Could PI Be A Rational Number?

This is not a normal post for me. Sometimes a combination of observations interest me and generate a train of thought that occupies my idle thoughts for a while. It often helps me to try to explain my thoughts as a way to guide my reasoning which is why I'm writing this post.

My mathematical skills are limited and I would guess that a mathematician could easily answer my simple question.

In math, a fractional number with no repeating number or recurring sequence of digits is called irrational. In decimal, PI is an irrational number. My question is: Is there any base/radix where would PI would not be irrational?

My guess is that the answer is no, PI will always be irrational. But working with computers has shown me that numbers that are irrational in decimal are not necessarily irrational in another base and that decimal numbers that are not irrational may be irrational in another base. (See further below for a further explanation of bases).

Here is a simple example comparing a fraction (one third) in decimal (base 10) and the same fraction (one third) in trinary (base 3).

The decimal fraction 1/3₁₀ is written in decimal as 0.3333...₁₀. The digits 3333... at the end recur an infinite number of times. When dealing with numbers in multiple bases the base is written as a subscript. The equivalent of the decimal fraction for one third (1/3₁₀) is written in trinary as 1/10₃ (again, see below). So one third is written in decimal as 1/10₁₀ or 0.33333₁₀ and in trinary the same one third is written as 1/10₃ or 0.1₃.

This is a simple example of an irrational fraction in decimal that is a simple number in trinary. My guess is that PI is probably always an irrational number, but I would like to hear why that has to be true. If there is a base where PI is not irrational, does that generate additional questions or novel solutions to other questions?

If there is a base where PI is not rational I'm guessing it would be a base that is a prime number or some exotic, non-integer base.

If you are interested, here are some further explanations of number bases.

As we learned in grade school, decimal numbers are made up of 10 digits, 0 thru 9.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, etc

Numbers in binary (base 2) are made up of digits 0 thru 1 and numbers in trinary (base 3) are made up of digits 0 thru 2. There are many other commonly used bases such a octal (0 thru 7) and hexadecimal (0 thru F). I would guess there are an infinite number of bases and some bases are not just sequences of integers.

When numbers are written the base can be shown with a subscript. 1₂ means binary, 1₃ means trinary and 1₁₀ means decimal. A number without a subscript is assumed to be decimal.

Here are the beginning numbers for decimal, binary, trinary, octal, hexadecimal so you can see the differences. Octal and hexadecimal are basically just different (more compact) ways to represent binary numbers and are shown here just because I wanted to show them. Decimal and trinary were used in discussions above.

decimal₁₀  binary₂  trinary₃  octal₈  hexadecimal₁₆
1        1      1     1     1
2       10      2     2     2
3       11     10     3     3
4      100     11     4     4
5      101     12     5     5
6      110     20     6     6
7      111     21     7     7
8     1000     22    10     8
9     1001    100    11     9
10    1010    101    12     A
11    1011    102    13     B
12    1100    110    14     C
13    1101    111    15     D
14    1110    112    16     E
15    1111    120    17     F
16   10000    121    20    10

and so on

While thinking about this problem, I used the simple number line from grade school to explore some issues.  (Sorry, the blog editor doesn't allow accurate placement of the arrow, but you get the idea.)

Below I show decimal 3 1/3 on a decimal number line and a trinary number line.

In school we used decimal integer number lines like
+4     +3     +2     +1     0      -1     -2     -3

|______|______|______|______|______|______|______|
     ↑
3 1/3 or 3.3333... in decimal approximately

We can expand the section of the number line from 4 to 3 and then sub-divide.

+4.0   +3.9   +3.8   +3.7   +3.6   +3.5   +3.4   +3.3   +3.2    +3.1  +3.0

|______|______|______|______|______|______|______|______|______|______|
                                                                                        ↑
                                                                    3 1/3 or 3.3333... in decimal approximately

We can expand the section of the number line from 3.4 to 3.3 and then sub-divide.

+3.4   +3.39  +3.38  +3.37  +3.36  +3.35   +3.34 +3.33  +3.32  +3.31   +3.3

|______|______|______|______|______|______|______|______|______|______|
                                                                                        ↑
                                                               3 1/3 or 3.3333... in decimal approximately

We can keep expanding and sub-dividing, but we would never find a place to draw a line for exactly 3.3333.... We could get closer and closer, but we can never get to decimal 3 and 1/3.

If we try a number line again using trinary, we easily find a precise spot.

+11.0  +10.2  +10.1  +10.0
|______|______|______|
                             ↑
Decimal 3 1/3 in trinary exactly

May The Best Fraudster Win

Elephant Shit

Kari Lake, a Trump acolyte and candidate for governor of Arizona, was asked if she would accept the results of the election.  She replied something like "I expect to win the election and I will accept the results".  When pressed if she would accept the results if she lost, she repeated "I expect to win the election and I will accept the results".

The fact the she basically refused to answer that simple question means she will not accept the results if she is not elected.  She obviously is a believer in "heads I win, tails you lose".

She logically can't object if her opponent makes the same election results statement.  That would mean that no matter who wins, someone is going to claim election fraud.  Of course that is insane. 

If Kari Lake really believes she can only lose if there is fraud, why is she campaigning?  Why try to convince voters you are the better candidate if you believe the other side is going to manufacture votes?  She can't know how many fraudulent votes her opponent could muster. 

If Kari Lake really believes the other side is going to produce fraudulent votes, wouldn't she be smart to tell us how the fraud will be committed so it can be prevented? Or maybe manufacture some fraudulent votes for herself? 

So extending Kari Lake's philosophy she believes both candidates will attempt to commit fraud and the winner will be the candidate that was the better fraudster.   

Kari Lake is wrong and should not be elected.  All candidates should commit to accepting the results of elections and to doing their part to make sure that elections are fair and accurate.   

 

MAGA Voting Sticker

 

On behalf of democracy loving Americans, we thank you!

 

Trump, Stay Healthy

It recently occurred to me that I didn't want anything bad to happen to former president Donald Trump. No cancer, no heart attack and certainly no death. It's not because I'm a supporter of his or I'm such a good person that I'm above evil thoughts.

I wouldn't mind at all if Putin soon met his maker.

I think Donald Trump was certainly the worst president in my lifetime.  He has done immense and possibly lasting harm to this country. I think he is despicable and I would be happy to see him gone.  Except that if anything does go wrong with him (other than his current incurable mental illness) his cult following will amaze us all with the astounding conspiracy theories they will concoct.

All the usual suspects will be accused of somehow causing him harm. The fact that they can't find any evidence that would prove that Democrats caused his heart attack will just convince them there was foul play. Probably some some deep state cabal used the most sophisticated science to clog Trump's arteries. Remember, Trump's world class doctor said Trump was so healthy he could live to be 200 years old.

I'm sure we'll hear that there is proof that Democrats promised aliens from outer space they could use earth as a second home for as long as they like if they would just find an untraceable way to get rid of Trump. Possibly by putting small doses of something in the greasy hamburgers he eats. And that may be one of the saner theories.

If the MAGA cult can work themselves into such a rage that they attack the Capital based on Trump's fake hissy fit over a lost election, what will they do when they are convinced the Dear Leader has been physically harmed?

What we need is for Trump to lead a long, healthy, quiet life at Mara-a-Lago reading his Kim Jung Un love letters. Once the MAGA crowd has moved on to the next autocrat, Trump is also free to move on.

 

Repeal The Second Amendment

The United States Constitution is in many ways an amazing document. It has allowed our country to become the envy of the world, but it was not and is not perfect. There have been amendments to correct some flaws and it is now time to fix a remaining flaw.

The Second Amendment needs to be repealed. Just take it out of the Constitution.

A well regulated Militia, being necessary to the security of a free State,
the right of the people to keep and bear Arms, shall not be infringed.

Repealing the second amendment doesn't mean all guns become illegal and will be confiscated. What it means is that the laws about guns must be legislated from the perspective of how we as a country want them to impact our society and not from the perspective of a constitutional right where we argue about what the founders actually meant. Gun ownership and use should not be some inalienable right. It should not be a foundational part of our country and enshrined in the Constitution. It's inclusion in the Constitution is one of the reasons we have such a terrible gun violence problem.

Most us drive cars and depend on cars to support our daily lives. But we also know that operating a car is not a right. We have to be licensed and provide insurance among other restrictions. There are rules for different types of vehicles and when and how they can be operated. Those rules are determined by legislation to fit our changing needs. For example, self-driving cars will require many new laws and regulations.

This is the way gun laws should be handled. Laws to define how guns can be used for self-defense, hunting, target shooting can be legislated just like laws that govern drivers, driving, cars, trucks, ATVs, motorcycles, etc., and without needing to quote the founding fathers or the Federalist Papers.

I believe the Second Amendment was written to provide the country with a service; citizen soldiers in militias to defend us from foreign threats which we no longer face. We now have well trained people in the military services, National Guard and Reserves that provide for our defense.

For the constitutional originalists/textualists, I don't see anything in the Second Amendment that provides guidance about personal self-defense or suggest that the amendment was meant to provide ready weapons for citizens to overthrow a misguided government. I also don't understand how the initial dependent clause can be ignored, although that is convenient since militias were male only institutions so the Second Amendment wouldn't apply to women.

And how well has the Second Amendment worked? Are we better off having it? How is our record on gun violence and the number of citizens killed by guns compared to the rest of the industrial world? We've been told for years that all we need is more good people with guns. Yet while gun purchases continue to sky rocket, gun violence increases and still more and more people are killed by guns. Clearly more and more guns are not making us safer or decreasing gun violence.

In my city we have an interstate highway where people are wary to drive for fear of being shot for an awkward lane change or just by a stray bullet. Kids are being killed in their homes from stray shots coming from outside their houses. Do we really want to live this way? And the situation keeps getting worse.

What does it say about us that guns are now the leading cause of childhood deaths? We should be ashamed. More importantly, we must do something about that.

Let's repeal the Second Amendment and start creating laws that allow reasonable ownership and use of guns.

 

Self Defense Begins At School

I suggest that the NRA rephrase one of its old sayings.

The only thing that will stop a bad man with a gun in a school, is a child with a gun.

 

Possible Concomitant Consequences

I have not been closely following the case of Kyle Rittenhouse.  Rittenhouse is the Illinois teenager who says he traveled to Wisconsin with his assault rifle to defend people and businesses from rioters.  He is accused of killing two people and wounding another person while he was in Wisconsin.  There is video evidence showing he shot them.

I did take notice when the trial judge, Bruce Schroeder, ruled that prosecutors could not use the term "victim" or "alleged victim" to describe the people who were killed or shot.  I believe I read he thought that calling these people "victims" would bias the jury against Rittenhouse when he has yet to be convicted of anything.  I don't think that explanation makes sense.

Dictionary.com defines victim as "a person who suffers from a destructive or injurious action...".  These victims were injured.  Whether or not they were injured by Rittenhouse, and if they were, whether or not the injuries were legally justified is what this trial is all about.  We won't know if these people were victims of Rittenhouse until the trial is completed so why not use the term "alleged victims".  The term "alleged" is just acknowledging that this is what the trial is going to determine.  

But the judge has legal training and experience that I don't have, so I was ready to give him the benefit of the doubt.  At least I was until he expanded his ruling.

He went on to say the defense could use the terms "rioters", "looters" and "arsonists" to describe the people who were shot if the the defense presented evidence supporting these descriptions.  

What?

These people have not been convicted of any crime so why can they be labelled by the court as law breakers?   The judge is not willing to allow possible negative bias against Rittenhouse during the trial, OK, but he has no problem labeling the victims as felons and giving Rittenhouse backing from the bench for any self-defense claims.  

I thought justice was supposed to be blind. I don't know of any charges the wounded person is facing. If there are any crimes, they would be alleged until he is convicted.  The two victims who were killed can't be tried or convicted.  They can't defend themselves, so why can't we at least call them "alleged victims"?

I would imagine the prosecutors will call Rittenhouse a murderer which presumes there are victims. So if prosecutors can't call them "victims" what do they call them?  They are not plaintiffs.  How about "possible concomitant consequences"?

 This blog was originally written in November, 2021, before Rittenhouse was acquitted.