Math Myth Shattered: There’s Always One Right Answer

This is Day 26 of 31 Days of Math Learning Success. Follow all the days here and check out others that are writing for 31 days here.

Float like a butterfly, sting like a bee.
~Muhammad Ali

At cocktail parties I usually hear one of two things:

“I’ve never been good at math. I can’t even balance my checkbook.”

“I love math – there’s always one right answer.”

Both of these are wrong.

I’ve attacked the first thing in many posts. But now it’s time for the second one: There’s always one right answer.

There may be no answer.

A lack of answer is sort of an answer, I know. But most people want a number as an answer. Not a comment.

But answers to questions like, “At what point does the graph of $y=x^2+1$ cross the x-axis?” has just that.

“Well, it doesn’t cross the x-axis at all.”

And how about solutions to problems like the 5 Five Room House Problem? I worked on that one for 15 years before realizing that there really was no solution.

There may be one PRETTY answer.

Here’s where most people are happy. The solution to $6x+1=13$ is $x=\frac{1}{2}$ .

But $x=0.5$ is also an answer. And depending on how you feel about fractions and decimals, you’ll think one of these is prettier than the other.

But $x=log_9 3$ is also a solution. And $x=3-\frac{5}{2}$ . They’re not pretty. But if you plug in $3-\frac{5}{2}$ for x, you get a true statement.

Which means $x=3-\frac{5}{2}$ is a solution.

So really, you have an infinite number of different-looking solutions (all the same value, but they could look different).

There may be MANY pretty answers.

How about the fun guys like $118x+308 = 30 x^3+76 x^2$ ?

This one has three distinct values as answers: $x= 2$ , $x=\frac{-7}{3}$ , $x =\frac{-11}{5}$ .

Of course, we can write them in many ways – some nicer than others. The fraction $\frac{-11}{5}$ can be very nicely written as -2.2.

There may be one distinct answer that can be written very differently.

I did a trig problem in class many years ago. My answer seemed solid (150 students were checking my work as I did it). But yet the answer in the back of the book was VERY different.

I validated the answer in the back of the book to make sure that one was correct. And I validated mine. A few times.

Oh how I wish I could recall that problem. The result ended up being like $2 sin^2x$ when the book showed $1-cos2x$ (a trig identity). I remember it being a far more obscure identity, though.

The point is that I got a perfectly legitimate, valid answer and it took me a while to figure out why it looked so different.

Float like a butterfly.

When you finish a problem – from the book or life – ask yourself if it’s the only answer. Is it the best answer? Would it be “nicer” written a different way?

Always look for more or less answers than you see.

Stay on your toes.