Category Archives: personal

Information Transfer

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I have some information in my mind.  I want to transfer it to another person’s mind.  This is a surprisingly subtle problem.

Obviously I can walk over to them and start taking to them.  Chances are good that I’ll be able to get a pretty respectable information transfer, especially if we both are competent users of the same spoken language.  But what if we’re not?

If it’s information about the physical world, I can probably gesture with my hands, or maybe even point at stuff if it has to do with things that are close by.  If not, maybe a picture is a better choice, especially if I can use color and/or rudimentary animation (a la comic book panels).  Or perhaps non-language sounds will fit the bill.

If my information is anything more interesting than whether I use 1% or 2% milk on my cereal, chances are good that this will evolve into a conversation, where we are both communicating information with each other (and requesting specific information from each other).  We might even move through different media as we go, adapting to the specific information being transferred at any given time..  But what happens when I only want to share part of the information in my mind?

Now we have something interesting.

Any given piece of information can be communicated in a huge number of ways.  For example, I don’t like dill pickles.  If you asked me, I’d tell you.  But let’s say we go have lunch together, and I leave my pickle spear behind.  The same information was transferred, though certainly with less clarity.  If I have lunch at the same place and I always leave my pickle spear behind, should an attentive server who waits on me numerous times stop bringing me a pickle spear?  Hmmm.  If I go in one day and get served without a spear, is the right thought “the server is stalking me” or just “oh, they must have forgotten”.  Perhaps the right thought is “they’re having supply chain problems, and are out of pickles”.  Hmmm.

Once I got asked “Do you eat ice cream?”  I answered like a smart ass, because a) I’m a smart ass, and b) everyone likes ice cream.  A split second later, my brain actually decoded the information being transferred in that question: “I don’t like ice cream, but I don’t know if you do, so I’m offering to get you some even though I don’t want any.”  Hmmm.

Somewhere in there is a line which delineates what is an appropriate use of transferred information, because we all transfer far more information out of our minds than we think it is acceptable for those around us to use.  Obviously the line is contextual: you would probably expect your spouse to react to the tone of “I’m good, how are you?”, but you would be appalled at the new supermarket checker who took it anything other than literal face value, regardless of inflection, tone, or body language.

To say that another way, it is shared context which determines the meaning of any communication.  Yet another way, language only governs a tiny part of the information transferred through the use of the language.  Let alone the information transferred through other media, like action.  Or through lack of action.  “Implicit consent” is the term for that one: they didn’t object, so it must be okay.

Our society runs on an economic basis.  Those with the money make the rules, those without the money can either accept it or get some money so they can help make rules. But in reality, money is a highly non-interesting economy.  The real economy is based on information.  The relative value is far less tangible, and it inherently has huge differences between different entities.  Having a currency just means you can interact indirectly – the lack of a currency means everyone is directly involved with everyone else.

If I know fact X about person Y, I can barter that information for information about other people.  This is called “gossiping.”  We all do it.  If you listen well, you get a double deal, because not only do you learn about the other people, but you also learn about the person you’re trading with.  What do they consider important?  What information do they value?

Police informants are good at this game, but they have a high-risk job with poor pay.  Which is, in itself, and interesting bit of information to internalize.  What it means is money is not the currency of the information economy, even by proxy.  The currency of the information economy is power, whether you’re in the board room, on the playground at a preschool, or running a siege of an enemy in war.  If you have power, you can buy information.  If you have information, you can trade for power, but the exchange rate is horrible.

What is truly amazing about all of this, is that every two-year-old grasps this concept.  “If you don’t give me the ball, I’ll tell mom you stole candy.”  Knowledge is power, and so it is reasonable to expect those with the most knowledge will have the most power.  But it’s not that way at all.  Usually the opposite.  The people with the least hesitation to use their knowledge to grab power will have the most power as long as the economy is stable.  If the economy is unstable, things are different; knowledge is power at the edges of the stability envelope.  When things are unstable people’s inhibitions fade, and they act more aggression and less restraint.

Between a constantly shifting set of shared contexts with other people (and other entities) and just their sheer number, it is effectively impossible for an individual to do the calculus of information transfer in real time.  Fortunately, the past few years have provided us with a wealth of tools (Facebook, Twitter, etc.) that can not only increase the number of people we’re frequently interacting with, but also increase the shiftiness of those shared contexts.

I will leave as an exercise to the reader what three conclusion paragraphs contain.  : )

Mercenaries

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I am a mercenary.  This is a weird revelation, as “mercenary” usually carries a “solder” connotation with it.  And usually across national boundaries and/or affiliations.  But it perfectly matches what I am: I solve problems the highest bidder can’t solve on their own.

There is no allegiance in either direction; there is nothing beyond utility.  No mercenary dies and has their employer mourn.  No mercenary succeeds and gets an award.

This seems at odds with the “craftsman” label I’d previously adopted as an apt descriptor, but I’m hesitant to disagree.  An traditional artist/craftsman who works on commission is exactly the same: a mercenary sans violence.

I have a certain set of skills.  They don’t happen to be martial in nature, but they are certainly worth people paying to have at their disposal.  Where is the line?  What is the differentiator?

Fractions!

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Fractions can be confusing.  But there there are a few simple rules which encompass everything you could ever want to do with fractions.  In order to use the fraction rules, you have to be able to add, subtract, and multiply ordinary whole numbers.  You’ll also want to keep in mind two rules about arithmetic which are very helpful with fractions.

Arithmetic Rule #1

Any number stays the same when zero is added to it or subtracted from it.

3 = 3 + 0 = 3 - 0

Arithmetic Rule #2

Any number stays the same when multiplied or divided by one.

5 = 5 \times 1 = 5 \div 1

Fractions have a few specific terms which are important to know as well.  Here are the big ones:

Glossary

  • numerator: the number on the top of a fraction
  • denominator: the number on the bottom of a fraction
  • improper fraction: a fraction with a larger numerator than denominator
  • reduced fraction: a fraction that cannot be equivalently written with smaller numerator and denominator
  • mixed fraction: a fraction with a whole number part and a fractional part

On to the rules, of which there are six, but only the last two are particularly interesting.

Fraction Rule #1

Any fraction with 0 as the denominator is undefined (illogical, nonsense, error).

undefined=\frac{2}{0}=\frac{0}{0}

Fraction Rule #2

Any fraction with the same number for both numerator and denominator is equal to 1.

1 = \frac{1}{1} = \frac{8}{8} = \frac{12345}{12345}

Fraction Rule #3

Any fraction with 0 as the numerator is equal to 0 (unless the denominator is also 0 - see rule #1).

0=\frac{0}{1}=\frac{0}{7}=\frac{0}{456}

Fraction Rule #4

Any simple number can be converted to a fraction by placing it over 1.

2 = \frac{2}{1}

This is the same as the division version of arithmetic rule #2, since fractions are just division problems.

Fraction Rule #5

Fractions with the same denominator are added by adding their numerators and copying the denominator. Fractions with different denominators cannot be added together directly (see below).

\frac{1}{5}+\frac{3}{5}=\frac{1+3}{5}=\frac{4}{5}

Fraction Rule #6

Fractions can be multiplied by multiplying the numerators and denominators separately.

\frac{2}{3}\times\frac{1}{6}=\frac{2\times1}{3\times6}=\frac{2}{18}

Using the Rules

Division

Division is often a useful thing to do with fractions, and it’s easy.  Just turn the divisor fraction (the second one) upside-down and multiply.  This is commonly referred to as “invert and multiply”.

\frac{1}{3}\div\frac{2}{5}=\frac{1}{3}\times\frac{5}{2}=\frac{1\times5}{3\times2}=\frac{5}{6}

Improper Fractions

Improper fractions can be made proper by repeatedly pulling 1 (expressed as a fraction) out of the fraction to create a proper mixed fraction.  Remember that every fraction has a “ones” place to the left of the actual fraction to make it a mixed fraction.

\frac{5}{2}=0\frac{5}{2}

The 1 that we want to pull out when converting an improper fraction to proper form is based on the denominator.  By addition rule #1, we can subtract 0 from something without changing it.  And we know that a fraction with the same numerator and denominator is equal to 1 by fraction rule #2.  So if we subtract a one-equivalent fraction and then add 1 back to it, it won’t have changed.

\frac{5}{2}=\frac{5}{2}-\frac{2}{2}+1=\frac{3}{2}+1=1\frac{3}{2}

In this case we still have an improper fraction, so we’ll repeat the process again (and continue until we no longer have an improper fraction).

1\frac{3}{2}=1\frac{3}{2}-\frac{2}{2}+1=1\frac{1}{2}+1=2\frac{1}{2}

So the improper fraction \frac{5}{2} reduces to the proper mixed fraction 2\frac{1}{2}.

Mixed Fractions

Sometimes you have a mixed fraction and want to make it a pure improper fraction.  Get rid of that pesky whole number part.  Improper fractions are usually easier to manipulate than mixed fractions, so it’s common to make everything improper, solve the problem, and then make the answer proper again.

The process of converting a mixed fraction to an improper fraction is quite simple.  Remember that a mixed fraction is the same as the fractional part plus the whole number part.

2\frac{1}{2}=2 + \frac{1}{2}

If the whole number is converted to a fraction, we can just add them and get a fractional result.  We know how to do that because rule #4 says we can slap a 1 under any whole number.  The denominators won’t match, but arithmetic rule #2 says we can multiply by 1 (a special form – namely the denominator over it self – see rule #2) without affecting anything.

2 + \frac{1}{2}=2\times1 + \frac{1}{2}=\frac{2}{1}\times\frac{2}{2}+\frac{1}{2}=\frac{2\times2}{1\times2}+\frac{1}{2}=\frac{4}{2}+\frac{1}{2}=\frac{5}{2}

This process is a special case of adding fractions with different denominators discussed next.

Adding Fractions With Different Denominators

Adding fractions with different denominators requires multiplying one or both fractions by a special form of 1 first (which doesn’t change anything – see arithmetic rule #2). The trick is in picking the right version of 1 to use.  The goal is to get both fractions to have the same denominator.  The easiest way is to multiply each fraction by the denominator of the other fraction over itself (which is a way of writing 1 – see fraction rule #2).  For the problem

\frac{1}{3}+\frac{1}{2}

we want to do this

(\frac{1}{3}\times\frac{2}{2})+(\frac{1}{2}\times\frac{3}{3})

Doing multiplication according to rule #6 yields this

(\frac{1}{3}\times\frac{2}{2})+(\frac{1}{2}\times\frac{3}{3})=\frac{1\times2}{3\times2}+\frac{1\times3}{2\times3}=\frac{2}{6}+\frac{3}{6}

Now that the denominators match (and they always will by following this process) we can add according to rule #5 without difficulty.

\frac{2}{6}+\frac{3}{6}=\frac{2 + 3}{6}=\frac{5}{6}

An there we go, adding two fractions with different denominators by multiplying the parts by a clever form of 1.

\frac{1}{3}+\frac{1}{2}=\frac{5}{6}

Reducing Fractions (Part 1)

Answers to problems, whether in math class or the real world, should almost always be given in reduced fractions.  No one likes to be told \frac{5}{10} when they ask how much pizza is left.  Much better is \frac{1}{2}, but how to get from one to the other?

Since you know that the above rules describe everything you’ll need to do with fractions, it’s probably not surprising that a special form of 1 is integral to the answer. Finding the right form is tricker than for addition above, but there is still a foolproof process.

The first step is to find a non-trivial common factor for the numerator and denominator.  A factor is a number that can be multiplied by another number to get the answer.  For example, 2 and 3 are two factors of 12.  The “non-trivial” means “not including one” since that’s a factor of every number and per addition rule #2 and fraction rule #4 dividing by 1 doesn’t change anything.  We don’t want just any non-trivial factor, however, we need one that is a factor of both the numerator and the denominator (a common factor).

factors of  5: 1, 5
factors of 10: 1, 2, 5, 10

Since 5 is prime, the only options are 1 and itself.  10 has a couple more options, but the only non-trivial common factor of both numerator and denominator is 5.  If you’re reducing a fraction and there are multiple choices for common factors, always pick the largest one.  If you don’t pick the largest one, when you’re done, you only have a partially reduced fraction, not a fully reduced one (and you’ll have to do the process again).  If there aren’t any non-trivial common factors, meaning 1 is the only common factor, then the fraction is already in it’s reduced form.

Now that we have 5 as our factor, we can rewrite our problem by “pulling out” that factor according to rule #6:

\frac{5}{10}=\frac{1\times5}{2\times5}=\frac{1}{2}\times\frac{5}{5}=\frac{1}{2}\times1=\frac{1}{2}

Thus we know that \frac{1}{2} is a reduced version of \frac{5}{10}.  Further, since 5 was the largest common factor, we know that it cannot be reduced any further.

Reducing Fractions (Part 2)

A quick way of describing the above process is to say “divide the same number out of top and bottom and repeat you can’t do it anymore”.  It’s still the same process, but mashing a few steps together and not worrying so much about doing it all in one pass.  As an example, reduce \frac{18}{24}.  Top and bottom are both even, so let’s divide out 2.

\frac{18}{24}=\frac{18\div2}{24\div2}=\frac{9}{12}

Still not reduced, but 3 will divide out of each.

\frac{9}{12}=\frac{9\div3}{12\div3}=\frac{3}{4}

Alright, no other numbers can be divided out, so it must be reduced.  If we’d picked 6 (2\times3=6) initially, we’d have been done in one step, because 6 is the largest common factor of 18 and 24 as discussed above.  Doing it this way meant we had to go through the process a couple times, but we didn’t have to figure out the list of factors and then figure out the largest common factor.

Both approaches are 100% correct and will yield the same result every time.  If you’re doing the reduction in your head while baking, chances are you’ll do the quick’n’dirty way.  If you’re doing the reduction while designing a bridge strut, you’ll probably do it the longer way.  Regardless which you choose, the answer will be the same.

Reducing Proper and Improper Fractions

Reducing fractions and switching from proper and improper forms are all simple conversions.  They’re not changing the fraction’s value, just the way it’s written.  Thus if you have to do both, it’s completely irrelevant which order you do them.

Since a proper fraction’s whole number part is always reduced (whole numbers are as reduced as they come), doing the reduction second is often easier because the numbers in the fraction will be smaller.  But it doesn’t change the math at all; the result is the same.