Choose for yourselves this day

In the last few weeks I have written about the basic building blocks of math:

• 4 main operations - add, subtract, multiply and divide
• symbols for those and other things, like the equals sign =
• a primary math tool - the pencil
• numbers themselves

and so on. I suppose one could exhaust one's self in trying to describe everything that makes up the mathematics world. Luckily I can do this as a small part of making a living. And I enjoy learning about math. That makes it a lot more fun. Serving a demanding master (the deadline) is still hard work, but at least the work is tolerable.

You may think math is about numbers ( it is ) but there are plenty of keen math words too.

Remember the patented process I mentioned in the last blog, where software chooses the font that best describes the content of an image? Well I don't have that kind of trickery at my desk. It was a patent application, not reality. And its downside is that it takes away the artistic and creative judgement that provides much of the fun of publishing.

I do my fontificating* the old-fashioned way, by thinking, reviewing choices, imagining the look, and applying the font. Here are a few math words, in fonts and looks that my imagination could cook up:

This is much more fun than creating a serious Glossary, which is what you will get if you click on the link.

When you write a glossary, the words have to spelled correctly, the concepts must be checked over and over again to make sure they 1) are correct, 2) make sense, 3) are illustrated clearly, 4) page references match, etc. It's lots of hard work.

With desktop publishing we can stylize words and give them a look and feel that express their meaning. The look of the words can be what ever you um, err, ah choose!

Why don't you take your one favorite word and just for fun, design it like I have done with the examples here. You won't be the only one to enjoy playing with the look and feel of a word; I found this quote in a typesetting book:

"we said at the outset of this chapter that character formatting was like recess; the fun part of word processing. If that's the case, fonts must be like the shiny new curly-cue slide that all the kids line up to slide down"

There are plenty of fonts around; most of them underutilized and unloved. And believe it or not, fonts and math are completely intertwined. But that's another story, for another day.

* - not a real word.

Take A Number

Have you heard this phrase before? It usually means WAIT YOUR TURN, rather than Here, help yourself to a pleasing number!

This is your lucky day. Take a number that pleases you. 

This is my total assortment of numbers on my Mac publishing machine.

All of these are shown in 18 point type. They are not emboldened or italicized or modified in any way. Please choose a favorite: (click on each image for a larger version)

 

Isn't it remarkable? Notice how although these numbers vary in height, width, outline, boldness, etc., they are still able to convey their messages of value.

Do you prefer numbers that reflect your interests?  Cowboy Lasso and We Are Aliens come to mind. Can you find them?

Which is your favorite? On what basis do you make a choice? Is it totally subjective?

Is it related to size (Look how BIG my number is) or discretion (notice how subtle and sophisticated my number is)?

You may be interested to know that 4 clever Australians have been granted a patent for a process to programmatically select a font based on the context in which it will be used. It parses through text that is associated with an illustration and then chooses the best selection from a library of fonts.

Here's the example from the patent document, showing how hyperlinks are converted to more interesting fonts - you could do the same by manually selecting each font but it would be lots of work.

Let's finish with a quiz - do you know which font this is?

Some are more equal than others

Some math concepts really aggravate people. One is the concept where we compare the members of a set to see if a member is greater than, equal to, or less than another member. Although this seems contrary to our democratic ideals that all are created equal - hey, it's math, not politics or human rights! Or science fiction.

Or e-commerce. There's actually a website called Equals.com that allows you to make Facebook party line calls - dial one number and get 5 people at a time. Wow! Of course, if you read the fine print, you learn that a subtle computerized voice will break into your conversation every five minutes with a sales pitch for something!

Before I give you the math sales pitch for equality or inequality, can I give you the signs?

> Greater than
>> Very much greater than
< Less than
<< Very much less than
≥ Greater than or equal to
≤ Less than or equal to
= Equal to
≠ Not equal to
≅ Approximately equal to; Congruent

I think those cover all the bases, value-wise. Bases, signs, pitch, get it? Ha ha ha. Maybe I should wait til baseball season starts to spring (training) these puns on you. OK, I know, don't quit my day job.

The symbol we use the most in math is the equals sign. It's been around for about 440 years. It was developed in England when its inventor, Doctor Robert Recorde, got tired of writing out longhand "such and such is equal to this" and "so and so is equal to that". He didn't get rich off this invention; in fact he died in prison a year later. But the symbol lives on.

Pretty soon we had a whole collection of shorthand symbols to save energy and ink.

These symbols are all related to one of the most important concepts in math - the notion of value. Does one value differ from another or is it the same? Here are some everyday examples:

• were you going faster than the speed limit?
• is this check going to be larger than the balance in my account?
• are there more people booked on this plane than there are seats?
• is it farther to the destination than we can drive on the fuel in this car?
• is this cup is holding less than or exactly 8 ounces of liquid?
  

Some of us were taught the alligator rule regarding the greater than/less than signs. Alligators are always hungry, so they are trying to eat the larger (greater value) number. Here he is:

There are lots of other ways that we use variations of these signs, especially in programming and logic. These are perhaps more closely related to identity than value, but I'll toss them in here anyway just for fun.

≡ Identical to
~ Roughly equivalent to
||  Incomparable to (you can't compare them)
<. Contained within ( Set A <. Set B means A is part of B)
<: A special type of ( Set A <: Set B means A is a subtype of B)

PS - my apologies if your browser does not render these symbols correctly

Reducing and Reworking Ratios

Ratio is a word used in the math universe to mean several things:

• comparative value of two or more amounts, such as 4 cups of water to 1 can of lemonade concentrate.

• relationship of a part to a whole, such as she took 1/4 of the quiche for lunch.
  

• you can also use a colon between the numbers to express a ratio, like this 1:4
  

Ratios can be reduced (like fractions). The 4 cups to 1 can of lemonade be converted to other units and reduced. But if the concentrate only comes in cans, there's not much value in simplifying to other, smallest terms.

The terms used in a ratio are not commutative or associative - they have to be in the right order. She wouldn't take 4/1 quiches for lunch.  Notice that the ratio does not imply anything about the value, ingredients, calories or even the size or weight of the quiche.

AN EXAMPLE
A common use of ratios is when making coffee - a frequently-quoted ratio of ground coffee to water is two tablespoons coffee to six ounces (by volume) of water.

Let's say a Coffee Thermos holds 100 ounces of brewed coffee. That's  about 16 times 6 ounces. If you need two tablespoons of ground coffee for 6 ounces, you need 32 tablespoons for 100 ounces.

You are making a bigger amount of coffee than a cupful, but the ratio stays the same.

(To be precise, it would be 16.66 times 6 but we don't need that level of precision, as the water amounts can vary slightly too. Some of the water you put in remains in the grounds and doesn't make it into the thermos anyway.)

If you want to, you can measure the ground coffee (and many other cooking ingredients) by weight. Ratios still apply. You have to be sure that you correctly convert the amounts being changed so the ratios remain equivalent.

Why would you want to do this? Well, it might be easier to pour a pound of coffee into a big pot than to do 32 tablespoon scoops.

How much does a tablespoon weigh?  It depends on the coffee. But let's say about 4-5 grams. So replacing our 32 tablespoons volume measure with 5 grams weight measure looks like this:

HERE COMES THE MATH
5 gm per tablespoon x 32 = 160 grams by weight

100 ounces of water weighs about 2800 grams.

Now our ratio is "cleaner" because it's in the same units.

160:2800 can be simplified to 16:280 or 8:140 or 4:70 or 2:35 or 1:17

Remember, this is by weight, so simply measure the weight of the water in your coffee cup, and add 1/17th as much coffee.

Conveniently enough, 6 ounces of water is equal to 170 grams. So you add 10 grams of coffee, and you are set. How many tablespoons is that? Two.

THE BOTTOM LINE
I think the days of the coffee cup holding 6 ounces are over. Adjust the quantities upwards since your mug is bound to be larger! Twelve ounce cups hold 340 grams of water so you can add 4 tablespoons of coffee. That will assure you don't fall asleep reading my blog!

Take away for six at seven?

English and Australian readers know that when I say Take Away,  I mean Food To Go.  I am asking about dinner for six people at seven pm.

In America, Take Away has many different meanings.

To deduct, remove, carry off, withdraw, to subtract. To divest, shuck, unwrap, strip, shell, scalp, cross off, strike out, delete, erase, weed out, detract, reduce, trim. Confiscate. Withhold.

As in T.A. your driving license, T.A. your X-Box privileges, etc.

I could go on and on. I found a site dedicated to improving the takeaway in my golf swing (takeaway in golf is when you pull the club back/up before swinging forward/down to hit the ball).

In math it means subtraction. The fourth of our primary operations. The inverse of addition.

So mathematically I could say seven take away six leaves one. No food involved!

There are a few special terms to learn for subtraction - the minuend (what you start with), the subtrahend (what you take away) and the difference (what's left). The special symbol is "-".

Subtraction eventually forces us to learn about borrowing (carrying, trading, regrouping).

I bought a bunch of stuff at the store.  I subtracted what I spent from the balance in my checking account. My balance is now less than zero. I have to borrow money to carry me over until next month.

That's not the kind of borrowing I was talking about (but it's a common result of subtracting).  I meant this kind:

Nowadays we usually say regrouping rather than borrowing. We take a group of ten ones from the tens column, add them to our existing, too-small group of ones, and then we can subtract. Like this, where we regroup the one ten so we will have eleven ones, from which we take eight:

Subtraction doesn't have all of the same special properties that addition does.

• Commutative - does changing the order of the numbers alter the difference? 2-3 ≠ 3-2. Subtraction is not commutative because changing the order of the numbers does change the answer.
  

• Associative - when three or more numbers are subtracted, does regrouping them alter the difference? (5-1)-4 ≠ 5-(1-4). Subtraction is not associative because regrouping the numbers does change the answer.

I think that's quite enough of these basic math operations, don't you? But wait, one more thing...

I don't know about your family, but if we get some take away, and try to take away some of the other person's fries, we always get caught. Notice that hand sneaking in there behind my fish and chips?

Money For Nothing, Multiplied

Nowadays, we seem to be enamored with the math operation called Multiplication.  Google the term and you see

1. Buy my book and multiply your effectiveness.

2. Master Your Muse and Multiply Your Blogging Effectiveness

3. Think in terms of three: three routes to every treasure, three solutions for every problem, three chances at every opportunity. You can't fail to multiply your effectiveness, reduce your frustrations and expand your income!

4. Multiply Your Business Productivity By Reprogramming Your Brain! Tap into that 6-inch battlefield between your ears, and tune your thoughts to the profit frequency.

5. You2: A High-Velocity Formula for Multiplying Personal Effectiveness in Quantum Leaps.

6. Use a Wiki to multiply your collaboration effectiveness! Check out my Wikipatterns How-To guide, with practical advice from a wiki expert. (That’s me!) Here’s what people are saying about my book:

• Create an idea-sharing environment where incomplete can be linked together and from this, solutions emerge. (sic)
  
• I’m going to recommend this without even reading it! Should be a must-read ...

Please stop! Did she really recommend a must-read book, without reading it? She did.

Multiplication is surely the internet blogger's favorite math concept. I loved the reprogram your brain page, but for sheer something-from-nothing-audacity, the award goes to Wikipattern How-To Guide.

This obsessive focus on multiplying effectiveness reminds me of the Dire Straits song,

Now that ain't workin' - that's the way you do it,
You play your guitar on the MTV.
That ain't workin' - that's the way you do it,
Money for nothin' and chicks for free

It seems the phrase Multiply your effectiveness is a politically- and economically-correct way of saying money for nothing and chicks for free.

☞  Most people want Something for Nothing. It's a book I found on Amazon. On greed, politics, etc.

☞ Did I say Something for Nothing? A different book, on kleptomania. Shoplifting. Same thing...

What does this all have to do with math? Math says this rags-to-riches strategy won't work, because

Something times nothing is equal to nothing.

Now if you weren't paying attention, you might think I said:

(Thanks to Molly's Grove)

But I didn't say that. I'll repeat the formula:

n x 0 = 0

means you don't get somethin' for nothin'.