Ethan

I am new at this . . . starting a thread I mean . . .

I have been logging on to this site for a while now and have enjoyed and even posted on some of the threads I've read.

Now I have a reason to start a thread because I have an algebra question that I do not understand and need help. Halx said in his thread about how to use this forum that this is the place for secular knowledge. . .

okay . . .

here goes. . .

How come 'n' to the power of zero equals one? No matter what 'n' equals (except maybe zero) that number to the zero-th power always equals one.

any thoughts?

lordjeebus

consider this:

k = whatever
for any non-zero x, x^n = x^(n+k-k) = (x^(n+k))/(x^k)
for instance, x squared is the same as x cubed over x.

when n = 0, x^0 = (x^(0+k)/(x^k) = (x^k)/(x^k) = 1, because anything divided by itself is one.

hmm. does anyone have a cleaner explanation? i have a feel for why x^0 = 1 but it's hard for me to articulate.

cliche

I always found a good way of understanding it was:

x^n * x^p = x^(n+p)

if p = 0, then we need it to come out x^n * x^0 = x^(n+0) = x^n
This can only work if x^0 is 1 (a number multiplied by 1 is itself).

Any help? I'm sure there are many other ways to show it - it just depends which works for you

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KnifeMissile

Hey, you're confused by x^0? Wait til you look at x^n when n = 2/3, or when n = pi (circumference / diameter), or when n = i (square root of -1).

It's just fun, all around!

TIO

KnifeMissile, that's nothing compared to the joys of the following statement:
If I offer you half a pen and ask you to choose one of them, there is approximately negative one third of a way you can accomplish that task.

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Strewth

JadziaDax

Here's a site that helps explain it a little better: http://mathforum.org/dr.math/faq/faq...to.0power.html

Sapper

Oh crap ... just wait until you consider infinity.

There have been some great discussions of the relative sizes of infinitiy on here.

Just know that infinity has different sizes (eg: infinity integers is smaller than infinity real numbers)

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phukraut

think of a group G with the following operator *.

say x is in G. now consider how x operates with itself, this is a general definition of integer exponents when talking about groups:

x * x * x * ... * x = x^k, for some positive integer k.

now every group has an identity element, call it I. if we try and find k such that, for any y in G,

x^k * y = y, then x^k = I by definition for some k. if k>0, then x^k * y may not be y, so x^0 = I. if G is the normal multiplicative group and * is just multiplication, you have that x^0 = 1.

here's another way to think of it, say we want to find the inverse of x^k. by group theory, for some z in G, x^k * z = I. the answer is z = x^(-k). what if k=0? then for some t, x^0 * x^t = x^(0+t) = I. but 0 is the additive identity, and so by definition, 0+t=0 only if t=0. therefore x^0 is its own inverse. therefore x^0=I.

it's just a useful convention for identity transformations. hope this sheds some light.

Ethan

Pardon my ignorance. . .

I'm still learning the basics. . .

What do we mean when we use the symbol ^ ? I just got more confused . . . but thanks JadziaDax for the link. I will try it and see if it helps. I was hoping that maybe some one could illustrate the equation with a word problem or diagram that would make more sense. At this point, 'n' to the power of 0 = 1 just seems like a fomula that is used even though it does not seem like a rational equation.

I must either be beyond help or still learning more elementary things than you guys thought. If so, then what I need are more elementary definitions. . . I think . . .

What do you think?

numist

the symbol ^ is translated as "to the power of" so 2^3 is two to the third power (2 cubed).

Kinda funny, this place is a huge resource for information, but sometimes the answers look harder than the questions!

Anyway, hope that helps.

The way I thought of it is like multiplying by 0. No one ever really was able to explain that to me, it just simply was 0.
Same thing with anything raised to the power of 0. It just is 1, most of the explanations always looked like advanced proofs to me too...

KnifeMissile

Wow, you are totally green! I didn't follow that link but I'm sure it will help. If it doesn't, it will probably mean that you haven't learned enough math yet to understand any specific explanation given to you.

Here's a simple plain-english explanation. It's consistent with the other rules concerining the power of numbers and it's convenient. So, we define n to the power of 0 to be equal to 1.

Oh, and ^ is used to represent exponentiation. So, 2^3 is the same thing as saying 2 to the power of 3, or 8.
You couldn't figure this out from context?

Poloboy

No dude, infinity doesn't have sizes. I know what you mean, I understand the different number sets, but the problem is that infinity is a concept, not a number. You can't think of infinity as a limitless set of values, such as natural numbers, certain fractional intervals, or even the set of real numbers. It's not a limitless set of values, but simply is the concept of a continuing function. There are no increments associated with it. Using infinity in algebra and calculus is ok, but you have to remember that when you write it, what you're writing is just a symbol representing the concept on the graph you're interpreting.

Sorry about the hijack.

KnifeMissile

Actually, I'm not sure that you do understand what the original poster was talking about. But, first thing's first...

Using infinity in algebra and calculus is not okay. There are some number systems that incorporate an infinity as an element but they are not typically used by scientists, engineers, or joe schmoe. They are certainly not groups, rings, or fields.
Calculus simply doesn't use inifity at all - at least, not as a number. The word is used colloquially, or as shorthand, but it is certainly not a number. I think this is what you were thinking of when you replied. Calculus is defined using limits. Values that are arbitrary or unbounded, but not infinite!

Now, back to the main point. There is a definition of the "size" of a set, even if it is infinite. Although we can't define the cardinality of an infinite set in terms of non-negative integers, we can define their relative sizes.
Set A is defined to have a strictly greater cardinality (or simply put, is bigger) than set B if there exists a bijection from B to a strict subset of A while no bijection between A and B exists.

For example, the power set of A, denoted P(A), is defined as the set of all subsets of A. Now, there is a famous (and very clever) proof that the set P(A) is strictly bigger than A.

Poloboy

Do you have a link?

phukraut

http://mathworld.wolfram.com/CardinalNumber.html

btw, for a finite set A with n elements, the powerset P(A) is of size 2^n.

KnifeMissile

Unfortunately, that's not a very good link, phukraut. It actually deals with something else and, only in passing, touches briefly with what we are talking about. Here's a better link . It's not perfect, either, but it is evidence that I'm not just making this up. It doesn't define greather than or equal to but I'm sure that when you see how equality (something it does define) is defined, you will find my definition quite believable.
I assure you, I know what I am talking about.

I applaud you, Poloboy, for your reply (although I do question why you presented yourself with such authority, in the first place). The empty and rhetorical responses from the Politics forum have turned me cynical. They often ignore what was said and replace content with insults. It's very refreshing to not see that here.

Now, do you think you are in a position to understand the proof that |P(A)| > |A| ? It's really cool but requires a firm understanding of these definitions, as well as mappings between sets...

Oh, and more to the point, there is an important proof that the set of real numbers is strictly bigger than the set of integers (despite how they're both infinite), but that proof is not too meaningful to someone just learning this stuff.
Also important, there is a simple proof that the non-negative integers have the same cardinality as the entire integers!

This stuff is pretty cool, eh?
Off topic, but cool...

Poloboy

Yeah it is really cool. I didn't mean to come off sounding like an authority on the subject, I'm very far from it. I just wanted to quash the thinking of infinity as a number I'll check out the link and see what I can glean from it, but I might not have the background schooling to get all of it.

Dilbert1234567

i cant give you a good reason, but it is a mathmatical law, anything rasied to teh 0th power is 1, infinity raised to the 0th power is 1


kind of like anything multiplied by 0 is 0

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cliche

Dilbert1234567 - you sure?

anything (positive) divided by zero is infinite (a/o=i); so (rearranging to i*o=a) should say that multiplying an infinite number by zero is undefined...

cue arguments about whether infinity can be used in this way

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sigma1042

ethan
you probably don't care to know all that theoretical "what is zero?" and stuff
ok to try and simplify and show you why n to the 0 power is 1
as the "power" number gets smaller, the answer gets closer to 1
when you start using decimals (actually fractions) as your power
think of that number as a root as in square root ("power" is 1/2) or cube root (power is 1/3) and so on
the square of the square root of a number equals that number

(n^(1/2))^2= n or (n^(1/5))^5 or (n^(1/x))^x
try writing it down to see it better
get a scientific calculator
we'll use n=2

2^3=8 , 2^2=4, 2^1=2

2^(1/2)= sq root of 2 = 1.414213 ; (1.414213^2=2)

2^(1/5)= 1.148698 ; (1.148698^5=2)

2^(1/1000000)= 1.00000069 ; (1.00000069^1000000=2)

this doesn't explain it totally correct, but you can see that the closer you get the power to zero, the answer gets closer to 1, so you can kind of assume that when you reach zero, you reach one

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Peetster

Actually, anything divided by a number approaches infinity as the number approaches zero. Mathematically, it can't actually get there.

Ethan

Thank you for all your replies. Not having a great mathematical background, seeing all this makes you guys look like whizzes.

I think the link that JadziaDax gave me helped a little but its going to take some practice before I totally get it. It is as if 0 is the 'landmark' of the number line and that everything is relative to it . . . except that the closer you get to it, the more undefinable it gets.

Anyway, its cool to see that you guys were willing to help.

Thanks again.

numist

well, Ethan, if we are whizzes, it looks like its rubbing off...

I never really got into the number line theory, my thing was always geometry and algebraic manipulation... undefinables and infinitismals kill me, especially in calculus...

shakran

OK, let's make this really easy.

Let's say we have 2^3 (2 to the power of 3)

that's 8, right?

And 2^2 is 4 right?

2^1 is 2 right?

So you can see that we're halving each number as we go down in powers (8, 4, 2)

Go down 1 more power. What's half of 2? 1. So 2^0 = 1

Another example:

3^3 = 27

3^2 = 9

3^1 = 3

3^0 = 1

because

27/3 = 9
9/3 = 3
3/3 = 1


No matter what number you pick, when you work it down in powers you always end up dividing the number by itself, which is 1