Some Number Line Intuition Needed

Post Reply New Topic

Ok, we have a simple number line, ranging from 0 to, say, 3 (for simplicity's sake).

so we have:

0 - 1 - 2 - 3

as our number line.

Let's say we want to find out, using the number line, what 3 - 1 is.

How does the number line show that the answer must be two? What does it mean to subtract 1 from 3 via a number line?

If I were to remove a unit of 1 from the number line above starting from the far right, what range would I be removing exactly? [2,3] or (2,3] or [2,3) or (2,3)? [ or ] indicating inclusivity and ( or ) indicating exclusivity

I hope this makes sense of what I'm trying to figure out.

Hm, is this a scientific question, or you trying to make sense of what they told you at school?

It's a matter of "precision". Trying to understand exactly why it works, and what exactly is being subtracted range-wise. I think there may be a bit of a metaphysical question to it as well.

I never thought of it as taking anything away from the number line. The way I think of it is that if you want to add x to a number, you move x number of places right, and if you want to subtract x from a number, you move x number of places left.

The number line is not the definition of the operation. It doesn't show that the operation itself is correct. It's just a notional way to think of numbers in a sequence.

On a number line, + = move right and - = move left.

3 move left 1 = 2

3 move right 1 = 4

etc. Works for a number line with whole numbers.

The "proof" that this works is simply the application of the definition in this case. That's not unusual in basic cases like this.

Ok, we're getting to what I'm trying to get at.

What about when it comes to distances between two points on the number line?

We know the distance between 2 and 3 on the number line is 1 because, mechanically, 3 - 2 = 1.

But how do we know intuitively that the distance is 1 and not slightly less than or more than 1?

If the 3 in the number line is inclusive then the 2 will be inclusivee, if the 3 is exclusive the 2 will be exclusive.
But it depends if the 1 is 1 or is infinitely close to 1 in that case the 2 will be inclusive even if the 3 is exclusive.
But I am not sure of that.

*Checks the age of the poster above and is greatly impressed*

I see what you mean, but I guess my problem is I don't even know how to express what I'm wondering about exactly.

Anyway, thanks to you all for trying to help. I'll try to figure it out on my own.

This is actually a cool question, OP, and one that makes me think I'm right to be in here. I've had the same confusion myself. Obviously I know basic math, but I had trouble visualizing it for a long time. Why is 300 - 200 = 100 rather than 99 or 101? It almost seems like you could set up the numbering system to produce any of those answers, and long practice in computer science with zero-based vs one-based arrays didn't help.

What did help, I think, was to view the numbers as something like an english ruler, with markings at every inch. But move the markings mentally so that they actually go in the center of each inch. So "1 inch" isn't a hash mark, but rather the entirety of the first inch-long section, ending at the hash mark.

So if you pull out "inch 2", you aren't removing a hash mark, you're removing an entire inch-long section and either leaving a blank in its place, or moving the following sections back by one to fill in the space.

I appreciate that this is an absolutely stupid way to look at things, and totally unnecessary, and yet somehow it does make more sense than thinking of the hash marks themselves as the units. Partly because 0 has a hash mark but no inch! 0 really doesn't have a hash mark, that mark is really just the beginning of the first inch, so it shouldn't be labelled 0 at all.

If anything were to make me think I was really on the autism spectrum, this post would be it.

The best is yet to come. I'm going to introduce a new stochastic number line in 2013. You never know where you are (every time you look, you see a different number), you don't know where you've been, and you aren't sure where you're going.

Cool.

The numbers in your example just identify the hyphens to their left. So removing four hyphens would leave you the hyphen between "4" and "5".

Like I said, my confusion arises because we label them wrong. The labels should be over or in the associated unit, not at the end of it.

So, rather than your example, try this:

12345
-----

Pull out four, and you're left with

5
-

Which is really just 1 unit.

Much clearer that way than being left with "4-5".

Or something. Heh.