Saturday, October 22, 2005

Counting and negative numbers question for all.

In mathematics there is a set of axioms called "Peano's Axioms" which are the basis for proof by induction. These axioms also build the set of natural numbers (1, 2, 3, .......) or counting numbers. There are some interesting studies by psychologists that seem to show that these axioms exist at an unconscious level in children and primates. The psychologists aren't referencing the axioms, it's just something I noticed when reading the papers. It's pretty cool and gives me something to think about when I'm driving.

So....here I am driving down the road and thinking about how we learn to count and when different number concepts were incorporated into arithmetic and mathematics. Which gets me to negative numbers. You can conceptualize the natural numbers and fractions (rational numbers), but how do you conceptualize a negative number? When numbers are thought of as quantities, what is a negative quantity? Any ideas out there?

If you have some uncontrollable desire to read about Peano's axioms, there is a good description here at Wikipedia.

Harvey said...

"how do you conceptualize a negative number?"

"I owe you \$10".

MathCogIdiocy said...

Harvey - Do you need my address to mail the \$10?

nellsquirrel said...

After I finished laughing over the \$10, I realized that's how I see it too - as something that is being "taken away" or "owed".

Hmmmm...

MathCogIdiocy said...

Nell - I suspect that most people see negative numbers in the way you do and it's a good way of visualizing negatives.

Mathematically, they're simply integers less than zero, an extension to the set of natural numbers in order to allow equations like x - y = z to have meaningful solutions for all values of x and y.
(Wikipedia http://en.wikipedia.org/wiki/Negative_and_non-negative_numbers)
I guess those darn mathematicians just had to try and make it more difficult for students to understand the concept. *bg*

Teresa said...

Oddly enough I have always visualized it on a ruler or number line if you prefer with 0 in the middle and the positives on the right and negatives on the left. But really I don't actually "think" too much about them - so far as I'm concerned... they just "are". You do a mathmatical formula and end up in the negative numbers... do a different one and end up positive. (You know this is much harder to explain than it is to simply accept *grin*)

MathCogIdiocy said...

Teresa - I have the number line mental model. I wonder if this is a result of studying math? Might make an interesting research question. hmmmmm......

Teresa said...

Future Master's Thesis?... Doctoral Dissertation? LOL!

MathCogIdiocy said...

Why does everyone suggest master's thesis or doctoral dissertation everytime I ask a question? Sheesh

Contagion said...

I'm more like Teresa, I see charts and graphs in my head with negatives and postives. I think this has something to do with classes I took in college(That you continue to give me flashbacks to, and not the good kind!).

As for the thesis, I think it's just due to the nature of the question. Most people I've met don't think in math.

Harvey said...

I tend to visualize negative numbers along a numberline, too. Mostly because that's how they were first presented to me in school.