Zero isn’t Bad – She’s Just Drawn That Way

This is Day 19 of 31 Days of Math Learning Success. Follow all the days here and check out others that are writing for 31 days here.
Zero is useful, classy and a lot of fun at parties, but she can get you in trouble if you're not careful!
Sometimes it is useful to know how large your zero is.
~A Deranged Graduate Student

Zero’s awesome.

She’s a super hero.

She comes to the rescue all the time.

But she’s slippery.

Zero sometimes ends up where she’s not supposed to be.

As a student, you’ll come to points where you don’t know what to do next. Since you’ve been following this series, you’ll know a bunch of stuff to try.

But one of the things you can’t do is set expressions equal to zero whenever you like.

We do it a lot, so it seems like one of the things to do. But there’s a specific purpose to it. And it’s not “because there’s nothing left to do.”

When CAN Zero step in?

Supposed we’re curious about the expression x^2-3x+2. We start playing with it a bit. We notice that it factors into (x-2) and (x-1). Kinda neat. But we want more.

We wonder (as Descartes did long ago), if we could have some kind of picture of it.

We pull out some graph paper and draw a Cartesian coordinate plane. We decide that the x-values we use will be along the, er…, x-axis.

And the number we get after substituting each number for x in the expression, well, that can be the y-value of the point we put on the graph.

So now we’re graphing y=x^2-3x+2.

Golly, we could figure out if there is a point with (some-number-here, 14.3). In other words, we could try to figure out what that x-value would be if the y-value was 14.3. We do this by solving for x in this equation: 14.3=x^2-3x+2.

But that’s kinda nuts. We might be here all day.

However, the charming and beautiful Zero is kinda nice. So instead of the Bon-Crowder-dumb-number 14.3, we use Zero: 0=x^2-3x+2.

Hmm… We factored it up above, so that’s also equal to zero:

    \begin{equation*} \begin{split} 0 & =x^2-3x+2\\ 0 & =(x-2)(x-1) \end{split} \end{equation}

And since any two numbers that multiply to get Zero require that one or both is Zero, we know that either 0=(x-2) or 0=(x-1).

A fact that isn’t true with my dumb 14.3, btw.

So x=2 or x=1 when y=0.

The kicker to all this is that we only set an expression equal to Zero when we’re looking for those points on the graph (the x-intercepts).

Don’t lose her, either!

I said she was slippery.

Suppose you’re rocking along with this equation: 5=3x+8.

You’re cool enough to know that it’s better to take 5 away from 8 than take 8 away from 5 (nobody wants negative numbers). So you do it:

    \begin{equation*} \begin{split} 5 = & \hspace{.05in} 3x+8 \\ -5 = & \hspace{.2in} -5\\ & \hspace{.05in} 3x + 3 \end{split} \end{equation}

Notice what happened there? Since many of us think of Zero as “nothing” we sometime just forget about her. But really it should look like this:

    \begin{equation*} \begin{split} 5 & = 3x+8 \\ -5 & = \hspace{.2in} -5\\ 0 & = 3x + 3 \end{split} \end{equation}

In a blink of an eye, you could lose everything. So be careful!


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Math in this post is elegantly produced by WP QuickLaTeX and the image is courtesy of KH.

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