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.