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)]

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

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