Something in Math HAS to Change

Convention is a beautiful thing. It allows us to use symbols to convey little things like direction or a sound. We can piece those things together to make larger things, and eventually use it to create something like what you’re reading now. There are no inherent meanings to these shapes we call letters, or the sounds we use when speaking. It all works because we agree, somehow, upon what they mean. Of course, over generations and cultures, and between even different languages, some things get crossed up in translation, but it’s still pretty powerful.

The structure of writing, punctuation, and the Oxford comma, they all work because we agree. We can look back and try to see the history of how the conventions have changed and sometimes find interesting connections. Sometimes, there are artifacts from our past that just don’t really make sense anymore. Either the language has evolved passed the usefulness, or the language adopted other conventions that conflict.

One example of this is the difference between its and it’s. An apostrophe can be used in a conjunction and can also be used to show ownership. Pretty simple rule to keep straight with its and it’s, but whose and who’s. Why is it whose, with an e at the end?

According to my friendly neighborhood English teacher there was a great vowel shift, which can be read about here, where basically, people in around the 15th century wanted to sound fancy and wanted their words to look fancy when written. So the letters e and b were added to words like whose and thumb.

Maybe we should take this one step further, and use thumbe. Sounds good, right?

But then, there’s the old rule, i before e except after c, except in words like neighbor and weight, and in the month of May, or on a Tuesday. Weird, er, wierd, right?

All said, not a big deal because those tricks of language will not cause a student to be illiterate. A student can mix those things up and still have access to symbolism and writing and higher level understanding of language.

There are some conventions in math that work this way, too. There are things that simply are a hold-over of how things were done a long time ago. The convention carries with it a history, that’s what makes it powerful. But sometimes the convention needs to change because it no longer is useful at helping making clear the intentions of the author.

One of the issues with changing this convention is that the people who would be able to make such changes are so well versed in the topic, they don’t see it as an issue. Or, maybe they do, but they believe that since they got it right, figured it out, so could anybody else.

There is one particular thing in math that stands out as particularly problematic. The radical symbol, it must go! There’s a much more elegant method of writing that is intuitive and makes sense because it ties into other, already established ways of writing mathematics.

But, before I get into that exactly, let me say there’s an ancillary issue at hand. It starts somewhere in 3^{rd} or 4^{th} grade here in the US and causes problems that are manifested all the way through Calculus. Yup, it’s multiplication.

Let me take just a moment to reframe multiplication by whole numbers and then by fractions for you so that the connection between those things and rational exponents will be more clear.

Consider first, 3 × 5, which is of course 15. But this means we start with a group that has three and add it to itself five times.

Much like exponents are repeated multiplication, multiplication is repeated addition. A key idea here is that with both we are using the same number over and again, the number written first. The second number describes how many times we are using that first number.

Now of course 3 × 5 is the same as 5 × 3, but that doesn’t change the meaning of the grouping as I described.

3 + 3 + 3 + 3 + 3 = 3 × 5

Now let’s consider how this works with a fraction.

15 × ⅕. The denominator describes how many times a number has been added to itself to arrive at fifteen. We know that’s three. So 15 × ⅕ = 3.

3 + 3 + 3 + 3 + 3 = 15

Three is added to itself five times to arrive at fifteen.

Let’s consider 15 × ⅖, where the five in the denominator is saying we are looking for a number that’s been repeatedly added to get to 15, but exactly added to itself 5 times.

In other words, what number can you add to itself to arrive at 15 in five equal steps? That’s ⅕.

The two in the numerator is asking, how far are you after the 2nd step?

3 + 3 + 3 + 3 + 3 = 15

The second step is six.

Another way to see this is shown below:

3 →6→9→12→15

Step 1: 3 → Step 2: 6 → Step 3: 9 → Step 4: 12 → Step 5: 15

Thinking of it this way we can easily see that 15 × ⅘ is 12 and 15 × 5/5 is 15. All of this holds true and consistent with the other ways we thinking about fractions.

So we see how multiplication is repeated addition of the same number and how fractions ask questions about the number of repeats taken to arrive at an end result.

Exponents are very similar, except instead of repeated addition they are repeated multiplication.

Multiplication: 3 × 5 = 3 + 3 + 3 + 3 + 3

Exponents: 3⁵ = 3 × 3 × 3 × 3 × 3

Do you see how the trailing numbers describe how many of the previous number there exists, but the way the trailing number is written, as normal text or a superscript (tiny little number up above), informs the reader of the operation?

Pretty cool, eh?

Now, let’s see some fractional exponents. They mean the same thing with one change...instead of asking about repeated addition they’re asking about repeated multiplication.

Just FYI, 3 times itself 5 times is 243.

15 × ⅕ = 3, because 3 + 3 + 3 + 3 + 3 = 15. That is, three plus itself five times is fifteen.

243^{1/5}= 3 because 3 × 3 × 3 × 3 × 3 = 243. That is, three times itself five times is two hundred and forty three.

You might be thinking, big deal... but watch how much simpler this way of thinking about rational exponents is with something like an exponent of ⅗. Let’s look at this like steps:

3 × 3 × 3 × 3 × 3 = 243

3→9→27→81→243

Step one is three, step two is nine, step three is twenty-seven, the fourth step is eighty one, and the fifth step is 243. So, 243^{3/5}is asking, looking at the denominator first, what number multiplied by itself five times is 243, and the numerator says, what’s the third step? Twenty-seven, do you see?

Connecting the notation this way makes it simple and easy to read. The only tricky parts would be the multiplication facts.

It depends on one’s experience with fractions. Most young people and adults as well are confused by fractions. That is why the decimal system is preferred in many instances. Youth must realize that mathematics is a progressing study. The square root symbol is one of the oldest symbols. So, we learn it first. However, when fractional roots are desired it can be awkward & confusing. When we used logs to compute such problems it was straightforward. Now we use calculators and the fractional approach is better. Bottom line….learn both methods because when batteries are not available to energize our calculators, we most go back to printed log tables and arithmetic.

By the way, I’m trying to find a previous blog early in the summer where you posted 4-5 -problems involving fractional exponents….how do I go back?