For What It's Worth

The Bible tells us that "the love of money is the root of all kinds of evil."  Money is just a medium of exchange -- physical or digital representations of our wealth.  When that becomes more important than worthwhile relationships, it becomes detrimental to our wellbeing.

But that wealth is also representative of our worth.  It is the measure of our physical and mental activity in the community around us.  Our money "pays forward" our work into the lives of our family and friends even as it provides a more comfortable life for ourselves.

Whenever we spend money, we normally "seek a bargain" to "get our money's worth."  In this, we are putting a value on those things that we want, drawing a line beyond which we don't want to cross. If we cross that line, and have to borrow money instead, then we have to rearrange our spending habits and pay extra in the long run.

The way to avoid the "credit" trap is to only spend money you already have.  For most of us, we buy large things -- like appliances, cars and houses -- against a promise that we will continue to make money in the future.  Assuming periodical improvements, this works well enough.  For smaller things, though, buying more than we can afford is a bad idea.

The best course of action for recurring expenses is to always "live within your means."  Money on food, utilities and travel should always from "money in the bank."  That is just "how money works."

This requires us to budget our resources with an eye on keeping money in the bank.  In this way, we might even have enough to let it "make money" for us.  Banks make money by charging more for other folk's money than they pay to keep our money.  The truth be told, the banks actually only have a fraction of the money they have control over; but as long as we aren't "rich," this does not matter much for us.

As soon as enough money is set aside for emergencies, the access acquired should be paid out to businesses that provide goods and services to allow them to stay in business.  This provides funds so that they can grow into larger businesses that are worth more.  When we do this, we become partners, sharing in the wealth that we hope will come.  It is best to buy into companies that pay back periodically, but good companies that increase in overall worth can prove a good investment when the time comes to sell your share to someone else.

If you are getting paid to remain a partner, these payments may be used to buy more of the company.  As you buy more "shares," the growing company becomes worth even more to you.  If at anytime the company begins to fail, selling off shares that still are worth more than they first cost provides funds to be used elsewhere.  Timing is important, so it is probably a good idea to let professional money managers make this decision.

So, What Do I Know?

• Money can get us into trouble.
• Money only represents value.
• "Credit" should only be used for big things.
• We should strive to use only our own money on recurring expenses.
• Excess money should be used to help others succeed.
• Wise investments can one day pay us well.

Redistribution of Wealth

Economics is a balancing act between selfishness and greed.  On the one end is the understandable drive for self preservation.  It is altogether natural to do what it takes to survive. Except for some enthusiasts, though, "survival" is not enough.  Most of us want to be "comfortable."  To the extent to which we extend this desire leads to the other end: Greed.

The poorest among us may be greedy most of the time.  When we get something, it is usually not enough to be comfortable. So, we seek more "stuff" so we can be more comfortable.  The problem for the richest among us is rarely selfishness, so even though they often are greedy for more stuff, they can soothe the guilty feelings by giving stuff away.

On a basic level, we humans depend upon others for our survival.  Self-sufficiency is very rare.  Starting with families, and then clans and tribes, we have learned to depend upon each other.  Most people have skill sets that can be used to supplement the comfort level of others.  And so, selfishness meets greed very nicely.

We marvel at the "stuff" that some people have -- stuff that just seems to flaunt their wealth.  Who "needs" a collection of antique cars, for example.  But they spend thousands of dollars acquiring, restoring and maintaining what is to most of us obsolete.  What is the point?  Could it be a way of redistribution of wealth?

Consider, for example, the original 1957 Chevy. The supply line to produce that car began a long time before the steel mills produced the frame.  A multitude of workers received compensation in several continents to provide the raw materials.  Then transportation workers got the materials to the factories.  Each of these were compensated as well.  The owners of plantations and mining companies sold those materials for a profit so they could buy stuff for their families.

By the time the car got to the showroom floor, hundreds of factory workers had been paid along the way.  The factory has spent money to send the car to a dealer responsible for selling it.  In some cases dealerships would buy stock, but often it was "on consignment."  Sometimes "Clearance Sales" had to be used to recoup the cost of having the car on the lot. Mostly, though, the car was sold with enough left over for a profit.  The salesman made his commission, and his family was able to live comfortably for another month.

Over time, the owner of the new car would spend thousands of dollars on maintenance for the car.  He would pay mechanics, or parts dealers and even car washes.  His hard earned wealth was being "redistributed' daily.  And then, he recouped some of that when he traded the car in for a new one.  The car would probably see another owner of a little lower class who would keep it for years.  He would spend what he could in keeping in up, further distributing funds to providers of services and products used.

Finally, after three decades, the old car is spotted and bought by a collector. Depending on its level of disrepair, hundreds to thousands of dollars are put into this classic vehicle to make it look like it was as it rolled off the lot.  The rich man who now owns the car spends hundreds of dollars in storage and display costs every month.  These funds go to those of a lower class, who in turn buy stuff to provide a comfortable living for their families.

So where is the greed and/or selfishness in owning that restored classic '57 Chevy?  Its survival for the last 60 years has provided the redistribution of hundreds of thousands of dollars through numerous years.  Its value, upwards from $50,000, may seem like a lot based on its original price, but it hardly represents a fraction of the money that has exchanged hands over the years.

Let us not envy our rich neighbors who have these fine and sometimes fancy things.  Consider instead how much the existence of these things has contributed to the well-being of those who have produced and maintained them.  Meanwhile, we can learn to be content with what we have.  In our having whatever we have, we ourselves redistribute the flow of wealth to countless others in all walks of life.

So, What do I know?

• Greed is a heart issue.
• Envy gets us nowhere.
• Economics is a study of the flow of wealth.
• Contentment is a beautiful thing.

What Electrons Do

It's What Electrons Do

Speaking of light (see last blog), that reminds me of a related, though unseen, thing that is still quite clearly the truth.  Electrons interact with other electrons in predictable ways.  I was never great in chemistry class, but lab was always my favorite part.

Simple experiments help to demonstrate that molecules and their accumulated atoms work together in dependably predictable ways. Having observed that caustic soda and powerful acids react in bubbly reaction to become harmless "salt water," I was able to safely neutralize a busted car battery that had spilled all over the driveway.  Though I used a whole box of baking soda mixed with water, I could have used a two liter Coke with similar results.

How is this related to light?  Well, those electrons, it turns out, are related to light.  While light is primarily "photons" that ride upon electromagnetic waves, electrons are bundles of that same energy that work by interacting with others in nearby atoms to create bonds that build chemicals of all sorts.  In the example above, bonds building Sodium bicarbonate together come apart "energetically" when introduced to Hydrochloric acid.  The resulting carbon dioxide (evident in the foaming liquid) produced the harmless salt water that I further diluted with with a garden hose leaving trace salt in the adjacent ground.

Electrons are probably best known as the power that "flows" through wires into billions of homes around the world.  These electrons are forced to line up and then are forced by magnetic pulses through the wires. Magnets have a mysterious relationship with the metals around them.  Their electrons flow in one direction until they escape briefly only to be pulled back.  This creates a "magnetic field" which is used to force electrons in the wires to line up and then to move a short distance, hoping from atom to atom.  The chain reaction perpetuates an electromagnetic field at near the speed of light towards a device that finally allows electrons to complete a circuit on the way to a positive ground.

I have learned that it is not a good idea to interfere with moving electrons -- whether in chemical reactions or electric circuits.  I've heard that chemical burns are worse than electrical burns.  But I'd just as soon not find out by experience.

So, what do I know?

• Electrons are tiny negatively charged particles.
• Chemical reactions happen when electrons meet.
• Electricity is when moving electrons create electromagnetic fields.

Light, Life and Information

Seeing Clearly

Millions of living things in the animal and plant kingdoms have a working relationship with light.  Any with sentience "know" what that relationship is.  Or, at least, they make decisions based on perceptions of the benefits of that light.

Light is a form of energy as is evident in the growth of plants.  Without light, green plants die.  Even with plenty of nourishment and hydration, the healthiest of plants need light.  This energy is used to combine water, carbon dioxide and various other nutrients to build stems, limbs, leaves and fruit.  This energy is then released when that material is consumed by other organisms or by combustion.

But the most remarkable thing about light is that it allows sentient beings to gather information from the environment remotely.  Direct contact is not necessary when we animals can see things around us.  Some of us need help as our eyes, wonderful light receptors that they are, become weaker.

I know that light moves so fast that, comparatively speaking, things close by and far away are equally accessible.  However, there are ways to slow light down just a little.  When this happens, light is scattered, absorbed or reflected.  In this way, over time, scientists have figured out that light travels through a vacuum at about 186,282 miles per second.  That is fast enough to travel around the earth at the equator about 7.5 times!

Since light moves so fast, information from all over the universe is accessible on a clear night.  Certain stars, along with the moon, allow navigation along the surface of the earth because the information is predictable.  Based on the apparent movement of the sun on the horizon at sunrise and sunset, we are able to discern the passage of time.

So, what do I know?

• Light is basically energy.
• Light travels very fast.
• Light enables access to information.

Falling is Normal

I move from basic math concepts which I "know" by instinct to other things that are harder to prove, but still just as much of my knowledge base.

Let's move to "gravity."  I won't get into the math behind it, for that is in the territory into which geniuses dare to tread!  But I will speak to the concept.

I know that things FALL.  How much more basic can you get?  If there is any space at all between my hand and the floor, when I let go of an object, it falls to the floor. I've heard, and I've seen video evidence to prove it, that all objects fall at the same speed in a vacuum.  That is convenient for making calculations if one knows the math.

Related to gravity is "weight."  This is the evidence of what in science is called "mass" and "density."  The fact that a feather takes longer to fall to the ground that a marble is because the air around us has both mass and density.  Because of this, we have weather as the air is heated to change its density.

Personally, I experience gravity's effect in whatever I am doing.  As I sit here typing this blog, I can feel my body pressing against the chair.  If I am not careful, my legs will grow numb based on pressure distributed upon my hips.  The keyboard rests securely upon the flat surface of the desk, making typing possible.  All thanks to gravity.

If I were to go outside to play a game of catch with my grandson, gravity would be there to challenge me.  I would take advantage of the fact that the ball is of such a mass and density as to be able to temporarily "defy" gravity due to the energy imparted to it by my releasing it in a forward motion.  That motion, though, would diminish with distance as gravity worked by the laws of motion. The ball would either end up on the ground or in the hands of my grandson.

In all likelihood, my grandson's return of the ball would take a curved line.  This line would put the ball further from the ground for a moment.  The arc it forms obeys the laws of gravity and I have to adjust to intercept the ball.

So, What do I know?

Things fall until they hit a surface.

Things have mass and density.

Things can "fly" when energy is exerted against them that is greater than the gravity that makes them fall.

Circular Reasoning

In some ways the circle is more basic to universal knowledge than the straight line.  More specifically, the curved line is most often incorporated in the art of toddlers given a crayon.  The youngster will rarely draw straight lines, but he will draw rough circles (ovals) as he doodles.

When he gets curious, he "looks around" and "goes in circles" to take as much of his world in as he can.  The space near him is "around" him and he is "surrounded" with things in every direction. To take it all in he goes in "circles."  The word "round" descends from the Latin word "rota," from which we get "rotate."  The word "circle" comes from the Latin "circulus," the diminutive of "circus" which it borrowed from the Greek "kirkos" which means "ring."

Even in ancient times, the Romans and Greeks built stadiums that were semi-circular or fully round (oval tracks).  They were built in such a way as to give a  view to as many people as possible.  And so, they built "circles."  The methodology naturally would have begun with the architect stretching a measuring line to the optimum distance and rotating the desired arch -- all the way to a circle.  On paper, the "circus" was a lot smaller, and thus a "circulus."

This line became known as the radius.  The resulting line forming the ring turned out to be about six and a quarter times radius (a bit over 6.28).  The radius, when doubled, became the diameter of the circle. The ratio between the widest part of the circle and the distance around it is an irrational number known as "pi" estimated as "3.14" or "22/7".  The area inside a circle is pi times the square of the radius.

The most ancient system of measuring the outside of a circle used what we call quadrants: the four directions measured by fixed points.  The common points were the sunrise and the sunset at the equinox.  The ancient Hebrews used East as forward, making that which was on the left "north" and tha which on the right "south."  Behind them was the unpassable sea, the "west" and sundown.  This way the circle has been divided into 16 common directions we know today.  However, this is not precise enough over long distances, so "degrees" of the arch use "base 60" with 60 times 60, or 360 degrees divided by "minutes" and "seconds."  For convenience, time also follows this scheme.

By the way, 360 is the result of multiplying 3,4,5 and 6 together. Seen as prime factors, this is 2*2*2*3*3*5.  This makes the factors of 360 a long list: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 54, 60, 90, 120 and 180.  All in all, the ancient math guys had it together.

So, What do I know?

The radius is a straight line used from a center point to inscribe a circle.

The diameter of a circle is twice the length of the radius.

The distance around the circle is about 3.1416 times the diameter and about 6.2832 times that of the radius. This is the ratio known as "pi".

The area inside the circle is pi (about three and a seventh) times the square of the radius.

A Square Deal

Geometry 4:

A Square Deal

When working with the right triangle, it is easy to build a "rectangle" by simply building a mirror image of the right triangle.  One might expect the term "right rectangle" to be used here, but that would be redundant.  This is because the prefix "rect" is a corruption of the original Germanic form of the Latin "rectus" which means "right."  The Old German was "reht" but the "h" was the hard "kh" sound that migrated up from the Greek "Chi" (looks like our "X").  The Old English form was riht, corrected (pun intended) by adding back in the hard sound.  This time, the hard "g" was used, rendering "ri-ght"  It did not take long before the hardness of the "chi" sound was abandoned.  From "rect" to "right" and back again!

Anyway, we have seen how a right angle can be constructed using the 3:4:5 ratio with the sides.  Using the largest side (the diagonal "5") a ratio of 5:4:3 becomes the mirror image forming a 3 by 4 rectangle.  The area of the rectangle is simple to determine in "square" units.  The area of a "right quadrangle" (four-sided with parallel sides) will be twice that of a right triangle.  Another way of seeing this is that the formula for triangle will be one half that of a rectangle.

The formulae for this are simple:

Area of a rectangle = base (width) x height (length).
Area of a triangle = 1/2 x base x height

The "square" is a special kind of rectangle.  Not only are all the angles inside a square equal, but so are the sides.  The ratio of the sides of the "half square" loses its whole number diagonal when "side a" and "side b" are forced to be the same size.  The diagonal becomes a multiple of the square root of 2! That works out to a little over 1.414.  I'd go out farther, but fourteen-fourteen works for me. But, when considering the area of the square, no irrational numbers need be involved.  I really prefer rational numbers.

When we move to other triangles and polygons, where the angles are not all right (no pun intended), then the "height" becomes a problem.  The height of a triangle is always figured in a right angle, which must be used to divide the shape in order to follow the rules of the square.  That is to say, everything must be reduced to right triangles and/or rectangles to be squared.  It's like playing with blocks!

I'll work out the bugs of those building blocks in another blog.  I know just enough geometry to be "dangerous."

So, what do I know?

Rectangle: Area = lw
Rt. Triangle:  Area = (1/2)lw = lw/2, where l and w form the right angle.

Square: area = l² (length squared) [length being equal to width)

All area within polygons are measured by the pythagorean theorem:  a² + b² = c²

The Triumphant Triangle

Geometry: Part 3

The Triumphant Triangle

As mentioned last time, in order to "square up" a patio, you have to make a triangle.  Of course, this shape is named by the number of angles it has.  After a straight line between two points, the next thing is getting to a third point.  The shortest path between the three points forms a three-sided enclosure with three angles: the "tri-angle."

The prefix "tri" bypasses the introduction of the "th" (theta) that replaced the original "t" (tau) in English.  The vowel sound changes depending on the context and language, but the original Indo-European "trei" is preserved in the Greek treis and its cognates in Western languages. From all I could find out, its meaning has always been "three" (2 + 1).

Though there seems to be a dualism in much of nature (good and bad, light and darkness, wet and dry, and so forth), there is also a triad of moderation that adds depth to reality.  These three dimensions define reality: past, present and future; liquid, solid and gas; width, length and height.  It is the latter of these that is of interest to us right now.

Without a third point, no shapes can be formed. This is true in plane geometry as well as solid geometry.  It is in two dimensions, at a time, that basic shapes are made on a flat surface.  On that surface, a triangle is the most stable of all shapes.  This is because once the three sides are connected, the angles are fixed in place.  To change an angle, one of the sides has to be compromised.  The changing of the length of a side changes the angles as the sides are reconnected.

Going back to the right triangle, the one with the set ratio of 3:4:5, let us say that side 'a' (3) is shortened to 2 units (let's use meters).  To maintain the right triangle, the side opposite the right angle now must be shortened as well.  The new length will be the square root of 20 (a little less than 4.5 meters). To save the 5 meter pole, all the angles change as well.  Rotating outward to about 108 degrees, the 5 foot side once again secures to an stable, though offset triangle. At a ratio of 2:4:5, the ratio is definitely not "right."

And now, let me show how the sum of the three angles will always be twice that of a right angle.  The reason I don't give this in degrees is because the presently defined circle of 360 degrees is based on assuming the "base 60" system of antiquity.  It makes dividing a circle into even numbers quite easy, but is not "known" in all cultures.  For the record, 360 is 12 times 30, or 3 x 4 x 5 x 6.

So, a right angle was formed halfway along an arch which will make a straight line.  If you take the original length of cord - that is 3 meters - you can make a semi-circle defined by a straight line.  This forms the base for two identical triangles, each with the ratio of 3:4:5.  Let's take the cord that forms the original triangle and move the second peg (point b) that was three meters out.  Moving it further from point c, the new position of the peg increases the angle of angle BCA while reducing the other angles.  All along, the length of the cord remains twelve (3 + 4 + 5) meters long.  The area inside the cord diminishes toward 0 until the two halves of the cord reach a length of 6 meters.

The cord is now twice the length of the perpendicular line that formed the right angle and the angle of that line is also twice that of the right angle.  At all times, the angles inside the changing triangles added up to twice the right angle.  If the peg had move toward point c, at some point the sides would have become equal, with each angle also being equal, or one third the angle of a straight line. In this case, 3 sides of 4 meters each.

So, What do I know?

A triangle is defined as a shape on a plane which has three lines intersecting at three angles.

The angles within any triangle will add to that of a straight line.  This total will be twice the angle formed by a perpendicular line bisecting that line.

In a triangle with three equal angles, the sides will be of equal length.  The angles will add up to that of a straight line.

In mathematical terms:

Right Angle BAC = 1/2 x (Angle ABC + Angle BCA)

Triangle ABC = Angle BAC + Angle ABC + Angle BCA

Given Right Angle BAC = 90 degrees, line segment BC will have 180 degrees.

Getting It Right

Geometry: Part 2

The Right Angle

In order to make sure your patio is squared, whether it is going to be a rectangle or a square, you will
need the cord three times the length of the distance to the distance to the first peg.

You really don't even have to measure the distance to the first peg to assure a square corner, but having a tape measure will make this easier.  Let's say your patio is to extend 12 feet (or, if you'd rather go metric, 4 meters).  This means you will need 36 feet (or twelve meters) of cord to easily square up your patio.  The cord will be a little bit longer to leave room for securing it to the pegs.

As pictured in figure #1, the tripling of the length is done by walking back and forth between the pegs with the spool of cord.

Once you have the full length (#2), walk that back to the first peg (#3) and then repeat (#4).  In the end, the four strands will be 9 feet (3 meters) long.  If you began against a wall, just lay the 9 foot piece against the wall and put the third peg down.

Fasten the 9-foot length to the third peg and the end of full cord to the first peg.  Taking the full cord, walk to the second peg, moving it to where it provides a taut line between all the pegs.

This works because in every triangle with a right angle the side opposite that angle has a length that is the square root of the sum of the squares of the other two sides.  It so happens that this ratio is found first in whole numbers with 3, 4, and 5.  And so, any multiples of these numbers produces similar triangles.  If you are going to be doing a lot of building, you could make your own "square" tool using a yardstick or even a twelve inch ruler.  Or, of course, just buy one when you get a chance!

So, what do I know?

A right triangle can be constructed based on the formula

a² + b² = c²

That is, when side c is opposite the right angle, the squares of sides a and b add up to the square of side c.

This manifests itself the ratio of 3:4:5.  That is to say, 9 + 16 = 25.  In whole numbers this only works with multiples of these three numbers.  The ratio using 1:2 would render an irrational number: the square root of 5.  Likewise 2:3 would need the square root of 13!  It only works with these three adjacent whole numbers!

Drawing the Line

Geometry: Part 1
Drawing the Line

We saw the "line" earlier, when we put numbers on it. I was working on an assumption that I knew what a line was.  But this is a blog about what I know, and I have laid down numbers about as far as I can go, so I will be moving on to geometry, the science of construction.

On a flat surface, the straight line is the shortest distance between to places.  On the surface of a sphere, that line is necessarily curved in relation to the radius of that sphere. I will begin with flat surfaces and straight lines in what is called "plane geometry."  That is not "plain," but "plane," as in flat.  It is what we usually think of when building things anyway.

I should begin with the zeroth dimension, which is the point.  A point has no dimensions, but it has coordinates.  The best way to envision a point, though, is as a very small circle, just big enough to see.  That way, we will know where we are and can see where we are going.

To make this practical, as a Greek guy named Euclid did, we will envision laying out a patio.  Making the sides straight will not be hard, and the next blog I will deal with making it rectangular -- or at least triangular, which is exactly half of a rectangle.  But first, the line.

It is quite easy to draw a straight line.  With nothing but a few pegs, a mallet and a measured cord, a perfectly straight line will be a cinch.  After placing a peg in the first corner, measure out a length of cord to a multiple of 4 that is equal to, or beyond the length of the patio.  Do not cut the cord, for you will need twice that much more to assure a square corner.

* * * three pegs
and a cord
---------------------------------


But first, the straight line must be drawn.  With one end secured to the first peg, pull the cord taunt. With a second peg connected to the end of the cord, etch an arch of maybe step in each direction, tentatively marking the middle of that arch.

If the cord is tight, you have your first straight line.  It is the shortest distances between the chosen points.  The slight curved line that you have made in the sand is the beginning of a big circle with a radius the length of that line, but that will come in handy later.

|------------)

So, what do I know?

A straight line on a plane is the shortest distance between two points.  The easiest way to make a straight line is with a string, cord or rope.  Even with modern tools, this is sometimes faster.

A point in space has no dimensions, and a line is in one dimension on flat surface in two opposite directions.

Next: A right triangle.

Thinking About Time

It's About Time

According to Albert Einstein, time is relative.  For the most part, there are two things that can change time: gravity and speed.  I will deal with these exceptions after I give a rundown on the obvious things I have learned about time.

First, the basic unit of time is the "second." Though the history of this unit is interesting, suffice it to say that its exact length was only settled in 1967.  It was then defined as a measurement "periods of radiation" of the caesium atom.  The old "second" was slowly changing since it was based on an average (actually a "mean") year.

A few years later, it was discovered that not only light was affected by gravity (as per Einstein's theory of Relativity), but so was time!  An atomic clock in Denver runs a bit faster than one at sea level (like in London).  Now only clocks at sea level are used to define the second.

Generally speaking, though, a second is 1/3600 part of a standard hour, of which there are 24 in a standard day.  That means there are 86,400 seconds in every day.  This is presently within two ten thousands of a second.  Close enough for me.  Given that each year is about 365.2425 days (rounded), the exact number of seconds per year varies, being corrected with "leap years" and "leap seconds" to keep things orderly.

I still remember playing with the numbers back before I reached a billion seconds -- about 31.7 years.  Now I am approaching two billion seconds!  In fact, by my calculations, I will turn 2 billion seconds old during the last week of May of this year at 63 years, 138 days +/- a day (leaving room for a miscounted leap day in there somewhere). Taking in account my conception being about 9 months before my birth, I have already exceeded the two billion mark!

A billion minutes ago, Justin "Martyr," a disciple of John the Apostle was preaching in the year AD 114.  Two billion minutes ago, in 1787 BC, Abraham lived with his grandsons Jacob and Esau when they were in their teens to mid twenties (allowing for differences in calendars).  What a difference of a factor of 60 can do!

It makes one consider just how big a billion is, but then, we are each made up of trillions of cells.  Numbers that big are dependent on what you are measuring.  It scares me that our national debt is measured in Trillions.. that is, a thousand billions!

But back to time... A million seconds passes really fast.  We each see our millionth second when we are ll.574 days old.  By the time your finished first grade, you had passed 200,000,000 seconds!  What a milestone! And you thought your daddy was old.  By 500 million seconds you were getting that driving license!  And then you just lost track of time.  Or maybe, you are in those years right now!  I forget who may be reading this blog.

So, to my young readers, count your days if you want, but leave the seconds to us old folks.  Time is, after all relative.  It passes a lot faster -- or seems to so, the older we get.  In my memory, things get crunched so as to seem like they were just last year, when they happened when I was half my present age.

Savor the days you have, and make lasting memories.  You don't want to forget the "good old days" as you age.  Even "bad memories" give you a perspective on life, so learn to get as much as you can out of them.

My goodness, I seem to have written quite a bit and said very little.  As the clock ticks away the seconds -- yes I have at least one clock in the house that ticks -- I will remember the days, some of the minutes, and an occasional second that made a difference.  But now, I think I will post this and go to bed.

So, What do I know?

Time is precious.  There are only 86,400 seconds and we sleep through about 30 thousand of them!  But then, we live to be over two billion seconds old!  At least on an average, we do.

Though time really is "relative," in everyday life each minute is the same for everyone in the world.  It comes, and it goes.  What we make of it is up to us.

Tomorrow, back to more practical things.

Prime Factors: Part 6

Prime Factors:

The Chart

Finally, Down to the basic formulas.  In this chart, or more specifically, diagram, one can find all the parts of the formulas used to determine whether a large number is divisible by a particular prime number.

Usually, a large number is reduced first by dividing by 2, 3, 5 and 7.  Then, an estimate of the square root of the remaining number gives the upper limit for searching for prime factors.  The chart works for the double digit primes as well as for "03" and "07."

The "rule" of the "sum of the digits" for the 3 can be derived from the X3 quadrant, assuming the format of "03."  Likewise, plugging "07" into the X7 quadrant gives the "trick" or "rule" for finding out if a number is divisible by that prime number.

To review:

For divisibility

 by 2: Look for the even number on the end.

 by 3: The sum of the digits will be divisible by 3.

 by 5: Look for the 5 (or 0, but see 2) at the end of the number.

 by 7: Using the formula for the X7 primes: For "07"10x+n = x-(n*(0+2))

          That is, for divisibility by 7, for any number "10x+n" take twice n from x until you get to a multiple of 7.

Double digit primes below 100 are on the chart.  Larger primes can be added remembering to put them in the right "family."

101 is in the X1 family, Therefore, plugging it into the formula: 10-(1*10) => K*X1.  Of course, with the zero in the middle, numbers up to 10,000 are easy to spot as multiples of this prime!

8989 => 898-90 = 808. Of course, that choice was influenced by my building the chart.  I put up a reduplicative whole number!

Really big odd numbers always give us a challenge.  Once divisibility by 3 and 5 are eliminated, the chart come into play.  I am going to blindly punch in 7 digits, making sure that the end number is not even or a 5.

7351613.  7+3+5+1+6+1+3 = 26.  So, not divisible by 3.

Let us try 7.

7351613 => 735161-(3*2)=> 735155=> 735155-(5*2)=> 735145=> 7351-(5*2)=> 734-2=> 732. 

732 is not divisible by 7.

What about 11?

7351613 => 735161-3=> 73515-8=> 73507=>7350-7=> 7343=> 431=> 42.  Nope.

So, 13?

7351613 => 735161+(3*4) = 735173 => 73517+(3*4) = 73539 => 7353+(9*4) = 7381 => 738+4 =

742 => 74+(2*4) = 74+8 = 82.  13 does not work. 

Which of the two attempts got closer?  42 is 2 away from 44, while 82 is 4 away from 78.  Perhaps the X1 family offers a better chance.

What of 31, which requires a multiplier of 3, two away from 1.

7351613 => 735161-3*6=> 7350-9*3=> 732-3*3=> 72-1*3 = 69.  69 = 62+7.  Further away!

So, let us first estimate a square root by dividing the seven digits into groups of two starting from the right.

7 35 16 13.  The square root will be a number between 2000 and 3000.  2,500*2,500 = 6,250,000.

So, my guess is that the limit of possibilities lay somewhere around 2750.  Ending in a 3, the number is not an exact square.  A lot of work to go, but I think I've written enough.

So, what do I know?

The search for primes continues past 97.  But, placing all possible primes in the four "families" makes the job a bit easier.  Once the one gets past division by 7, at least without a calculator, having derived a family formula helps to discover larger primes by elimination.

I have not memorized the above chart, but I did use simple math to derive the formulae.  The exercise of my brain has staved off mental deficiencies for at least a few months!

I don't know if finding the largest prime number yet is worth the effort, but perhaps using this chart will help someone in the search.

I know that the chart will help an inquiring mind find proof that 7,351,613 is not prime.  I challenge a reader, whether a friend or an random reader, to send me the prime factor.  You DO have a clue.

  

Prime Factors: Part 5

Prime Factors:

The 19 Family

Rightfully, this should be called the "9" Family, but alas, 9 is not a prime number.  However, as a member of the 19 family, 9 come holds an honored place in mathematics.  In the  similar "warped mirror" 11 family the 1 performed a similar place of honor.  1 is not a prime, for it does not have another factor.

Like 1, whose difference from 0 is a +1, 9 is different from 10 by 1, but it is a negative difference.  As such, the key to discerning the 19 family of primes is seeing a difference of 1 per 10 following a plus sign.  As you may recall, with the 11 family the multiplier followed a minus sign.

And so, starting with 19, let's build that table:

19 => 1+(9*2) = 1+18 = 19
29 => 2+(9*3) = 2+27 = 29

and so on. I will throw in the "09" to show the full sequence.

09 => 0+(9*1) = 0+9  =  9 -- not prime
19 => 1+(9*2) = 1+18 = 19
29 => 2+(9*3) = 2+27 = 29
39 => 3+(9*4) = 3+36 = 39 -- not prime
49 => 4+(9*5) = 4+45 = 45 -- not prime
59 => 5+(9*6) = 5+54 = 59
69 => 6+(9*7) = 6+63 = 69 -- not prime
79 => 7+(9*8) = 7+72 = 79
89 => 8+(9*9) = 9+81 = 89
99 => 9+(9*10)= 9+90 = 99 -- not prime

Half of the family are not prime numbers.  This no surprise when the "father" of the family is the one odd number among single digits that is a square.  Note that there is one more square in the family: the number 49.  A blessed family indeed.

As before, this blog would not be complete if we did not have and example or two.  I will start with a random 5 digit number and a random family member.

The random number: 54,977 -- my finger bounced on the 7, but let's go with it. Seeing the 54, let's use 59.

54,977 = 5497+(7*6) = 5497+42=5539 => 553+(9*3)=553+27= 580

58 is not divisible by 59, but came so very close!

Now, I will take the number 5497 from step two above and multiply it time 59.

324,323 => 32432+(3*6)=32432+18=32450 => 32450=>3245
3245 => 324+(5*6) = 324+30 = 354 => 35+(4*6)= 35+24 = 59.

That took some time, but we didn't have to use a calculator.  Well, I used one to get the original number, but I didn't have to.  Working with the "9 family" all I had to do was multiply by 60 and subtract the original number.

      5,497

         x 60

   329,820

    - 5,497

   324,323

I love "short cuts."  Looking at the number another way, I could have taken 5500*59 - 3*59.  That is, (5500-3)(59).  But that would have taken longer.

Now, no more nonsense.  One more example, using a manufactured number multiplying some simple primes, the last one being an unknown member of the 19 family.


33,495/5 =
6,699/11 = 
609/3    =
203 => 20+(3*3) = 29.  203/29 = 7

5*11*3*7*29 = 33,495.

Okay, I could have used the test from the 7 family first, but  please note:

20+(3*x) = 29
           3x = 29-20 = 9
             x = 3

So, what do I know?

The patterns for the "families" of prime numbers are easy to discern.  There are four such families.  The 3 and 7 families are "mirror images" of each other, as are the 11 and 19 families.  I will include a neat chart on my next blog post.

For now, here is the chart for the "19" family.


19 => 1+(9*2) = 1+18 = 19
29 => 2+(9*3) = 2+27 = 29
59 => 5+(9*6) = 5+54 = 59
79 => 7+(9*8) = 7+72 = 79
89 => 8+(9*9) = 9+81 = 89

Note the form:

X+9 = (X/10)+[9*(X+1)]

Prime Factors: Part 4

Prime Factors:

The Royal Sevens

It is said by those who study such things that the number seven is the number of completion.  This is, of course, based on the creation of the world being finished in six days and God resting on the seventh day.  I accept this, for it was codified along with the fourth commandment on tablets of stone.

It is for this reason that I have decided to call the related primes ending in the number 7 the "Royal Family" of primes.  Seven is a combination of three and the first square number, that is, the number 4.

However, the important feature is the relationship with the number 5.  It just so happens that 7 is a "mirror image" of 3. Where 3 is two less that 5, 7 is two more.  This translates into a chart very much like that of the threes.

However, in this case, subtraction back to the 5 will get the answer we need. I will start with 17 and backtrack back to 7.

What relationship does 1 have to 7 to get a something divisible by 17.  It so happens that 5 times 7 is 35 and 2 times 17 is 34.  That looks good, since they are just 1 apart.

The bigger number is the 35, so the operation will have to be subtraction.  It will look odd, but the multiples in our present format will all be negative integers.  But, then I know that -1 is just as good a factor as +1.

Here is the chart (with seven as "07" inserted)

07 => 0-(7*2) = 0-14  =  -14
17 => 1-(7*5) = 1-35  =  -34
27 => 2-(7*8) = 2-56  =  -54    
37 => 3-(7*11)= 3-77  =  -74
47 => 4-(7*14)= 4-98  =  -94
57 => 5-(7*17)= 5-119 = -114
67 => 6-(7*20)= 6-140 = -134
77 => 7-(7*23)= 7-161 = -154
87 => 8-(7*26)= 8-186 = -174
97 => 9-(7*29)= 9-203 = -194

Just like with the threes in the mirror, the multipliers increase by 3.  In this case, they start with 2 instead of 1, so the "mirror" is a bit warped.  The columns are not as neat as the numbers on the left don't match those on the right.  We actually have to do some math to see what is happening here.

When you put a candidate for division in the left column the format makes more sense.  However, with the multipliers getting into the twenties, we need to step back and concentrate on getting our math right.

Let's pick 37 as our prime.  I will use a very simple multiple of 37 to illustrate the formula:

111 => 11-(1*11) = 11-11 = 0. As a matter of fact, 37 times 3 is 111.

To get a more obvious answer, let us use the square of 37.

1369 => 136-(9*11) = 136-99 = 37.  Yep, 37 is a factor of 37!

One more thing, since 7 is 5+2, if you need to multiply something by 7, you only need to know the twos table and addition.

789 x 7 = ???
789 x 10 = 7890
7890/2 = 3945
789*2  = 1578

                          3945                         

+1578

    5523 

Or, 7 = 10-3.  With 3 = 2+1. Good old twos tables and subtract. Or, of course, just learn the seven's table. Oh, yeah, you can use a calculator!


So, what do I know?

Because of the special relationship between 5 and 2, the tables for the 3 and 7 families look much alike.  Practically mirror images, one finds a "positive" difference by adding, while the other uses the "negative" difference by subtracting.  Each progresses by a factor of 3 between decades.

And so, the chart with only the primes:

07 => 0-(7*2) = 0 -14 =  -14
17 => 1-(7*5) = 1 -35 =  -34
37 => 3-(7*11)= 3 -77 =  -74
47 => 4-(7*14)= 4 -98 =  -94
67 => 6-(7*20)= 6-140 = -134
87 => 8-(7*26)= 8-186 = -174

Prime Factors: Part 3

Prime Numbers:

The Amazing Mr. Five.

Of the prime numbers under 10, only 5 stands alone.  He has no other primes in his family.  As such, recognizing multiples are easy. Just look for a 5 at the end.

This is true, because any number that is divisible by 10 -- that is has a 0 at the end -- divisible by both 2 and 5.  Further determination is made after removing the 0.

For instance:

1234560 is divisible by 2 and 5 (that is, by 10). That removes 5 from the running, at least temporarily.  Here is the progression:

1234560/2*5 = 123456
123456/2    = 61728
61728/2     = 30864
30864/2     = 15432
15432/2     = 7716
7716/2      = 3858
3858/2      = 1929
1929/3      = 543
543/3       = 271

My guess is that 271 is prime. It is one over 270, which has obvious factors.  Further investigation bears me out.

The "Five and Dime" Store.

Being that 5*2 is 10, and inversely 10/2 is 5, multiplying and dividing by 5 is easy. You only need to multiply by 10  and divide by 2 to get any number multiple of 5.

123*5 = 123*10/2 = 1230/2 = 615

Using the decimal point, it works the other way:

(123/2)(10) =61.5*10 = 615

Finally, it is easy to spot numbers divisible by 25 (5^2) and 125 (5^3). This is because 100 and 1000 are multiples of 10.

For 25, there are 3 multiples: 25, 50 and 75.  For example:

123,450 is divisible by 5 and 25. 50 is 2*5*5. So 2, 5, 10 and 25 are factored out.

123,450/2*5 = 12,345
12,345/5    = 6,171
6,171/3     = 2057, an odd number.  Is it prime? Stay tuned.

Another, example:

987,625 is divisible by 125 (5*25=625).  This would also place a 5 on the end of the next factor, raising the power by another 5.

8*125 = 2^3*5^2 = 10^2 = 1000

This leaves 987,000.  Removing the zeros, we then look for the factors, if any, of 987.  3 works, though 9 doesn't.

987/3 = 329.

So, dividing an odd by an odd, we've ended up with an odd number.  Further calculations are needed, but our odd number is within reach.  20*20 is 400, setting a limit.

So, what do I know?

I know that 5 and 10 are closely related by the prime number 2.  And this makes multiplying and dividing by 5 very easy. There are no other prime numbers related to 5, making the possibility of a random number being prime at less than 25%.

Even so, mathematically, there are an infinite number of prime numbers.  This is because "infinity" can not be divided. There are an infinite number of fractions.

Very odd, so to speak, that when not all odd numbers are prime, there can be an infinite number of primes anyway.

I know my brain hurts contemplating that.

To restate the FACTS:

1. The natural number 5 is prime.

2. 5 = 10/2

3. 5*2 = 10

4. Therefore, all numbers ending in 5 or 0 are multiples of 5.

Prime Factors: Part 2

Prime Factors:

The 3 family

We have taken it as a maxim that the sum of the digits will let us know if a number is divisible by 3.  But when we put it with its family, we see a clear pattern, while at the same time "proving" the maxim.

03 => 0+(3*1) = 3.

That is quite obvious, but using the formula on bigger numbers, removing the digit to the far right and adding it to  a tenth of the digits to its left, we can see WHY our maxim is true:

2853 => 285-9 = 276 => 27-18 = 9 (3*3)

2+8+5+3 = 18 (3*6); if in doubt, keep going: 18 => 1+8 = 9

Amazingly, the multipliers progress by 3, and in the third and fourth columns below, the family relationship is abundantly clear!


03 => 0+(3*1)   = 0+03 = 03
13 => 1+(3*4)   = 1+12 = 13
23 => 2+(3*7)   = 2+21 = 23
33 => 3+(3*10) = 3+30 = 33*
43 => 4+(3*13) = 4+39 = 43
53 => 5+(3*16) = 5+48 = 53
63 => 6+(3*19) = 6+57 = 63*
73 => 7+(3+22) = 7+66 = 73
83 => 8+(3*25) = 8+75 = 83
93 => 9+(3*28) = 9+84 = 93*

Building this table is easy, especially when you remember to add 3 to the multiplier, but then again, this is the "3 family."

Column 1, start with 0, add one per row.
Column 2, 3 all the way down
Column 3, just like #1 followed by "+"
Column 4, just like 2, but proceed with "("
Column 5, start with 1, add 3 per row
Column 6, same as #3
Column 7, start with 03, add 9 per row. Close with ")"
Column 7a repeats the 3 and goes on to 8
Column 7b starts with {1}3 and goes down to {0}4
Column 8, start with 03 and add 10 per row.

Wow.  I'm not sure if that is "easy" or not.  The second half presents us with a problem of large numbers as factors.  However, notice that one of them is 13 (in the family).

A second, easier, factor presents itself on down the line.  Since 30 is obviously in the family, and when added to 13 gives us 43, we have an alternative to multiplying by 13.

43 => 4-(3*30) = 4-90 = -86  (-2*43).

The other factors for primes -- 53, 73 and 83 -- use factorable numbers: 16, 22, and 25 (two squares and 2*11).  The 3 family is a math friendly family all around.

So, what do I know?

I know that fabulous patterns emerge if one looks closely.  These patterns are demonstratable using the properties of math, and can be proven.  However, it is much easier just to experiment with the results.


Here is the "chart" with only the primes:

03 => 0+(3*1)  = 0+03 = 03 (special case 1*)
13 => 1+(3*4)  = 1+12 = 13
23 => 2+(3*7)  = 2+21 = 23

43 => 4+(3*13) = 4+39 = 43 (special case 2*)
53 => 5+(3*16) = 5+48 = 53 (16 = 4*4)

73 => 7+(3+22) = 7+66 = 73 (22 = 2*11)
83 => 8+(3*25) = 8+75 = 83 (25 = 5*5)

*1 Adding the digits and reducing to a recognizable multiple of 3 works for this.  The square of 3 is 9, leaving a bonus indicator for divisibility by 9.  If the number is even, then both 2 and 3 are prime factors.

*2 Beyond 39 (3*13), so an alternative using 30 (3*10) presents itself:

43 => 4-(3*30) = 4-90 = -86 (-2*43)

Happy calculating!

Prime Factors

Though multiplication and its inverse, division, are performed easily with all whole numbers, the principle of equivalency leaves an easier solution to working with many of them.  The practice of "finding the factors" need not end with big numbers. In taking on large numbers, it is perfectly alright to break them up into the numbers from which they came by way of multiplication.

Even Numbers.

One half of all natural numbers are "even," this is to say they can be divided by the number 2.  Two is the first "prime" factor.  Almost everyone remembers the cheer: "Two, four, six, eight, who do we appreciate?"  Well, those are the first four "even" numbers, which hypothetically go on forever.  Along with the the beginning whole number "0," these provide the clue that a number has at least three factors: 1, 2 and the number in question.

Whether you multiply an even or an odd number by an even number, the answer will be even.

For example:  1234 is an even number, and therefor is not a prime number.  Its factors are 1, 2 and 617.  Is 617 a prime number?  Well, it isn't even.  Two is the only "even" prime number, so let us move on to other "prime suspects"

Odd Numbers

The other half of all natural numbers are odd.  This does not mean they are prime, but it makes task of finding big prime numbers a little easier.  I am not one to pursue such a task.  Suffice it to say that an odd number needs to be approached with care.  It can have hidden factors just waiting to be discovered.

Taking the factor of 1234 -- 617 -- the first thing is to find that number's square root.  This is best done with a calculator, but I recognize this as close to 25x25, that is 625.  This sets the limit.  A prime factor would have to be under 25, but not by much.  23x23 has a product 529.

Dropping back to the basics, then, we start with 3.  By theorem, the sum of the digits of any number must be divisible by 3 if the number is divisible by 3. 6+1+7 = 14.  14 is not divisible by 3.

The next prime number is 5. "Counting by fives" is easy, and it reveals to numbers, one even and one odd.  Every even number ending in 0 is divisible by 2 and 5.  This is two for one!  So, 617 is not divisible by 5 either.

The last odd number under 10 is the number 7.  There is no easy way to tell if a number is divisible by 7.  In the case of 617, we see a seven, but the first two numbers return a remainder of 5, leaving 57 (not divisible by seven.  Knowing the multiples of 7 up to at least 9x7 is advisable.

Note that multiplying an odd number by an odd number will get an odd number:

1x1 = 1 5x5 = 25 9x9 = 81
3x3 = 9 5x7 = 35
3x5 =15 5x9 = 45
3x7 =21 7x7 = 49
3x9 =27 7x9 = 63

With the factors 11, 13, 17, 19, and 23, only 11x17 even remotely comes close. However though 11x17 ends in 7, it is far from 617. What about 11x27?  That gets closer, but is far short as well (270 + 27 = 297).

So, 617 is indeed prime.

The most important products to know are those of the prime numbers 2, 3, 5 and 7.  Note, standing by itself is the number 49!  The "new" answer to the universal question!

2|  4

3|  6  9

5| 10 15 25

7| 14 21 35 49

      2  3  5  7 

Finding prime factors:

Starting with 2, what are the prime factors of 7,984,356 (a totally random seven digit number!)

Immediately 2 "works."  Trying 4, we get 1,996,089. Not even, so on to 3.  These digits are 1+9+9+6+0+8+9.  Added this gives us 42; reducing further to 6

So, with factors 2x2x3, we can divide by 12 to get 666,563  Is ths as far as we can go?  Not divisible by 5, so we try 7, 11, 13 and 17.  Seventeen works, yielding 39,139.

This leaves factors of 1, 2, 3, 4, 6, 12, 17, and 39,139.

Using a handy calculator, I know that the square of that large number is just under 198.  197 is a prime number having as its square 38,809. 197 times 199, the next prime number, equals 39,203. This means there are no more prime factors of our chosen number.

The prime factors of 1,996,089 are 2,3,17 and 39,139.


So, what do I know?

Odd x odd = Odd number
Odd x even = Even number
Odd + even = Odd number
Odd + odd = Even number

A prime number is a natural number that has exactly two natural divisors: 1 and itself.

2, 3, 5 and 7 are prime numbers under 10.

Even number 2 is the powerhouse of the primes, affecting ALL even numbers.

Odd number 3 can be seen to be a factor if the sum of the digits add up to a number divisible by 3.

Odd number 5 shouts out from the end of one of its products.  If the even number 0 is there, the 2 and 5 are instantly known.

Seven times seven is forty-nine (7x7=49). Every digit between 1 and 9 shows up as the final digit of multiples of 7.  So, don't look for an easy out here.

All prime numbers larger than 2 are odd numbers. About one in four numbers is prime (in the first 200 natural numbers, at least).

The factors of any number start with one and end with the square root of that number.   

Inverse Behavior

Everything in math has an "opposite."  It is sort of like a religion, with its light and dark, yin and yang, or whatever.  In math, opposites "cancel" each other out.  Sort of like matter and antimatter.  But in doing so, these reactions make math a lot easier.

There are four things one can do with numbers: Add, subtract, multiply and divide.  The latter two are "short cuts" of the former two. That is to say, multiplication is just adding; and division is just subtracting until you come out even or with something left over.

Addition and Subtraction

Opposite Numbers

So, just how do the "inverse" reactions help in math?  First, numbers live in "parallel universes" on either side of the "neutral zone," aka zero.

    -10 -9 -8 -7 -6  -5 -4 -3 -2 -1|    |+1 +2 +3 +4 +5 +6  +7 +8 +9 +10

<-------------------------------------- 0--------------------------------------------->

Addition

The trouble with numbers, is that they only get "stronger" the farther they get from the "neutral zone." This is a great truth, for a number can increase indefinitely by adding just one unit at a time.  The end is not in site, for it is always beyond the biggest number.

But many times, that power is diminished as necessary reversals happen.  If too much momentum is lost, the number changes sides and wears the sign of the opposing "universe"

Inverse reaction (subtraction)

 But for this illustration, numbers are stubborn, they "teleport" into the other universe!  But alas, they are inexplicitly drawn to their counterpart, resulting in annihilation of both of them.

-6+6

=0

Fractions

Trying to get to the other side of the zero by just turning around is not a good path for an signed number either.  As they try to approach zero, they get weaker and weaker. Finally, they reach the "event horizon," (+/-1) and they begin to break into pieces, but never quite die.  Just as they could always advance away from zero, they will be stuck between +/-1 and 0 unless they again turn around and progress away from "absolute" zero.

  < 1/20  2/19  3/18  4/17  5/16  6/15  7/14  8/13  9/12 10/11 

0=======================1

Scotty, now would be a good time.  Beam me up, now!

Back to reality

Alright, enough fun.  Back to reality.  Hopefully, it might help someone "see" how negative numbers fit in to the scheme of things.  You really can add one more forever in each direction.  That is known as infinity.  It is also true that there are an infinite number of divisions between each of the whole numbers and their opposing negative selves.

Going in a negative direction along the line is called Subtraction, the inverse of Addition.  If one were to go over into the opposing "universe" the inverse would be Addition.  When you just lay a negative number next to a positive number, you have a "subtraction" problem using the bigger of the two.  The sign of the bigger number "wins" as the "difference" is determined,'

-8+5 = -(8-5) = -(3) = -3    Notice the subtraction was done inside a negative parenthesis.

+17-13 = +(17-13) = +4     Just for consistency, the same format is used. Positive wins!


I've read that the first functioning "computer" did not add, but subtracted.  It was a mechanical "difference" machine.  That brings us to Division.  Not like a battalion in the war of numbers, but in the process of dividing whole numbers by "natural" numbers.  Only these numbers can be "trusted" for they all follow a fast rule:  To be "rational" a number must be able to exist in the following form

a

b

(where a is an integer and b is a natural number)

What is a natural number?  It is any positive integer.  This excludes 0. which is neutral.  In other words, you cannot divide by 0.  In fact, you cannot divide by anything but a positive integer.

Integers:  {. . . -3, -2, -1, 0, +1,+2, +3 . . .}

Called "signed" numbers, these are all numerals, be they negative or positive.  It includes the "supernatural" number 0  -- just kidding.  As opposed to "natural,"  zero stands apart, but has powers beyond ordinary integers.

Whole Numbers:  {0, 1, 2, 3 ...}

Zero holds it's own as a "whole" number.  It is healthy, and stands in places all other numbers do, but without any voice.  It is barely noticed.  But when missing, the other numbers are greatly diminished.  The zero is second only to the ruling integer, being his "right hand" man.  I know, weak analogy, but alas, it fits.

Natural Numbers:  {1, 2, 3 ...}

One thing zero is forbidden to do is divide.  It is against all the "natural" laws of numbers. It is beyond logic to say that a whole number can be divided into pieces that have NO value.  If  zero were used as a divisor, then math would be impossible.  It is easy to see this.  Consider this:

For x = 0, verify  2x/x = 1.

2x/x = 1

2x*(1/2)= 1*(1/2)

x = 1/2

0  =/= 1/2

2*(x/x) = 1

2*1 = 1

2 =/= 1

Since by definition x/x  is always 1, and 0/x is by definition 0, then 0/0 would become 1!  Something out of nothing?  Nope, not happening.

Division cannot be by a fraction either. In the case of "division by a fraction" the inverse of division is used.  That is, the "fraction" in the denominator is turned over to become its own "reciprocal" and then the inverse of division, that is, multiplication, takes over.

So, what do I know?

+A-A = -A+A = 0

A-A = 0

A/A = 1

A = A/1 (a rational number)

1/A * A/1 = 1

A/B * B/A = 1 

Next: In Their Prime: Factors that matter

Making Arrangements

Okay, I've used some of this earlier, but to increase my knowledge, I  have had to "relearn" he technical terms.  While the labels we put on don't really matter, the concepts do.

The most important thing I like to stress about math is knowing how things work.  However, if someone ever asks, there are three "properties" in math that can make things a whole lot easier for those wishing to take control of the numbers they face every day.

Commutative Property

A+B = B+A

A*B = B*A

Yes, this is the very first thing of which I said I am quite certain.  It is one of the undeniable facts of nature.  When it comes to numbers, it makes no difference what order they are in when adding or multiplying.  If you are ever asked what this property is called, just remember the "commute" to work (or school, or wherever).  The numbers are just "moved around."

For this reason, those really scary tables with scores of numbers in them can be reduced in half!  It is a very convenient and time saving fact.  Besides that, it helps you around most multiplication roadblocks.  When memorizing the tables, it is no coincidence that you get the idea that you've seen that "answer" before.  I like to just use the bottom triangular half of the table.

One number says it all on the multiplication table: 49

I almost made that number a separate blog.  It is the square of the number 7, which means it is 7x7.  Anyone who has ever memorized the dreaded "Times Tables" has faced the difficulty of "the sevens."  It is hard to visualize, and difficult to "count by," multiples of 7.  They just about HAVE to be memorized.  That is why the commutative property is so cool.  You don't have to use the "sevens" if you use the OTHER number instead.  Except for one time.  Memorize the fact that 7x7 = 49.

Associative Property

(A+B)+C = A+(B+C)

The associative property is an extension of the commutative property.  It takes advantage of the fact that we can work only with two numbers at a time.  By getting to "round" numbers one can be more confident with the answer.

2+47+64+31+72+99+44+63 = (47+63)+(31+99)+(64+2)+44 
= 110 + 130 + 66 + 44 = 240 + 110 = 350

7 x 9 x 3 x 2 x 5 x 6 = (7x3)x(9x6)x(2x5) = 21 x 54 x 10 = 540 x 21 = ????

So, with multiplication, associative properties only get you so far.  As numbers get larger, another concept is needed to reach the answer. Old fashioned arithmetic has us stack the numbers and then distribute the task using on digit at a time.  That brings us to the next property.

Distributive Property

A(B+C) = AxB + AxC

This property, in which multiplication is spread out over several steps, is the method used practically on scrap paper across the world.  Take the unsolved product of  "540 x 21" for example.

540

x21

The process of distribution is not evident, but it happens as you multiply by 1 and then by 20.  Written in distributive form it looks like this (54)(20 +1). 

(540)(20+1) = (540*20)+(540*1) = 10800 + 540 = 11,340 

Where the distributive property comes in real handy is when calculating products near 10 (8,9,11 and 12).  Though 11 and 12 are derivative of basic facts, they can be bypassed when using a little math. Some math "short cuts" are as follows:

3 = 2 + 1

7 = 5 + 2

8 = 10-2

9 = 10-1

11= 10+1

12 = 10+2

So the take away is that using this property of multiplication, one can "divide and conquer."  One example and then I will let the reader's brain rest.  I will now "randomly" chose two large numbers to multiply.

Let me see, lets do a three digit number by a two digit number, but a little harder than the one above.  I'm randomly picking "794" and "69."

So, a slip of a finger (I meant "8") and blind luck give me 794 x 69.

794 x 69 = (800 - 6)(70 - 1) = [(800x70) -  (6x70)] - 800 -(-6) {I confess, I took a short cut here}

= 56000 - 420 - 800 + 6 = 56,006 - 1220 = 54,786

I really didn't mean to be that complicated, but this is about "what I know."  So using both distributive and associative properties, I took the long way around.  Using rounding, I was able to redistribute the numbers and work inside my head.  

Using the old fashioned arithmetic would be faster, but I showed that I know these three handy properties of numbers!

So, what do I know?

AxB = BxA

A+B = A+B

(A+B)+C = A+(B+C)

A(B+C) = AxB + AxC

Next: Inverse Behavior

Strange Facts about Exponents

Sometimes things just have to be proven.

We are taught to accept some things as fact, but they are not as evident as the facts I know intuitively.  These are the "math facts" that must be derived from the basic things.

This can be done using the "shorthand" known as the exponent.  It is like a "component" but set over to the side.  It is not part of the number but is a reminder that the number has been acted upon by multiplying it by itself.  The common exponents are the "square" [2], and the "cube" [3], drawn from the construction industry: flat panel and a "box."

When working with "whole" numbers, that is to say, zero and all the numbers we count with (aka "natural numbers), the exponent tell us at a glance that things just got a lot bigger.

Exponents also work with what are called "rational" numbers and even with 'irrational" numbers.  And lo, and behold, exponents work with "imaginary numbers" -- all these are things I know, but they need to be shown to be true.

Squares, Cubes and Beyond

First an illustration:

Consider that you are going to build a box, not quite as big as Noah's box (ark) but perhaps about the size of what that ancient Jewish guy, Bezaleel, did with some acacia wood (the box: the ark).  We'll skip the gold.  Anyway, the point is, measurements are made for the box that included its "footprint" (length and width) and its "body" (adding height).  The ends of the box were square, having the same height and width.

Given a width, call it W, then the board will be cut that wide in two (2) directions to get a square.  Or, in this case, W squared.

W x W = W^2

Well unlike Bezaleel or Noah, let us say the instructions to make the box the same measurements in all three dimensions -- put on your 3D glasses here.

Then, keep the measuring using the same mark on the stick, and you construct a cube!

W*W*W = W^3

No, not that "WWW" -- just a cube.  You know, like the description of the Holiest of Holies.  You don't know about that?  (leave a comment, I'll explain).

If you divide a cube by a square, you're back to a line the length of one of the edges.  So how does that help anybody?

Consider this algebraic express:

w^3/w^2 = W*W*W/W*W.

Going a step further:

W/W * W/W * W = 1 * 1 * W = W

OR

W^2/W^2 * W = W

Now, notice the relationship between the cube and the square:

W^3  = W^1.  W^1 = W^[3-2] or simply W.
W^2

The principle is, when dividing exponents, subtract those in the denominator from those in the numerator.

So, when when using the same base, the form N^D/N^D = N^[D-D] = N^0.  But wait, there is that zero again. N^0 = 1.  No matter what N represents!

All that to get to the first know fact of the day:

N^0 = 1.

Then, what happens if the exponent is larger on the bottom?

N^3   = N^[3-5] = N^[-2].
N^5

What in the world?

Simple, really. Look at it the "long" way:

      1*N*N*N     = 1/N*N.  The N^2 is in the denominator.
N*N*N*N*N*1

Think of it using the number 2.

2^3 = 8
2^2 = 4
2^1 = 2
2^0 = 1
2^[-1] = 1/2
2^[-2] = 1/4
2^[-3] = 1/8


So, what do I know?


N*D = multiplying N times D
But
N^D = multiplying by N, D times

N^0 = 1

N^[-D] =     1   
                 N^D

Coming Next: Making Arrangements

The One and the Only ONE

So if you can't use 0 to solve a multiplication problem, what is there that sort of "disappears" when you multiply or divide?  There is only one answer to that.  The answer is one, and one is the answer.

Here are the facts in algebraic expressions:

1*A = A

A/1 = A

A/A = 1

There you have it, the secret to "higher math".  Well, not really, but it helps a lot in problem solving.

Remember how you can rearrange multipliers?  Well that is where "reciprocals" come in.  Stated with variables that looks like this:

A/B x B/A = 1.

Got that? It looks better with pencil and paper, but my scanner is asleep right now.

Anyway, "A over B" times "B over A" equals "1".

It works out to something like this:

A*B/B*A
= A/A * B/B
= 1 x 1
= 1

Funny how that works out, huh?

Now we can use this unique upright integer to solve a problem.

Solve for A:

3A = 9
3A/3 = 9/3

Let's redo that second line:

(3/3)*A = 9/3
1*A = 3
A = 3

I really do need to get a scratch pad for Windows.  Or Something.

Anyway, here is what I know so far:

A+B = B+A
A-0 = A
A-A = 0
A/A = 1
A*1 = A
and
A*B = B*A
A/B x B/A = 1

Aren't I smart?


Well?

Perhaps I need to check to see if my hat is getting tight.

Tomorrow: Strange Facts about Exponents

Working With Nothing.

The Zero was a great idea.  Take nothing and make something out of it.

No, this is not a theological statement. Nor is it a statement based on logic.  In math the zero represents nothing.  And that is very important in solving equations.  When you have a zero, you have reduced the possibilities by whatever it was you took away to get to that point.

We can illustrate this in two ways:

Negation:  A - A = 0.  Or  +A-A = 0.

Basically, you take something away and you have nothing left over.  In theory this is invoking the "inverse" operation or the "opposite" interger.  But hey, what works is realizing the truth, the absolute truth, that when you take something away, it is gone.

Put a different way, this is saying:

A +/- 0 = A.

That's right, taking away, or adding, nothing leaves you with what you started with.  Here is how it works in an equation:

715 + A = 984.
715 - 715 + A = 984 - 715
0 + A = 984 - 715
A = 269.

In an equation, you MUST take away or add the same thing to both side. Using the inverse, this works with adding back what is has been taken away.

A - 888 = 111
A - 888 + 888 = 111 + 888
A - 0 = 999
A = 999

Sure, most of you look at it and know A is 999.  But your mind is just that fast. It adds back what was taken away to get to the original value of A.

But what if you are doing something for free and told that those in charge were going to pay you twice as much for today's work? Twice nothing is nothing!  Exactly. In math that looks like this:

0 x 2 = 0.

It works that way no matter what you multiply by zero.  So, the way it looks in algebra is:

0 x A = 0

One last thing to remember:  You cannot DIVIDE by 0.

I know this to be a fact, and can prove it.  But to try to explain it using the word "Nothing" gets a bit bizarre.  Take my word for it.

Next: The One and the Only ONE

Getting it togeher

First, I know that mathematics is the purest of sciences.  The basic truths of math cannot be altered.  Surprisingly there are not as many truths as one might think.

First out, though we can make assumptions -- like working in base 10 or base 2 (binary) -- using "algebra" with variables works in whatever number system that is used.  Algebra is  just basic problem solving, balancing an equation.

The Equal sign about says it all -- you have to be "fair" to both sides of the equation.  That has great applications in "real life," but I'll get to that later.

But let us start with the first thing we know.  I will be using letters in place of numbers (called variables) because it does not matter what number you use, the answer will be the same in these equations.

WORKING TOGETHER

A + B = B + A.  

This cannot be denied. When adding things together, it doesn't matter which direction you go.

It comes in handy when adding a lot of numbers together.  Order doesn't matter, so you can regroup so you get to numbers you can work with more easily.  Most people like 10 and 5, so here is an example:

8 + 7 + 1 + 2 + 3
= 3 + 7 + 2 + 8 +1
= 10 + 10 + 1 = 21

A x B = B x A.

Multiplication is just shortcut addition.  In the example above we find 10+10.  That means we have two tens, or 2*10.  So, the direction doesn't matter, and you can group any way you want to.

4 x 8 x 7 x 2 x 3 x 11
= 7x3 x 2x4 x 8x11
= 21 x 8 x 88.
= 21 x 704
= 14080 + 704 = 14,784

Yeah, I know, a student might need scratch paper to see what I did there, but basically, I used arithmetic the old fashioned way.

Next: Working with Nothing - aka Zero

A New Blog?

I sit in my office -- well, in a corner of the dining room -- and wonder how much of what passes as "knowledge" is even verifiable.  It is not that I'm a skeptic, for I relish my faith in the unseen reality all around me, but I also have writer's block

One of the primary rules for writing is "write about what you know."  And so, I lay aside my other long neglected blogs to start a fresh one.  From this point on, I will write about what I know, not what I think.

Well, at least in this blog.

In the coming months I will start with the most obvious things and branch out.  First, since I am what some call a "Math Guy," I will begin with the "purest" of the sciences: Math!

So, I hope this gets someone's attention.  If not, it will break that writer's block to bits!