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!