can anyone tell me the uniqueness of the number zero? Mathematically.?
484386
0 as a number\n\n0 is the integer that precedes the positive 1, and follows −1. Zero first appeared as a number in Brahmagupta's work dated to 628. Prior to that Babylonians used a space marker that played one of the functions of zero. Babylonians did not have a special symbol for zero. In most (if not all) numerical systems, 0 was identified before the idea of 'negative integers' was accepted.\n\nZero is an integer which quantifies a count or an amount of null size; that is, if the number of your brothers is zero, that means the same thing as having no brothers, and if something has a weight of zero, it has no weight. If the difference between the number of pieces in two piles is zero, it means the two piles have an equal number of pieces. Before counting starts, the result can be assumed to be zero; that is the number of items counted before you count the first item and counting the first item brings the result to one. And if there are no items to be counted, zero remains the final result.\n\nWhile mathematicians all accept zero as a number, some others would say that zero is not a number, arguing one cannot have zero of something. Others hold that if you have zero dollars in your bank account, you have a specific quantity of money in your account, namely none. It is that latter view which is accepted by mathematicians and most others.\n\nAlmost all historians omit the year zero from the proleptic Gregorian and Julian calendars, but astronomers include it in these same calendars. However, the phrase Year Zero may be used to describe any event considered so significant that it virtually starts a new time reckoning.\n\n\n\n0 as a numeral\n\nThe modern numeral 0 is normally written as a circle or (rounded) rectangle.\n\nOn the seven-segment displays of calculators, watches, etc., 0 is usually written with six line segments, though on some historical calculator models it was written with four line segments. This variant glyph has not caught on.\n\nEarly Europeans hesitated to consider zero as a numeral or number. Leonardo of Pisa or Fibonacci says the following in 1202 AD when the Indian number system arrived in Europe.\n\n"After my father's appointment by his homeland as state official in the customs house of Bugia for the Pisan merchants who thronged to it, he took charge; and in view of its future usefulness and convenience, had me in my boyhood come to him and there wanted me to devote myself to and be instructed in the study of calculation for some days. \nThere, following my introduction, as a consequence of marvelous instruction in the art, to the nine digits of the Hindus, the knowledge of the art very much appealed to me before all others, and for it I realized that all its aspects were studied in Egypt, Syria, Greece, Sicily, and Provence, with their varying methods; and at these places thereafter, while on business.\n\nI pursued my study in depth and learned the give-and-take of disputation. But all this even, and the algorism, as well as the art of Pythagoras, I considered as almost a mistake in respect to the method of the Hindus. (Modus Indorum). Therefore, embracing more stringently that method of the Hindus, and taking stricter pains in its study, while adding certain things from my own understanding and inserting also certain things from the niceties of Euclid's geometric art. I have striven to compose this book in its entirety as understandably as I could, dividing it into fifteen chapters.\n\nAlmost everything which I have introduced I have displayed with exact proof, in order that those further seeking this knowledge, with its pre-eminent method, might be instructed, and further, in order that the Latin people might not be discovered to be without it, as they have been up to now. If I have perchance omitted anything more or less proper or necessary, I beg indulgence, since there is no one who is blameless and utterly provident in all things."\n\nLeonardo of Pisa \nHere Leonardo
443911
I do not think you can.\nSo Bhaskara tried to solve the problem by writing n/0 = ∞. At first sight we might be tempted to believe that Bhaskara has it correct, but of course he does not. If this were true then 0 times ∞ must be equal to every number n, so all numbers are equal. The Indian mathematicians could not bring themselves to the point of admitting that one could not divide by zero. Bhaskara did correctly state other properties of zero, however, such as 02 = 0, and √0 = 0.
540459
something close to zero
187658
it's defined \n\n0^0=0
367158
(0! + 0! + 0! + 0! + 0!)!\n\n\n0^0 is undefined.
1109270
Once upon a time there was no zero. Of course people knew if they had nothing, but there was no mathematical notation for it. Zero was independently invented only three times.\n\nThe first recorded zero is attributed to the Babylonians in the 3rd century BC. A long period followed when no one else used a zero place holder. But then the Mayans, halfway around the world in Central America, independently invented zero in the fourth century CE. The final independent invention of zero in India was long debated by scholars, but seems to be set around the middle of the fifth century. It spread to Cambodia around the end of the 7th century. From India it moved into China and then to the Islamic countries. Zero finally reached western Europe in the 12th century.
18903
There is no number that will multiply by zero to not equal zero because for example 5x5 would be the quantity of five added five times, 5x1 would be the quantity of five added one time, and 5x0 would be the quantity of five added zero times meaning there is nothing there or, if you prefer, that the quantity (Five) is not addded at all.\n\nBTW- even infinity multiplied by zero equals zero\nBTW- There is NO NUMBER!!! that will multiply by zero to to equal anything but Zero. Not Infinity or undefined!! 0x0=ZERO Infinity x 0=ZERO, ETC.
348468
This 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, 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. Tus the expression 0*0^(-1) is undefined in a field.\nProof by contradiction is very powerful for many types of statements.\nAs to the limit argument, 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. 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.
682438
Infinity.Any number divided by 0 is infinity.
778163
The problem is to show that f is identically 0 using the density of the rational numbers. I would strongly suggest to the asker that they do their own homework at this level. If you don't understand the questions or techniques, you will have a *lot* of trouble later on.
579225
Division by zero is undefined, even if you divide zero into zero.\n\nhttp://en.wikipedia.org/wiki/Division_by_zero
797722
AARGHHH!!  I withdraw my earlier attempt.  It must be dementia, I'm afraid.  It's still asking about 1000! (1000 factorial), of course, but a careful accounting is needed, not only of the effect of all numbers ending in: \n\nA. 1, 2, 3, ... 9 (I did this by a variety of ways, always coming up with a final non-zero digit of 6 for the product of all such sets), \n\nbut also of the following two other types of number, which I regrettably neglected the first time around:\n\nB. 10, 20, 30 ...90 (as often as sets of numbers like that occur), and finally, of types:\n\nC. 100, 200, 300, ... 900 (this set occurs but once).  The final, unique 4-digit number (1000) only adds three zeros at the end and doesn't affect the final non-zero digit.\n\nO.K., here goes:\n\nNumbers of Type A.\n\n1 x 2 x 3 x ... x 9 = 362880; final non-zero digit is 8.\n\nThere are 100 such "basic sets" of type A altogether.  \nMultiplying them together, set after set, we get the \nfollowing repeated cycle of non-zero ending digits \n(top row is # of basic sets used so far; bottom row is the corresponding final digit.)\n\n1   2   3   4   5   6   7   8   9 ...\n8   4   2   6   8   4   2   6   8 ...\n\nNotice that after every four basic sets have been used, we end up with 6 as the final non-zero digit.  Since there are exactly 100 such sets (a multiple of 4), we end up with that same last non-zero digit for the product of ALL numbers of type A.  (But we're not done yet.) \n\nNumbers of Type B.\n\nThere are exactly 10 sets of numbers of Type B, i.e. ending with  exactly one (but not two or three) zeros.\n\nMultiplying just these sets together, we'll have the following analogous results:\n\n1   2   3   4   5   6   7   8   9  10  \n8   4   2   6   8   4   2   6   8     4   \n\nSo, the last non-zero digit in these "10's products" is 4 for this lot.\n\nNumbers of Type C.\n\nWe have just one set of "100's".  For this, the last non-zero digit is 8 (it's just our "basic set" with a large number of zeros following it); only the last non-zero digit is of interest.\n\nSo, to summarize, the different sets we've divided our numbers into, when multiplied together, give us final non-zero digits as follows:\n\nType A:   6\nType B:   4\nType C:   8 \n\nFinally, then, the last non-zero digit of the product of everything is the last digit of:\n\n6 x 4 x 8, or 2.  (PHEW!!)\n\nWell, it's now the value of 2 (as reported by zanti3 from the 100 factorial website at:\n\nhttp://www.justinwhite.com/big-calc/1000.html)\n\n(Finish off this URL with 1000.html, instead of 1000... The Yahoo site would not reproduce the full URL from my draft.  Alternatively, just go to Yahoo or Google with 1000 factorial 1000 - this worked for me.) \n\nBoy; what a problem!  Evidently the jury is still out on the question of my impending dementia.  (See zanti3's kind words, below about my earlier fiasco.) \n\nThanks again to zanti3 for bringing the existence of a (1000 factorial) web site to my attention.\n\nLive long and prosper, everyone.
152638
It is the number 36 in binary.\n\nFrom right to left:\n0 = 0\n0 = 0\n1 = 4\n0 = 0\n\n0 = 0\n1 = 32\n0 = 0\n0 = 0\n\n32 + 4 = 36
945266
Does 0.9999999999999999999999999....(continued) = 1.0\n\nA great topic in maths which will actually interest people because the above statement is true.
288098
There are a countably infinite number of rational numbers between 0 and 1.  (Countably infinite means it's the same order of infinity as the counting numbers 1,2,3,4....)\n\nTo see that the rational numbers are countable, enumerate them like this:  0/1, 1/1, 0/2, 1/2, 2/2, 0/3, 1/3, 2/3, 3/3, ... This counts some numbers multiple times (like 1/2 and 2/4), but that's OK. The point is that any rational number between 0 and 1 will eventually get reached with this kind of counting.\n\nBy contrast, the real numbers are not countable - any way that you try to count them, you'll end up leaving some out, no matter how long you count.
64293
No, it is not true that is e^n a transcendental number for all values of n.\n\nConsider the value n=0.\n\ne^0=1, and 1 is a rational number.\n\nSome mathematicians include zero in N, the set of natural numbers, and some do not. If you exclude zero as a possible value for n, then the answer to your question will be different.
1040908
Zero is not that intuitive and was ‘invented’ for at least two reasons (and not at the same time). It is a place holder like in 10 but we cam also say that 10 is X in which case we do not need a place holder. It other application it represents 'nothing'. What is nothing? cayou imagine 0 pencils? What color are they, what shape? If you think carefully it becomes an uncomfortable concept. It took zero to appear before mankind for quite sometime.