbers, if ‑1 is used as a prime number. This theorem can be called the fundamental theorem of primary and composite integers, to avoid confusion with the conventional definition.
Additional examples of the concept are presented below, based on the formula:
Fundamental Theorem of Arithmetic, Modified for all Rational Numbers
With the ideas presented below, the Fundamental Theorem of Arithmetic, can be modified so it will apply to all rational numbers, including positive and negative fractions. The modification is based on using negative one, (‑1) and positive one (+1) as prime numbers. Then the Fundamental Theorem of Arithmetic, can be modified as follows: Every rational number is either a prime number, or the product and/or quotient of prime numbers, if negative and positive one are used as prime numbers, as indicated by the following formula:
With this formula, P and F represent prime numbers
To avoid confusion with the conventional definition, the modified theorem can be called the fundamental theorem of primary and rational numbers.
The concept is illustrated with the following formula and related examples, below:
With
the ideas presented below, the Fundamental Theorem of Arithmetic, can be
modified so it will apply to all real, rational, irrational, and imaginary
numbers, and related variables. This can be
achieved by using any NUMBER OR VARIABLE that
CANNOT BE FACTORED, as a PRIMARY NUMBER.
This includes, all of the following: 
This is illustrated with
the following example:
Based on the above the Fundamental Theorem of Arithmetic, can be modified as follows: Every rational, irrational, and imaginary number is either a prime number, or the product and/or quotient of prime numbers, if any number or variable that cannot be factored is treated as a prime number, as indicated by the following formula:
The modified theorem can be called the fundamental theorem of all numbers and variables, to avoid confusion with the conventional definition.
In
the formula presented above, N is the resulting number, which will be
composite, unless all the factors are equal to one.
When the exponent w, on this factor 
is
zero, the resulting number N, is a real
number. When w is equal to one, N is imaginary. Similarly, when the exponent
X on this factor 
is
zero, N is positive, and when X equals one, N is negative.
The above based on conventional concepts of exponents. This is demonstrated with the following calculations carried out by Microsoft mathematics, for Word:
Below there are a number of examples, based on the above concept, and the following formula:
To go to the first page of this
chapter left click on these words.
Below is the hyperlink table of contents of this chapter. If you left click on a section, or subsection, it will appear on your computer screen. Note the chapter heading, the yellow highlighted sections, and the blue subheadings are all active links.
To Access Additional Information with Hyperlinks
Basic Numbers Sets, and Related Concepts
The Set of Irrational Numbers.
The Sets of Real, Imaginary, & Complex Numbers
Complex Numbers, and Quadratic Equations
The Sets of Odd, and Even Numbers, and Related Concepts
The Set of Prime and Composite Numbers
Devising a General Formula, for Odd Numbers That Are NOT Prime Numbers
The Fundamental Theorem of Arithmetic, Extending the Theorem, and Related Concepts
The Fundamental Theorem of Arithmetic, Modified for Negative Numbers
Fundamental Theorem of Arithmetic, Modified for all Rational Numbers
To go to the first page of this
chapter left click on these words
To go to the following chapter
left click on one of the following links
www.TechForText.com/Ma/chapter-6
www.TechForText.com/Ma/chapter-6/PDF.pdf
If you want to see a list of all
the chapters in this e-book go to