We’re taught that exponents are repeated multiplication. This is a good introduction, but it breaks down on 3^1.5 and the brain-twisting 0^0. How do you repeat zero zero times and get 1?
what about irrational exponents?
Great question. At an intuitive level, I see an irrational exponent as a different amount of time, in between known rational numbers, and ultimately approximated by a rational number.
When we write “3 * sqrt(2) = 4.24” we “know” that we’re really taking an approximation, and that 3 * sqrt(2) goes on forever.
Similarly, “3^sqrt(2) = 4.72” is just an approximation and the real decimal result goes on forever.
There is a small problem with your 2^3^4 = 2^12 example: exponentiation is (usually?) understood to be right-associative (ie top-down), therefore: 2^3^4 = 2^(3^4) = 2^81
I’m pretty sure your “repeated exponentiation” section is not quite right - when there are repeated exponents like that, in the standard notation, we work from right-to-left. Thus, 2^3^4 means 2^(3^4) which is neither 2^(3*4) nor 2^(4^3)…
2^3^4 is in fact 2417851639229258349412352 while 2^(3*4) is 4096 and even 2^(4^3) is only 18446744073709551616.
@Steven, sabik: Thanks for the comments! My mistake, I didn’t realize it was right-associative. I’ll add in the parentheses to make this more clear.
Amazing work once more Kalid. You make learning fun again.
Forgive me if I ask a dumb question. It’s been awhile since I’ve been in any sort of math class.
If I put 3 in the expand-o-tron for 0 seconds, I get 3 because there was no time expanding. If I put it in for 1 second, shouldn’t I get 9 since it spent one second expanding? But I though 3^2 was 9. Maybe I’m reading it wrong.
@Todd: Great question – I should make the operation more clear. Normally, we start off with 1.0 and see how it grows, so with a setting of 3x power for 2.0 seconds, we’d get 1.0 * 3^2.0 = 9.0.
If we started with 3.0 and ran it for 1 second, we’d get: 3.0 * 3.0^1 = 9.0.
Most of the time we want to see what happens to a “unit” amount.
It’s great to see another article. As with many of your articles, it helped me realize that I have not understood something I’ve been using for so long.
About evaluating i^i: it appears that I have the problem half-way done.
So, I put 1 in with i times growth for i seconds. Meaning, each second I rotate 1 counter-clockwise 90 degrees in the complex plane. However, how do I look at i seconds intuitively and thus complete the computation?
Nobody: Unfortunately the “^i” operation is a little confusing. You should start by understanding e^i and e^(a*i) where a is a real number. Then remember that x = e^(log x), and i = e^(log i)
so i^i = (e^(log i))^i = e^(i*log(i))
Analogies only go so far!
@Nobody: Great question! (Tomer, thanks for the details).
It’s hard to think about “i” seconds, just like it’s hard to think about -1 seconds – it’s easier to think about what the transformation does. Negative numbers flip the direction, and i brings items into the complex plane. When you take i as an exponent, it changes the direction in which you are growing (instead of growing in the real dimension, you start growing in the imaginary dimension). There’s a lot more to say, but that’s the intuitive approach I take with it :).
I really appreciated this article, and just want to suggest that you make it more clear at the top that you ALWAYS start with one.
I really struggled with your explanation of 0^0 until I got down into the comments and saw that that was the case. Thanks.
When you take i as an exponent, it changes the direction in which you are growing (instead of growing in the real dimension, you start growing in the imaginary dimension).
Umm, what? When you have i as an exponent, you go in circles, getting neither larger nor smaller.
Is there a Nobel prize for explaining math 
@Sabik: It’s tricky – consider i^i. It “seems” like it should stay on the unit circle (that’s where it started, magnitude of 1), but it doesn’t. I should be more clear: i as an exponent changes your instantaneous rate of change by 90 degrees.
@NF: Heh, I think the rumor is Alfred Nobel hated mathematicians so there’s no prize for pure math :).
Thanks for the great visualization of how exponents work. I’m a huge fan of your articles.
The step I can’t intuitively grasp is in the multiplying exponents section: “What if we want to two growth cycles back-to-back? Let’s say we use the machine for 2 seconds, and then use it for 3 seconds at the exact same power”.
To me, if I put something in the microwave for 2 seconds, then again for 3 seconds, intuitively this would be like adding together two doses of microwaving:
x^2 + x^3 instead of x^2 * x^3 (i.e. running the microwave for 2 min, and then again for 3 min is 2 + 3 = 5 min, not 2 * 3)
I’m not sure how to picture what x^2 * x^3 means in microwave terms…maybe something like increasing the wattage by 3 times.
Anyway, keep the great articles coming!
yet another “a-ha” moment when reading this…=)