Understanding Exponents (Why does 0^0 = 1?)

That’s a great explanation.
But this make me wonder of how other people brain operate. For me most of the time I understand math as rule of the game and I don’t need interpretation to digest this. How about others?

Have to re-read this but when i entered 0^0 into my calculator and indeed “1” came-up i know i mis-comprehended a critical point in high school math class…

Excellent explanation. I teach a high school Algebra 2/Trig course and I will definitely be using this approach when presenting exponents to my students this fall.

@watchmath: Thanks – I guess it depends on your personality. I need math to make sense at a deep intuitive level, it just doesn’t feel right if the rules seem “arbitrary” to me :).

@x: Yes, I think many calculators may be programmed with that result.

@DJ: Great, glad it was useful!

thios is a realy good website about exponents

@billy: Thanks, glad it helped!

This material is incredibly easy to digest and apply. Do you have any material on logarithms in general?

Here is a possibly more simple understanding of why 0^0 = 1 (depending on your point of view).

Take a total map to be a function that is defined on every point in the domain.

Then,
We can think of n^m as the number of total maps (ahem functions) from the m element set to the n element set. To see this draw 2 circles, but n dots in one and m dots in the other (it is advisable to pick small n and m!). Then very methodically draw all possible total maps. Maybe make a tree.

Now, the empty set is a subset of every set, thus there is always a map from the empty set to any set. Moreover, this map is unique. It follows that there is a map from the empty set to the empty set, and this map is unique. Then the number of maps from the 0 element set to the 0 element set is 1. Thus

0^0 = 1.

@Jonathan: Thanks for the comment! Yes, that’s another, more detailed/combinatorial way to think about it. Depending on the context of the problem, growth or combinations may make sense. Often times people see e^x and are trying to figure out the meaning of a 0 exponent; other times, you have n^m (n choices, m decision) and this interpretation helps too.

Hi… Is your book available anywhere in India? Thanks.

@Lampan: Hi, you can get the ebook anywhere that supports PayPal, Google Checkout, or credit card. Thanks!

Hi Kalid,

My math teacher brough up the 0!=1 parodox today but wasn’t able to explain it so I thought I’d research it myself. It makes sense to me now except for I cant see why the expand-o-meter always starts out at one. If I accept that fact everything else is clear I just dont see why it should be and one. Thanks!

@Alley: Great question! Here’s how I see it – exponential growth is about, well, growing stuff with multiplication! So we need to start with something to grow and compare. The reference point is 1, our original amount, and we see what we transform it into.

If we don’t do any changes, then our input = output, so we finish with 1.

Another way to think about it “times 3” really means “1 times 3”, that is, start with your original amount and then scale it up. We often drop the implicit “1”, but when doing exponents it can help to explicitly talk about it. Hope this helps!

What will happen if I place a base number of 1 to the expandotron, set it to zero growth and turn back time for 1 second?

@Lee: Let me see if I get the question. With the base number 1 in the expandotron, we’re saying that after a unit of time, we have the same result. In fact, with 1 as the base, any amount of time will have the same result (no change). If we use imaginary numbers, strange things can occur, but that’s another article :).

Turning back time for 1 second is asking “1 second ago, given this rate of growth, what was our number?”. Since the rate of growth doesn’t have any effect, the number is the same as the original. When you aren’t changing, 1 second ago you are still at your present amount.

In math terms, 1^x = 1 for any real x. Hope this helps!

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Will you see:
30. Asked Is 0 to the power of 0 equals 0 a final answer 27 Oct 2010 05:54
31. First answer to Is 0 to the power of 0 equals 0 a final answer 31 Oct 2010 12:29
on WikiAnswer,
as my comment on your proof 0^0 = 1.

Thanks that really help though i still have a question in mind

by the way i do A-Level maths curently

i saw in a cartoon an equation that read

46x*87y over the sq root of 90?

i mean is that even close to possible cos i know you said that ‘once you work out the impossible, anything can happen’.

0 to the power 0 doesnt equal anything
it equals infinite or negative infnite

i will explain everything

if you have zero, its a value, aint positive nd aint negative
when you power that ‘nothing’ by ‘nothing’, as a result, you dont actually get a number
0^0 does not equal 1 or 0
it equals infinity OR negative infinity

0^0 wont equal 1, i can believe that
when 2 rules clash, neither would be valid
e.g. if you type 0^0, the calculator would be saying ‘syntax error’ ( if you have a scientific calc) when what its trying to say is that it cant have an answer because well the only time you can have 2 or 4 answers is in quadratic or sim quad equ

and calc cannot display 2 answers because that we cannot then perceive a calc as a correct instrument of mathematics.

and by the way jjeigers, infinity isnt a number
its a CONCEPT, its a way that we think of number
or bacially a limitless thing. It cease to exist as a number. However, google and googleplex are, google is 10^100 and google plex is 10^google(this is a huge number. probably close to infinity? no.)

0^0 is one of those questions that prove we haven’t found out everything about maths. we may know a heck of a lot, but not everything