This certainly explains exponents better then I’ve ever encountered explanations before.
(In essence it doesn’t differ much from the repeated multiplication explanation but introducing “time” in the Expand-o-tron 3000 simplifies imagining non-integer powers and their effect on outcomes.)
@CL: Great question! I had the same thoughts myself, I may need to go back and make that section more clear.
Exponents act like multiplication, but the amount you multiply by has 2 inputs: growth rate and time (vs regular multiplication, which is just growth rate).
You can combine regular multiplication: Multiplying by 3, and then multiplying by 15 is the same as multiplying by 15 (3x5) all at once.
With exponents, using x^2 and then x^3 is the same as multiplying by x^(2+3) = x^5 all at once.
The tricky thing is to remember that the time is being combined, but it’s happening in the exponent you raise it to.
Hope this helps!
@mike: Glad it happened 
@Theo: Thanks! That was the goal – to help expand our insight so things like fractional (or even negative) exponents can make sense. Negative counting is confounding.
[…] Kalid at Better Explained likes to find intuitive ways to explain difficult mathematical concepts like, Why does 00 = 1 ? […]
Thank you Kalid. Another reminder that math concepts are easy to teach and very hard to explain. Your analogies and patient decomposition of the problems are wonderful to read.
Excellent article! You are good. You are on my RSS feed.
@Stuart: You’re welcome – I agree, I think the difficulties in understanding math is more to due with complex explanations, not complex ideas. Thanks for the kind words :).
@CJ: Glad you liked it!
Here’s my own little explanation for powers to the i, though it’s not nearly as well-written as yours, of course :3
Let’s start with this famous equation: e^i*pi=-1
It’s called Euler’s Identity, and is presented by purists as e^i*pi+1=0. We can prove it’s true, but it doesn’t seem to make any sense. After all, what the hell do i, e, and pi have to do with each other? Let’s see if we can boil this down a bit.
For this example, we’ll work with 2^i, which is equal to about 0.769+0.639i, if you plug it into a calculator.
First, let’s remember that, as Kalid put it above, we can turn an exponent into a grower-oriented one by changing it. So 2^i = e^ln(2)i.
Euler’s formula tells us that e^xi = cos(x)+sin(x)i
If we plug in ln(2) to this, we get cos(ln(2))+sin(ln(2))i, which is about 0.769+0.639i, the answer the calculator gave us before!
However, this doesn’t really help us much unless we already have a really good understanding, intuitive of what Cos and Sin are (and I’m not going to explain all of that here). Instead, we can look at this another way.
e^x is another way of saying e^(rate*time), correct? Well, rather than say we grow for i seconds, how about we use i as our rate of growth, since switching the order in which we multiple doesn’t change the result?
Multiplying by i is simply doing a 90 degree turn. So as Kalid previously said, we just start at 1 and start growing in a circle. So when we say “we grow for ln(2) seconds”, we mean traveling around the unit circle on the complex plane a length of ln(2) radians! (Read Kalid’s article on radians for more info) If you already know what Cos and Sin are, this should be making sense by now. We can check our work with this as well.
The circumference of a circle is 2pir, where r is the radius. Since we start at 1 before we start growing in a circle, the radius is 1, which means the circumference is 2pi. This means moving ln(2) radians around the circle is about the same as rotating 39 degrees.
Let’s return to Euler’s Identity. With what we’ve just established above, e^pi*i means moving pi radians around a unit circle with a radius of 1, right? Moving pi radians around the circle is the same as doing a 180 degree turn, since it’s half the circumference of the whole thing (2pi), right? Which means we go from 1 to -1.
Which brings us to e^i*pi=-1.
Hope this cleared up some confusion…
I would LOVE to hear Kalin do an article on Euler’s Identity or really on Euler’s formula. I have forever wanted to understand the relationship between e, i, pi, and rotational position. I appreciate your comments Apple but I’m just not quite there yet.
It would also give Kalin an opportunity to bring together many of his articles into one giant mathematical relationship that has many practical applications.
So here is the heart of all my Euler’s Identity questions:
Lets just look at e^pi for a moment:
1*e^pi means you start at one and then move in the positive direction at an increasing “unit rate” for pi seconds. You will end up at a seemingly arbitrary location (23.1407).
However e^ipi means you start at 1 and ROTATE at an increasing “unit rate” for pi seconds. You end up making exactly 1/2 circle! This means there is some relationship between e and pi. 2^ipi or 5^i*pi will put you at some random spot on the circle, but a growth rate of e takes you exactly pi radians in pi seconds. Doesn’t that seem strange?
@Apples, Jeff: Thank you both for the wonderful comments! Yes, I’d love to do a follow-up on Euler’s formula, and seeing i as “rotating” your rate of growth is how I’m learning to make intuitive sense of it :).
Thanks Kalin I am looking forward to it! At the very least can you help me understand this one point? I feel like the answer is staring me right in the face but I can’t wrap my mind around it.
1*e^pi means you start at 1, “grow” by a factor of e for pi seconds and you end up at 23.141.
Now if we throw a little “i” in the mix then it gives us 1e^(ipi). If this means our growth takes a “left turn” so to speak and we rotate our growth, then why don’t we travel 23.141 radians in pi seconds?
Thanks again Kalin for all your work.
I find thinking about exponents as “how many patterns can I make with some objects if the objects come in x varieties of color and I have y objects” for x^y.
For instance, let’s say we’re lining up a bunch of eggs.
Our eggs come in 2 colors: brown and white.
x represents the variety. x = 2.
y represents the number of objects (eggs in this case). y = 3
If we line up 3 eggs in a row, how many different patterns could we make with white eggs and brown eggs?
white, white, white. white, white, brown. white, brown, white. white, brown, brown. brown, white, white. brown, white, brown. brown, brown, white. brown, brown, brown.
That’s 8 patterns. 2^3 = 8.
If we had one egg, we’d only have to possible patterns: white. brown. 2^1 = 2.
If we didn’t have any eggs, we’d have exactly one pattern: no eggs. (the one and only pattern is the absence of eggs).
If we had -1 eggs, then how many patterns would we have? So -1 eggs means we take one egg away from however many we have. If we started out with 10 eggs, we have 1024 patterns (2^10). If we took one egg away, we’d have 512 patterns (2^9). So taking one egg away gives us exactly half as many eggs if we hadn’t taken that last egg away. 2^-1 = 0.5.
Now lets say we have (2^3)^4.
Inside the parenthesis, we have 3 eggs, each with 2 possible colors. We have 8 possible patterns. If we laid 4 groups of eggs side by side, how many patterns could we have? Now we will use a group of 3 eggs as our new object instead of using just using one egg as our object.
We know each group of 3 eggs has 8 patterns (2^3). So each group of 3 eggs has 8 possible variations. So our new x is 8. Our new y is 4.
(2^3)^4 = 8^4.
But instead of thinking in terms of groups, we can just count the number of eggs. If we have 4 groups of 3 eggs, then we have 12 eggs in a row.
Each egg has two colors. Thus, (2^3)^4 = 2^12.
I find this pretty intuitive. What do you think?
@Jeff: Great question. I’d like to do a more complete follow-up on Euler’s theorem, but consider this:
e^(ipi) means we are growing at 100% (rotated) for pi seconds. Growing at a 90 degree angle does not change your magnitude, it only changes your direction. So we always end up growing at 100% (without the compounding interest) and end up going only 100pi or 3.14 radians around the circle. We aren’t “growing” in the correct direction to increase speed; instead, we are growing to change our velocity.
@Jeremy: Thanks for the explanation! I think the set interpretation has some merit for whole numbers, but is too far removed from the traditional “repeated multiplication” explanation to help with repeated fractions (what does 0.5^2.3 mean in regard to sets with a number of elements?).
But, this may be because I’m not well-versed with it (and I think that may be the ‘modern’ definition of exponents). I always welcome other mental models as they may map better to different problems :).
Don’t get a big head (-: but I think you would make a very gifted math teacher - putting many so called “math teachers” in the US to shame, shame, shame.
Will you be writing any books?
If you have something like SQRT(8) + SQRT(2), the numbers underneath the square roots are not the same, but you can combine the terms. EXPLAIN?
Dear Sir,
I am sorry that someone had not brought this to your attention sooner, but you are plain wrong. 0^0 is an indeterminate form. Defining 0^0 = 1 is not only misleading, it is a mistake so deep that when mathematicians finally realized how much damage it was causing, our understanding of mathematics was greatly changed for the better.
Consider the following.
Choose ANY positive number ‘c,’ and for x>0, let
f(x) = x^(-c/log(x))
Naively “plugging in” x=0 looks like we should have
0^(-c/(-infinity)) = 0^0
But we need to be more careful. Taking the log base e (or “ln” if you like) of f(x) and using properties of logs gives
log(f(x)) = (-c/log x)*(log x) =-c
so that f(x) --> e^(-c) as x --> 0 (from the right). A similar argument works for negative c. The point is, c can be any number you want. Thus,
0^0 = any number you want.
Sometimes defining something which does not have a definition to begin with is OK, as in the case of a PARTICULAR function, such as g(x) = (x-1)/(x^2-1), which has a singularity at x=1. It is perfectly fine to let g(x) = 1/2 (or anything else, if you don’t mind violating continuity). However, to define 0/0, 0^0, 1^infinity, infinity - infinity, or any other indeterminate form will lead to many, many problems in mathematics. Historically, this has happened quite a lot, and it took us mathematicians quite awhile to realize the source of the problem, which was that people kept trying to define these things.
I urge you to please correct your article, and publish your mistake.
@Anon: Thanks for the details, I’ve updated the article to be more clear about this point.
It seems the debate about 0^0 is still not complete; the goal for the article is to present an intuitive (non-rigorous) interpretation which explains why 0^0 = 1 “makes sense” in many circumstances.
Typo: At the start you describe addition as “combing numbers”, you meant “combining”. Great site! Thank you so much- your take on maths is so intuitive and interesting.
Also, the URL “http://betterexplained.com/articles/intuitive-guide-to-angles-degrees-and-radians/” is not an actual link (Under subheading “Advanced: Rewriting Exponents For The Grower”). Probably because of the preceding colon.
Happy Math!