What is the number line? It is a visual represesntation of all real numbers arranged in the form of a horizontal line. What is a real number? ONE THAT ISN'T IMAGINARY, OF COURSE.
That wasn't a joke. Imaginary numbers are a thing.
Anyway the number line looks something like this. It's basically a graph with only one axis, or a "one-dimensional graph" if you want to sound like a mathematically knowledgeable douchebag about it. (I will of course be calling it a one-dimensional graph.)
The right side goes all the way to positive infinity, and the left side goes all the way to negative infinity, as indicated by the arrows on either end, and zero sits there in the middle being way too fucking smug about the whole thing.
The cool thing about the number line is that it's a really handy visual for explaining concepts! Remember that post about how subtraction is a lie? The number line serves as a way to show that. I am probably way more excited than one man should ever be about a one-dimensional graph. I just do not understand why teachers glaze over this and show it once and then never bring it up again - IT IS SO USEFUL FOR ILLUSTRATING SO MANY IDEAS.
HOW TO GRAPH POINTS ON A NUMBER LINE: Start at 0, first off. You're starting out with nothing.
Then look at the point you're trying to graph. If it's positive, you move to the right (positive direction), and if it's negative, you move to the left (negative direction.)
This is 5 and -3:
Five steps in the positive direction
Three steps in the negative direction
This totally doesn't sound as cool as advertised, does it? That's just putting points on a line, what is there even to get excited about?
God, quit being such a killjoy. You jerk.
Look here's how addition works on a number line. Take 5+3. Let's plot that on the line. We start at 5 and then move 3 spaces in the positive direction:
Daaaang that was an unnecessarily convoluted way to show something you already know how to do! LET'S DO IT AGAIN! This time we'll subtract, 5 - 3 = 2. Start at 5 and then move 3 spaces in the negative direction:
Okay whatever, what is the SIGNIFICANCE of that? Well, when you add you move in the positive direction, and when you subtract you move in the negative direction. What happens when you add a positive number (right of zero) and a negative number (left of zero)? Let's check it out:
5 + (-3)
(-3) + 5
Oh shit, did I just prove that subtraction is addition of negatives and that it is TOTALLY commutative? I THINK I DID.
(But Guindo nobody was even questioning you about that--SHUT UP)
Now let's talk about something that most people think they understand but they actually don't because it was explained to them in a very simplistic, watered down way: absolute value. "Wait, I know how absolute value works!" you are thinking to yourself. "That makes things positive!"
And you would be WRONG.
If 3 - 5 is really 3 + (-5), and absolute value makes things positive, then wouldn't |3 + (-5)| become |3 + 5|, making |3 - 5| = 8 ? No. No, that is not correct at all. (This is not a wholly made-up example by the way, I have had college math students come to me thinking that this is how it worked because of the shitty "makes things positive" explanation their teachers gave them.)
What absolute value actually means is "distance from zero," which brings us back to the number line! First you perform whatever operation is inside the absolute value bars, in this case we're using 3 - 5:
3 - 5 = -2
Then, to figure out the absolute value of that, count how many spaces away from 0 your answer is:
3 - 5 = -2, which is 2 spaces away from 0, which means |3 - 5| = 2.
And now you understand what absolute values are and what your math teacher is actually asking you for when you see those goofy straight-bars in a problem! Absolute value isn't telling negative signs to take a hike, it's giving a distance.
This post was maybe still not as exciting as advertised so here is the absolute value of my cat's awesomeosity:
NEXT TIME: The behaviour of this blog as time approaches infinity.
* "Why all the purple in this post?" BECAUSE PURPLE IS AWESOME FUCK YOU
why, purple IS awesome
ReplyDeleteInteresting post. It is amazing how difficult students find it to build the 1-D graph or scale in the first place. The need for equal lengths to represent equal number of units is often not grasped, even by undergrads at top universities.
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