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

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

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