Why is a number to the zero power always one?
963510
So that it is consistent with power rules - 1 = x^a / x^a = x^(a-a) = x^0.
494525
Think of it in terms of logarithms, since it is the inverse of exponents.\n\nStatement to prove:  1^infinity = 1\n\nRewritten in logarthmic form:  log[1](1) = infinity\n(Statement above is read: log base 1 of 1 equals infinity)\n\nWell that can't make sense, right?  It is a law of logs and of exponents that log[b](1) = 0 for any positive base b.  This is the same as saying b^0 = 1, for any b.  Also, log[b](b) = 1, for any positive b.  This makes since in b^1 = b.  Both of these laws would conflict this reasoning.  In fact, when b = 1, they conflict with one another.  Therefore, lets look at it with a variable instead.\n\nProve:  1^infinity = X\n\nRewrite in logartithmic form: log[1](X) = infinity\n\nNow, what value of X will give you infinity?\nCan you determine that?\n\nI would say it is indeterminate :)
999313
0^n = 0, where n is not equal to zero\nn^0 = 1, where n is not equal to zero\n0^0 = undefined. why?\n\nExpress the exponent 0 into n - n (n is not equal to zero)\n0^0\n= 0^(n - n)\n\nUse the property b^(x + y) = b^x · b^y\n= 0^n · 0^-n\n\nUse the properties 0^n = 1 and b^-x = 1/b^x\n= 1 · 1/0^n\n= 1/0\n= undefined\n\n^_^
238404
thanks 4 voting my answer as best but i didnt understand what u said\no divided by any no is always zero and with exception of zero where i guess it's not defined(aint that also zero iam not sure)
230430
The square root of -1 has no relation to zero.  It requires an imaginary number.  Division by zero is, unfortunately for our number system, impossible.  You cannot cut something into zero pieces, and in fact the very statement makes no sense.
7873
well in the demostrantion of the Real numbers...\n\nall the numbers and relations depend of the existence of 1... suck that there are some rules..\n\nso we even demostrate that 1 exists... and that 1 > 0... after that we can prove the existence of any other Real number..
600176
This is a Frequently Asked Question. Following is my previous answer:\n\nThis hypothesis is false:\nSuppose 0/0 = 1. Since 1 is the multiplicative identity, this implies (0/0)*a = a for all real numbers a. But (0/0)*a = (0*a)/0 = 0/0 = 1. This contradicts our hypothesis that 0/0 is the multiplicative identity, so the hypothesis must be false (proof by contradiction). QED\n0/0 is indeterminate as a limiting form. See http://mathworld.wolfram.com/Indeterminate.html .\nAs an algebraic expression over the real field, it is undefined: http://mathworld.wolfram.com/Undefined.html .\nThis is a Frequently Asked Question; the following is my previous answer:\n\nThis question may be asked so frequently (along with the infamous 0.999...) because no concrete axioms or definitions are ever given for the real number system until one studies higher algebra or real analysis; it is assumed that everyone knows what a real number is until then. The proliferation of these questions show that this assumption is no good.\nFor the sake of conciseness, I will not define what a real number is here, as only the algebraic structure of the real numbers is necessary to answer this question. The algebraic structure of the real numbers is what we call a field. A field is an algebraic structure that follows a finite set of axioms. These axioms define what statements are true and false in this system based on the laws of logic. You can find the field axioms here: http://mathworld.wolfram.com/Field.html .\nEach line is preceded by "For all real numbers a, b and c, the following is true...". Note that the proper form of the additive identity and multiplicative identity is that it prescribes the existence of some number 1 and some number 0 such that the equations hold. These same numbers are then used to prescribe the existence of additive and multiplicative inverses.\nDivision a/b is just shorthand for a*(b^(-1)). So in your question, you are looking for 0*0^(-1). It is easy to show that there is no such number as 0^(-1) in any field. Suppose 0^(-1) exists.\nThen 0*0^(-1) = 1. But multiplying both sides by 0^(-1) gives 0 = 0^(-1), which means 0*0 = 0 = 1. This is obviously false.\nSuppose 0 = 1. 0*a = a for all numbers a. But 0*a = (1 + (-1))*a = a + (-a) = 0. Thus 0 is not equivalent to 1 and our sole assumption that 0^(-1) exists must be false. Thus the expression 0*0^(-1) is undefined in a field.\nProof by contradiction is very powerful for many types of statements.\nAs for trying to define 0/0 using limiting behavior, note that you are evaluating a limit over a plane of the function f(x,y) = x/y. As we approach (0,0) along the x-axis, this limit is 0, but as we approach (0,0) along the y-axis, the limit diverges. Thus, the limit of f(x,y) at (0,0) does not exist, so f(0,0) remains undefined with no continuous way to complete it. The limit is not indeterminate for this function; it does not exist.\nThe expression 0/0 is said to be indeterminate as a limiting form because one can have many functions that approach the form 0/0 that have convergent limits of different values. For example, the limit as x approaches 0 of [sin(x)]/x yields the indeterminate form 0/0, but L'Hopital's rule reveals the limit to be 1. Similarly, multiplying the above limiting form by any real number will yield that number as the limit, even though the form is still 0/0. The expressions listed on this page: http://mathworld.wolfram.com/Indeterminate.html are called indeterminate because they do not have a fixed behavior when studied as limits of expressions.
2756
A to the power of n is:\n\nA * A * A ... n times.\n\nor\n\n1 * A * A * A ... n times.\n\nsince n is 0, w're left with\n1.\n\n--\n\nbut really it's defined to be 1.
497395
1.\nBecause e to the power minus infinity is 0.
955603
x+0=x is always true. This is the definition of the identity property of addition.
827218
Mathematics wouldn't be as advanced as it is today without the zero. This is because you may have noticed that 0 is very 'powerful' meaning that if for example i had the following expression:\n\n999999x8575x75238x652938x753928x7539042x34173x0\n\nthe answer would be 0. \nIt is this type of number that helps in simplifying alot of equations in calculus. Believe me, without the number 0, planes would not have been to fly!
39584
If you derive a constant, you always get 0
465001
Zero is known as the additive identity because when you add it to anything (say "x"), it doesn't change the value of "x".
189562
This is not simply a quirk of our usual number system: in any number system that satisfies certain basic properties such as distributivity and associativity, 0 multiplied by anything will give 0. So if you start with any non-zero number x, there cannot be any number which when multiplied by 0 gives you x, so there can be no answer to the question "what is x divided by 0". Mathematicians say that "division by 0 is undefined", meaning there is no way to define an answer to the question in any reasonable or consistent manner.
1073227
Zero is such a complicated number.   If you look at the way numbers are sequenced -3, -2, -1, 0, 1, 2, 3 etc. it would appear to be even.  There is a problem, however.  Zero came from the Arabic Hindu system and was considered a placeholder.\n\nZero is used as a place holder.  Numbering systems start with #1 and progress to 9.  1 starts over in the tens position and is followed by the placeholder 0 so we know one is in the tens position and not the single unit position.  The same is so for 20 and 30 and so on and so forth.  Hundreds, thousands, tens of thousands, hundreds of thousands, so on and so forth.\n\nNow back to the numbering sequence -3, -2, -1, 0, 1, 2, 3.  Zero is a placeholder and is neither odd or even.  It is a null value.  You have heard of zip, zilch, nada?  That is zero.\n\nZero was brought to the West by a mathematician named  Fibonacci some time in the 1200s.  It made numbers a lot easier to write.  Most numbers were written with a space where a zero would go today.  Made for lots of problems.  The numbers could only be understood in the context of the writing and could never be used by themselves.   1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 21 22 23 etc.  These are writen in sequence so you know the 2nd 1 is really a ten.  But standing alone, it still looks like 1.\n\nThe computer age really stirred the pot.  Programing in the binary system put 0 before 1.   Auuuuuuuuuugh! as Charlie Brown would say.  The computer needed continuity in the old on/off programing in binary language.  When 0 followed 1 the computer wouldn't compute.\n \nWhen people count, it is assumed that zero (or nothing) comes first before we have something (1).  But we couldn't use it as a placeholder, because one is a single unit.  There is nothing to hold a place for.  10 is another situation.  Zero is holding an empty unit spot.\n\nYou can have the shovel now, I think I've dug a deep enough hole. (0)  Chuckle.
145210
They call ZERO an even number, who are they that mention like this?\n\nActually, odd numbers are start with 1, then 3, 5, etc., even numbers are start with 2, then 4, 6, etc.\n\nIf you check at the numbers system, then you can understand what meant by odd number, even number, negative number, natural number, whole number and others. \n\nSo, here the point is, ZERO actually is not a even number (the smaller even number start with 2), then there was no reason to tell you that why can call it an even number.
467241
Add zero to anything and you get the same number you started with. It does not matter at which number you start with add 0 to it, it will still be zero. For example, you can start with 0 and add any number (n), lets say, you will still get that number n. Thus the notation for the previous example would be [0+n]. Ex. [0+1]=1 and again in the notation that you are asking about [0+n]=n, in the example above i just "substituted" 1, or just let [n] that represents any number for instances equal one. \n\nThe same goes for [n+0]=n, let [n] equal 1. add 1 to zero. What do you get? You should get one. see they (n+0 and 0+n )are the same thing.