Abstract, Concrete, Imaginary, and Just Plain Dumb Numbers

(Reviewed by a Master of Science in Mathematics I am lucky enough to know and would count knowing her as lucky even if she knew nothing about math (Yes, girl, if you read this, it is a compliment :-) ). )

I was recently in a discussion started around one of those on-line Mensa tests. Parts of the discussion traveled into word problems and math problems.

It went like this:

All problems are word problems.

1+1=?

Like you, when I see 1+1, I automatically think 2. But, does it really equal two?

Answering that led to thinking about numbers in general. (PurpleMath is a good site for thinking about numbers: http://www.purplemath.com/modules/index.htm )

Even though you may have looked at the many types of numbers at PurpleMath, there are two primary types of numbers.

Abstract numbers represent nothing but themselves.

Concrete numbers represent something more than themselves. An apple, an orange, a complex mix of units like horsepower.

I threw in Imaginary numbers because they sound neat. Despite their name, they are a subset of concrete numbers. They represent something even though that something cannot exist in the real world as we understand it. The item represented involves the square root of a negative number.

Just Plain Dumb Numbers is a phrase I made up because they represent by their very nature either a false impression or a lie. We sometimes see them appearing in breathless medical news reports. “Eating more than one unicorn horn a day doubles your chances of cancer.” Frightening, isn’t it? Double your chances of cancer. Your chance of getting cancer from eating unicorn horn is 1 in 1,000,000,000. Doubling your chances of unicorn horn induced cancer raise them to 2 in 1,000,000,000 or 1 in five hundred million. (Your chances of finding a unicorn horn to eat, being zilch, are never mentioned in the report.)

As you can imagine, abstract numbers aren’t much use to us beyond learning very basic math. Math so basic we begin to leave it behind about the third grade when a math book first dropped a question with concrete numbers on us. Your first contact probably went something like this:

Johnny has 1 apple. He has 1 orange. How many apples does Johnny have?

If you are like me, you probably got that one wrong when you saw it presented in the form:

1 apple
+1 orange
-------
_2_ apples

That’s fine for a grade school student to get wrong a few times.

We expect scientists, engineers, even inventory takers, to instinctively recognize the simple concept that numbers without words are pretty much useless.

Imagine the inventory taker coming back to the office. “I counted 849,322 today.” What? Where are the cans of beans, bags of rice, cups of soup? Here’s 1. Your pink slip.

Engineers, even if it’s not instinctive, should know just from their everyday work that words are essential to math.

“My car has 360!” Yeah. So? Degrees, lug nuts, horsepower? In my case, the answer is horsepower.

Horsepower (HP) makes an even more obvious example of why math without words is pointless. Without words, a horsepower is 550.

We have to have the words for HP even more than apples and oranges. HP is a mixed measurement. One HP means moving 550 lbs one foot in one second. 550 foot-pounds per second, ftlb/sec. A HP is 550 foot-pounds/second.

If you are still here, you’re probably asking if there is a point or reason for this rant.

Yes, there is. That any student could get into college believing math and words are exclusive is a sad statement about our educational system.

That the advancement of that student is the result of elementary Ed teachers who don’t know enough about basic math is an indictment of a failed system. It has failed the teachers and now fails their students.