can zero have an exponent of zero?
IT is NOT true that 0^0 =1 This is false.
For some real number “a” not equal to zero:
a^n = a^n*a^0
Thus we should be able to divide both sides by a^n and get: 1=a^0
IF a=0 or a^n=0 then we run into problems.
a^0 is defined only if a does not equal zero.
I found this by googling what I thought everyone would want to know, and as usual a question that keeps me up nights isn’t even a part of the known universe…here it is. I had googled “why the exponent goes down one” and I found this excellent intuition which I plan to study more carefully. My application is the power rule of differentiation, used to find the derivative. They say that the power rule is just a trick, I say if it is equal to something and the something it is equal to is differentiation, then the power rule is not just a trick but an identity of the facts behind differentiation. If that is so, then there should be an intuition behind the power rule that gives an explanation of why, for example, 2x^3 diff-ates to 6^2. I don’t believe in coincidences. Is there a way you can use the expandotron to develop an intuition for the power rule that reveals the heart of differentiation to a beginner like me? I believe it involves considering dx the UNIT, and dy/dx the RATIO or proportionality factor, and y the function or dependent variable, and of course x as the independent variable. My guess about the heart of calculus is that it is the linear equation but I have to go over this page and try to see what I think you would say and see if we come up with the same answers. Thanks!
I really like how this article got me thinking. I read it a long time ago, and recently had an AHA moment. Example: 9^1/2 = 3. As you grow from 1 to 9, half way there you are at 3. This is obvious with a base, or growth factor of 3x. 3^0=1, 3^1=3 and 3^2=9. What if we have a growth factor of 2, or 4? Does this still hold true? We always start at one and grow, grow, grow. 2^0=1, 2^1=2, 2^2=4, 2^3=8, and 2^4=16. 2^x = 9 so x = ln(9) / ln(2). so x=3.16993, or 2^3.16993 = 9. Does 2^(3.16993/2) = 3? You bet!!
@jj: Yep, 0^0 is indeterminate but it’s useful to pick a value for it, to simplify many equations.
@Scott: Thanks for the suggestion, I’d like to do an article on the power rule. The main intuition for 2x^3 going to 6x^2 is the number of combinations that arise.
Intuitively, think of a cube (3 by 3 by 3). When you make it bigger in all dimensions by 1 ( to a 4 by 4 by 4) you can consider this “adding a layer” to the 3 outer sides. That’s why x^3 => 3x^2. There is a tiny piece in the corner which is “ignored” in the Calculus sense (it becomes negligible as your changes get smaller).
For x^2 => 2x, it’s similar (to make a square larger, add a strip to 2 sides). I’d like to write more about this!
@Laura: Yes, I always consider 1.0 going into the expand-o-tron to start. I see exponents as a “scaling factor” that have to operate on something else. So even
3^2
can be considered
1 * 3^2
That is, we start with something (1) and make it 3x bigger for “2 seconds”. When the duration is 0, it’s like we never used the machine, and we get the original amount out (scaling factor of 1). That’s the intuition that clicks for me ;).
@kalid: Agreed. Look at this link for a good presentation. Kudos to University of Waterloo in Canada
https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node14.html
So when you say that the scaling factor is always 1, we can also think of this as the original number we are putting into the expand-o-tron that will be affected by both the growth and the duration. This is why when the exponent is 0(the duration=0) and no changes are made, we get an answer of 1?
Thanks Wesley!
From what I understood from this article, and correct me if I am wrong, when you calculate 0^0 and get a result of 1, the “1” is actually the scale factor, or the number by which the 0 is multiplied. so basically you multiply 0 by 1, and still get 0… so why is it that you would write 1 as the answer, and not just the answer itself of 0?
@Patrick: Great question, something I need to clarify. My analogy for exponents is:
original * growth^duration = new
so if we have 0^0 we get
original * 0^0 = new
original * 1 = new
original = new
I.e., we didn’t change at all. The scale factor is multiplied into the original amount, so 0^0 = 1 is another way of saying “we didn’t change at all”.
For something like 3^2, we write
original * 3^2 = new
original * 9 = new
which is to say, our new value will be 9x our original one. Normally we don’t think about the original amount and write “3^2 = 9” but then it becomes harder to see why 3^0 = 1. So in my head, I even see 3^2 as “1 * 3^2”, i.e. we have a starting point and begin to change it.
wtf why did it say x^3 * x^4? why did it jump to multiplication for “two growth cycles back to back”? what is the logic of that? You do this again when you go a^1.5 then a^1.5. why is it multiplication all of a sudden?
also, why does 2 growing for 1.5 amount of time give 2.82842712475?
0/0 is indeterminate because there are an infinite number of correct answers through the definition of division. 0/0=1 is correct because 1x0=0. 0/0=2 because 2x0=0 is correct. Since there is no unique answer, 0/0 is indeterminate. 0³/0³= 0° = 0/0.
Therefore 0° is indeterminate machine or no machine. Remember figures don’t lie, just liers figure.
I should correct myself, 2 posts up (wish you could edit on this thing) you take the inverse of the chance that youre approaching the second cut in a fashion that will halve one of pieces before you stop
Faces (6) x corners (8) x edges (12) = 576
any unique combination for multplying 2 of those “aspects” of the cube = 6x6, 8x8, 12x12, 6x8, 6x12, 8x12 the sum is 480
480 possibilities out of 576 = 1/6, the inverse 5/6ths multiply by the 3 planes (XZY) = 2.5 or 1 + your 1.5 cuts.
I haven’t got far enough in math to fully analyze and integrate pi and e, but I am assuming somewhere in the mix they come into play since you can still create assymetrical lines that would divide the cube in “half” as far as volume/mass was concerned without the end result of the quartering process being identical “looking” peices.
Instead of thinking about growing think about dividing it that about of times. If you have 1 block of cheese, and you cut it in half you have 2 blocks of cheese. If you keep those pieces jammed together, turn it sideways, and cut it in half again, you have 4 blocks of cheese, if you keep it together and flip it onto the Z plane (XYZ) and cut it in half again you have 8 peices, but if you break it apart and XY cut the peices you get and you “quarter” them again you get 2^2 x 2^2 16 blocks of cheese
I should also add a note on the “to the 1.5th” power, or 3/2 power. If your intention is to halve the block of cheese, you can choose to approach that “halving” process from any angle possible on the block so long as you are cutting it “symmetrically”. If you cut something in half from one angle, and then halfway through the process of cutting it from another angle (it doesn’t have to be "perpendicular on cut 2 so long as the resulting cut effectively quarters the volume upon completion, but you stop halfway through the cut…sometimes you will only have 2 blocks of cheese equal in volume with gouges cut into them, and other times you will have 3 peices because one of the halves was effectively “halved” when you were halfway through the cut. The chance that you will have of making 3 instead of 2 is approximately 82.84271…% of the time, so that is why it equals 2.8282871…
a block (cube for arguments sake) has 6 faces, 8 corners, 12 edges. be mindful of the operator you’re using in any given application. it has 6 faces, because it has 3x2 faces per plane (XYZ, 3D), it has 8 corners because it has 2^3(XYZ) corners, it has 12 edges because it has 3x4 sides per (XYZ) plane
If you take a block of cheese, and you cut it zero times from all 6 faces of the block of cheese, you still have 1 block of cheese.
amazing explanation