and the UNeXpLaiNed ©Copyrighted by Dave Ayotte & Caty Bergman

SCI: Mathematics

TOC (Table Of Contents)


          01.01 SUMMARY
          01.02 DEFINITIONS

    02.00 HISTORY

          02.01 EARLY MATH
          02.02 CALCULUS AND BEYOND
          02.99 TIMELINE


          03.01 REVIEW
          03.02 DIVISION BY ZERO
          03.03 ALGEBRA
          03.04 GEOMETRY
          03.05 TRIGONOMETRY
          03.06 PRECALCULUS
          03.07 CALCULUS

    06.00 FOOTNOTES
    09.00 etc...



          01.01 SUMMARY
          01.02 DEFINITIONS

01.01      INTRODUCTION: Summary

Math is not everyone's cup of tea. As a matter of fact in a calculus for stupid people book that we're reading right now, it jokingly explains that some people hate math so much that they'd rather get kicked in the face by a mule than learn math. Although I personally wouldn't go so far as to get kicked in the face by a mule to avoid math, there are definitely a lot of things I'd rather be doing instead, but here we are and since there are many things we love that are better understood with a rudimentary knowledge of math, it seems like we really have no choice in the matter, but to learn it as best as we can. Thus, we have started this webpage to do just that. Maybe, it will do some good and we'll actually learn something. There really is only one way to find out for sure and that is to just do it. So like we just said, here we are, so let's get started.

We'll start off with some definitions of words we ran across while researching for this webpage. Those are listed below. We'll add to them as run across more words that need to be explained. Next we'll provide a little history and a timeline so it makes better sense. Then, we'll review a little of the math you probable already know so you won't be completely lost. The, we'll explain a possible explanation on how to divide by zero which Dave discovered a few years ago.

Then finally, we'll explain math from Algebra up to Calculus. Hope you enjoy reading this as much as we do writing it, and maybe you'll even learn something. Like we said earlier, there is only one way to find out and that is to just do it.

All that's left to do before we start is wish you luck and happy reading.

01.02      INTRODUCTION: Definitions

COEFFICIENTS: Are the actual numbers in an equation that are not varibles. For example 2x . y - 1.5, where 2 and -1.5 are the coefficients, and -1.5 is a constant. The coefficient for y is considered 1 (1 . y = y).

EXTREMA: The high and low points of a curve (maxima and minima, respectively).

FACTORING: Factoring is just reverse multiplication.

FACTORS: The numbers that you multiply together to get your product (answer). For example, 2 . 3 = 6, with 2 and 3 being the factors, and 6 the product.

GCF: Greatest Common Factor

INTEGER: A number that isn't a decimal or a fraction. 3 and -6 are integers, while 10/3 and 3.1 are not.

IRRATIONAL ROOT: Is an x-intercept that is not a fraction.

MAXIMA: The high point of a curve.

MINIMA: The low point of a curve.

PRODUCT: In multiplication, the answer (product) is arrived at by multiplying the factors together. For example, 2 . 3 = 6, with 2 and 3 being the factors, and 6 the product.



          02.01 EARLY MATH

               02.01a PYTHAGORAS

          02.02 CALCULUS AND BEYOND

               02.02a ISSAC NEWTON

          02.99 TIMELINE

02.01      HISTORY: Early Math

Humankind has probably been using some form of math even before symbols were invented. It's actual beginnings are buried under the sheer weight of antiquity, but some things we do know, just not how some of them work. For example, can animals count? It's obvious that on some rudimentary level, they can tell when there is only one of something and when there's more than one.

There are other things we know also. The Greeks were the first to get really serious about mathematics and as a result, much of what we use in math is built on that foundation of knowledge:


"The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns and the recording of time. More complex mathematics did not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy. The systematic study of mathematics in its own right began with the Ancient Greeks between 600 and 300 BC... "

[02.02a]:     Pythgoras                 TOC

Pythagoras, (570-495 B.C.) a Greek mathematician, is generally credited with discovering the Pythagorean theorem.


02.01      HISTORY: Calculus And Beyond


[02.02a]:     Issac Newton                 TOC

In 1687, Issac Newton published Philosophiae Naturalis Principia Mathematica (or The Principia). There are many mathematicians and other academics that believe it is the very beginning of calculus, and probably just as many who believe it began with the works of Gotfried Wilhelm Leibniz, which were published in 1684, three years before Newton's Principia was published in 1687. In his Principia, Newton even credit's Leibniz with developing a similar method to his own.


"Leibniz is credited, along with Sir Isaac Newton, with the invention of infinitesimal calculus (that comprises differential and integral calculus). According to Leibniz's notebooks, a critical breakthrough occurred on November 11, 1675, when he employed integral calculus for the first time to find the area under the graph of a function y = (x).


"Leibniz did not publish anything about his calculus until 1684... "

02.99      HISTORY: Timeline

3,000      B.C. - (Approx.) Simple mathematics (1+1=2) begins to 
                   get complex (1+x=2, x=1)

600-300    B.C. - (Approx.) The Greeks begin to take Math to a 
                   higher level

570-495    B.C. - Pythagoras is generally credited with discovering 
                   the Pythagorean theorem

1684            - Gotfried Wilhelm Leibniz publishes many of his 
                   ideas which were similar to the ones published by 
                   Issac Newton in 1687

1687            - Issac Newton publishes Philosophiae Naturalis
                   Principia Mathematica (or The Principia)



          03.01 REVIEW
          03.02 DIVISION BY ZERO

               03.02a ARE ALL ZEROES EQUAL?
               03.02b THE TWO DIFFERENT ZEROES
               03.02c DAVE'S THEORY
               03.02d PROOF

          03.03 ALGEBRA

               03.03a LINEAR EQUATIONS
               03.03b FACTORING
               03.03c SLOPES
               03.03d QUADRATIC EQUATIONS

          03.04 GEOMETRY
          03.05 TRIGONOMETRY
          03.06 PRECALCULUS
          03.07 CALCULUS

03.01      ADVANCED DEGREE: Review

The building blocks leading up to an understanding of Calculus begin with an understanding of Algebra, and then Geometry, and finally Trigonometry.

One of the first basic skills we'll need to master in order to understand Calculus is the ability to read graphs, and the simplest graph to work with is the line, which leads us to linear equations, but first let's talk about dividing by zero for a minute if you don't mind, or if you want, you can skip right to linear equations. We won't mind either way.


03.02      ADVANCED DEGREE: Division By Zero

Division by zero was just an outside the box exercise Dave thought through a couple of decades ago while thinking about a subject he was researching for an article he was writing for a monthly newsletter he was publishing at the time. When Dave thinks through some math problems, his mind acts weird. He is able to come up with the answer without knowing exactly how he does it. Division by zero was just one of those things. The answer came to him out of the blue. What if all zeroes weren't all the same?

[03.02a]:     WHAT If All Zeroes Aren't Equal?                 TOC

He showed it to a friend of his that was in the process of getting his Bachelor of Science Degree in Mathematics, and that friend thought it made sense, and was possibly a true discovery, but because of the way mathematicians thought of zero, that they wouldn't see it the same way Dave did and would resist it vehemently.

Dave's friend hit the nail right on the head when he explained how mathematicians saw the number zero. All zeroes were all the same and worked the same way when doing mathematical problems, but what if that weren't true? What if there were two kind of zeroes, the number zero and the placeholder zero?

[03.02b]:     The Two Different Zeroes                 TOC

What if these two different zeroes represent two different things? What if one of those zeroes was really just a placeholder, and the other was what most people think the real number zero meant?

Let's take that one step further by asking, what if the placeholder zero was different because it didn't really mean the same thing as what people generally thought the real number zero represented? In other words, what if the placeholder zero was only there to help represent the larger numbers, like the zeroes in 10, 100, and 250 do, and what if the other zero (the number zero that stands alone, the one that separates negative numbers from positive numbers), what if that zero (the number zero) really did represent what most people think it represents?

What if the number zero (rather than the placeholder zero) really did represent nothing. This was the key to Dave discovering the answer to the division by zero problem.

Multiplication in math is accomplished by multiplying two or more factors together to produce a product or answer. With me so far? The current theory (or law) in math is that it doesn't matter how the factors are multiplied (or in which order they are multiplied), the answer will always be the same. For example:

          Five fours equal twenty (5 . 4 = 20)
          Four fives also equal twenty (4 . 5 = 20)

[03.02c]:     Dave's Theory                 TOC

What if this theory is true in all numbers EXCEPT ZERO. According to almost ALL mathematicians:

          Five zeroes equal zero (5 . 0 = 0)
          Zero fives also equal zero (0 . 5 = 0)

The first word problem is correct as far as word problems go, but it is also the reason why division by zero is "seemingly" impossible. This is because instead of the way the word problem is written above, it should be written (and thought of) as follows:

          Five Times Zero Equals Zero (5 . 0 = 0)
          ZERO TIMES FIVE EQUALS FIVE (0 . 5 = 5)

In order to solve the problem of division by zero, you have to remember that zero can also be thought of (if you think outside the mathematical box, like we explained earlier) as the numerical representative of "NOTHING".

FOR EXAMPLE, five times zero becomes five times nothing, or nothing is being multipled by five, which can also be written as nothing is being multipled five times which means zero remains zero.

Conversely, zero times five becomes nothing times five, or five is being multiplied by nothing, which can also be written as five is not being multiplied at all, which means five remains five.

[03.02d]:     Proof                 TOC

That last sentence makes division by zero possible. Since division is the reverse of multiplication, reversing the above operations using division should result in getting the original equations back, which would definitely help prove that division by zero is very possible.

Let's use the same example from above and do it right now to see what happens:

          (5 . 0 = 0) = (0 / 5 = 0)
          (0 . 5 = 5) = (5 / 0 = 5)

When you divide zero by five you get zero because no matter how many times you try to divide up zero (five zeroes are still zero), it still remains zero:

          0 + 0 + 0 + 0 + 0 = 0

Consequently, when you divide five by zero, it's like five is being divided by nothing, or not being divided at all, and thus remains five.

Another way to check this is with addition. Since addition is merely a long form of multiplication. In short, when you multiply something, you are merely adding the same number together as many times as the number you are multiplying. It doesn't matter which factor you use as the multiplier, at least according to current mathematical theory anyway:

          5 . 4 = 4 + 4 + 4 + 4 + 4 = 20
          4 . 5 = 5 + 5 + 5 + 5 = 20

Once again, we (Dave more than me) say this is true except when the number zero is involved.:

          5 . 0 = 0 + 0 + 0 + 0 + 0 = 0
          0 . 5 = 5 = 0 . 5

The first equation is obvious, while the second one is where the debate lies. Until mathematicians realize that the placeholder zero (10, 100, 250...) is different from the number zero (...1, 0, -1...), the result of the debate will probably always be, at least amongst most mathematicians anyway, that division by zero is impossible.

Only time will tell for sure whether Dave's theory is ever accepted as fact, or proven wrong absolutely. In his mind anyway, Dave doesn't think it ever will be proven wrong absolutely because zero is merely a concept (a theory if you will) and not an absolute fact, at least not at this writing anyway. Only time will tell, he says.

I personally don't know what will happen with his theory. On the face of it, the way he explains it makes sense and could possibly be true, but I don't know enough about math to really say for sure or even put a number on it that's represents how probable it is. Right now though, Dave's theory makes sense to me in a way because, like him, I too think of zero as nothing and that's what helps me to buy into the whole thing. Which brings us to one of the reasons we started this page.

One of those reasons was to complement our Cumputer History page, and the other was to learn enough about math to prove or disprove his theory. Either way, it's been fun so far, and we hope you are enjoying it also. Thank you anyway, either way, for at least giving us your time by reading this and if you have any comments, positive or negative, they would be deeply appreciated. You can contact us by going (or clicking) HERE.

03.03      ADVANCED DEGREE: Algebra

One of the simplest of all the complex equations is the linear equation. Working with linear equations is one of the first things you learn in Algebra. With that said, we'll start our advanced degree program off with a discussion about linear equations.


[03.03a]:     Linear Equations                 TOC

The most common form of the linear equation is

          Ax + By = C

with the variable terms on the left and the answer on the right. This is known as the standard form. Other rules apply also. Coefficients A, B and C must be integers, with A being positive. The standard form is ultimately helpful, especially to instructors, in determining that the correct answer is arrived at by their students.

[03.03b]:     Factoring                 TOC

The first thing that needs to be done with almost all equations (after they've been created of course, which we'll talk about when we get to Slopes) is to make sure that they have been simplified to their simplest form, and the most common way to do this is to factor it. Factoring is just reverse multiplication, and it's done by using the equation's Greatest Common Factors (GCF) to reverse the multiplication process.

Before you can do any kind of factoring, you first have to find all the GCFs. The multiplication process has at least three parts. There are two factors and a product (answer). The product is arrived at by multiplying the two factors together. For example,

          2 . x = 6

2 and x are the factors, and the product is 6. To simply this equation using factoring, first you need to find the factors of the two known numbers, 2 and 6, and these factors are as follows:

          2 (1, 2)
          6 (1, 2, 3, 6)

You then figure out which is the greatest (largest) common factor. In this example, the 2 is GCF.

You then divide the GCF (2) into the two known numbers (2, 6) to end up with the following equation:

          1 . x = 3

To further simplify this equation, all you need to do is multiply the x by the 1, which will keep it x:

          x = 3

You really can't simplify an equation any more than getting the actual answer to all the unknown varibles, but this is a good example of factoring in it's simplest form. You will very rarely, if ever, see an equation like this to factor. Usually, it'll be a little more complicated, and sometimes way more complicated than this one:

          2x2 . 3y = 4a . 6x2

The next question you obviously must have is how do you simplify the above example using factoring? Good question. The answer unfortunately requires a bit more in depth discussion about factoring.



[03.03c]:     Slopes                 TOC

[03.03d]:     Quadratic Equations                 TOC

03.04      ADVANCED DEGREE: Geometry


03.05      ADVANCED DEGREE: Trigonometry


03.06      ADVANCED DEGREE: Precalculus


03.07      ADVANCED DEGREE: Calculus


"In mathematics, differential calculus is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus.

"The primary objects of study in differential calculus are the derivative of a function, related notions such as the differential, and their applications. The derivative of a function at a chosen input value describes the rate of change of the function near that input value. The process of finding a derivative is called differentiation. Geometrically, the derivative at a point equals the slope of the tangent line to the graph of the function at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function at that point.

"Differential calculus and integral calculus are connected by the fundamental theorem of calculus, which states that differentiation is the reverse process to integration... "





Complete Idiot's Guide To Calculus, The
         Copyright © 2002 by W. Michael Kelley





Differential Calculus - Wiki:

Gottfried Leibniz - Wiki:

Leibniz, Gottfried - Wiki:

Mathematics - Wiki:






Etc... is for subjects that just don't fit in any of the above categories.

LAST UPDATED: March 5, 2013
by myself and Caty.