[Math] How to think about decimal exponents

Question

I don't know if this is the right place to ask a question about middle school-level math, but I'll give it a shot. I don't have the background to fully understand related questions (like How to calculate a decimal power of a number), but my son, who's currently taking Algebra I, has been asking me how logs affect other functions. I don't have much of an answer, but in exploring, we found something kind of weird. The graphing calculator at GeoGebra plots $\log_2(x^{2/5})$ with curves on both sides of the y axis but $\log_2(x^{0.4})$ only shows up on the right side. (We used $\log_2$ instead of $\ln$ because it was easier for us to think about.)

We get that fractional exponents work like this:

$$
x^{\frac{a}{b}} = \sqrt[b]{x^a}
$$

And exponents with even numerators have positive y values for positive and negative values of x, so log of them is defined.

My guess about why $\log_2(x^{0.4})$ is only on the right side is that if you were going to vary the exponent continuously, the evenness of the numerator would oscillate extremely. And expressing the exponent as a decimal implies that kind of continuous variation rather than clearly distinguishing the numerator and denominator.

Is that kind of the right explanation (for me and a 7th grader, that is)?

Thanks in advance to anyone who's willing to talk about this at our level.

Answer 1

As you know, 0.4 and 2/5 are the same, so the plots should (apart from the vertical shift by 1) be identical. The cause of the discrepancy is due to the way the calculator was programmed, so without looking under the hood I can't say for sure, but I have a guess:

When you type $x^{2/5}$, GeoGebra probably recognizes the 2 as an integer in the fraction, whereas when you put in $x^{0.4}$ it's probably treated as a floating-point number (a different way for a computer to store numerical values using something like scientific notation), so in the latter case, GeoGebra doesn't "realize" that the $x^{0.4}$ will always be positive.

Does that help? P.S. You were on the right track with the idea about evenness; a full answer as to what happens as the exponent varies continuously would require complex numbers.

EDIT: Unfortunately, I don't remember ever seeing any (3Blue1Brown is outstanding at such explanations, though), but I can give a try.

We think of real numbers as being on a line - right is positive and left is negative. Complex numbers are what we get if we think of numbers on a plane - right is still positive and left is still negative, but now we also have numbers extending up and down. This is the complex plane; it's just the x-y plane but we've identified the y-axis with the "imaginary part" of the number, so we can still think about it geometrically. Now, what happens to a number when we exponentiate it?

Two things happen - the distance of the number from the origin changes, and the number rotates. We're specifically interested in the rotation part. The argument of a complex number $z$ is the angle between the line from $0$ to $z$ and the posite part of the x-axis; let's call this argument $\theta$. When $z$ is raised to the power $r$, the angle of $z^r$ is $r\theta$ - the angle is rotated by the number in the exponent.

Now, what happens when we consider specifically numbers on the real line? Positive numbers are on the positive real line so the angle is $0$, and of course $r\times 0=0$ so they stay as positive numbers. Negative numbers, however, are on the opposite side - their angle or argument is 180°. If you multiply 180° by an even number, for example, you get a multiple of 360°, so you're back on the positive side or the real line. That's what we're used to - negative numbers to even powers give us positive numbers.

A fractional exponent means you're not rotating by a whole number multiple of 180°, which means you could end up somewhere that's no longer on the real line - that's how you can get into complex numbers with exponents. In fact, the imaginary unit $i$ is what we get by taking $\sqrt(-1)$, or halving the angle with the positive real axis, so we go from 180° to 90° - directly above 0, not to the left or right at all; in other words, the number $i$ which is 90° off the real axis has no real part.

There are more details involving roots being multivalued - for instance, both $1$ and $-1$ square to one, so you could say that both are the square root of 1 - but I won't get into those details here.

Also, the definition of an irrational exponent is essentially just as a limiting case of rational exponents - this question answers it.