What is the smallest number?

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: Whilst the question simply asks what the smallest number is, mathematicians define and discuss numbers in different ways depending on the needs of the problem. We classify numbers into sets, so the answer to "what is the smallest number" would depend on which set of numbers we are using. :

: The smallest natural number is 1. :

: If referring to the set of whole numbers, or integers, or rational numbers, or real numbers the answer to the question is in my opinion is zero. Zero is a whole number. Zero is an integer. Zero is a rational number. Zero is a real number. As far as mathematicians are concerned zero is a number. It is a member of several different sets of numbers. To have no value, or no quantity, no magnitude is to have less then, or a smaller amount of any other amount you could have. :

: If we don't like zero because we decide we want the smallest number that has a value or has a quantity or has a magnitude we are making a rather arbitrary decision. : If we want the number with the smallest quantity (other than no quantity)…an awkward question indeed…then zero is not the answer to the question, but unfortunately there would be no answer to the question. : If we try to answer the question "what is the smallest number with non-zero quantity", we might explore rational numbers. A rational number is defined as a number that can be represented by a/b where a and b are both integers and b is not zero. The rational number 2/3 for example is a single number. When defined as a rational number it does not need to relate back to 1, it does not need to be a fraction of 1, it does not need to be defined as a fraction of a whole though it can represent a fraction of a whole, it does however have a magnitude. 2/3 is an example of a number that is smaller than 1. Of course we could write all sorts of increasingly smaller rational numbers. In the set of rational numbers there is no smallest number with a non zero quantity. :

: The set of real numbers includes the rational numbers, so there is no smallest real number. :

: Irrational numbers like e or π or √2 are an interesting set. You can make increasingly smaller irrational numbers, such as π/10 or π/100 and so on, but whether π/10 is a number or it is merely an irrational expression, I am not sure. I suspect expressions like π/10 are not numbers. :

: There is another set of numbers mathematicians use called the set of hyperreal numbers. Infinitesimals are an example of these. Infinitesimals are discrete quantities. They are numbers. But the value of a single infinitesimal when converted to a real number is zero. There is no smallest infinitesimal because you could take any given infinitesimal and divide it by 2, or by 100….. : To summarize: :

: If we don't put any limitations on the question and take it for how it is written, then the smallest (real) number is zero. By Occam's razor this seems to be the best answer. :

: If we limit the question to natural numbers the answer is 1. :

: If we limit the question to real numbers or rational numbers or integers or whole numbers and rephrase the question to say "what is the smallest number with a non zero quantity…" then there is no answer. :

: If we take other number sets (like hyperreal numbers) then we can come up with things like infinitesimals that have discrete quantities and values and magnitudes, but again there is no answer to the question. I would be very interested in a post by a real mathematician regarding the possibility of a smallest number in more obscure number sets. : The Guinness book of records states that Graham's number is the biggest number that has actually been used in a mathematical proof, so perhaps we should ask "What is the smallest number that has ever been used for anything?".

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Instead, please use the discussion area by using the View Discussion button shown below this answer section. The text below consists of the answers which were given before the Discussion about this question was opened. == == * Zero. Any other number, regardless of sign, has a magnitude. Zero has no (zero) magnitude. * Anything beyond zero in either direction becomes an increasingly large number. Negative infinity, if it is to be regarded as a number, is simply a large negative number. It is important to differentiate between the concepts of 'small' and 'less'. Negative infinity is certainly LESS than zero, but is larger than zero NOT smaller. * First, take the largest number possible and put a negative sign in front of it, and you get the smallest number. Which is much lower than zero * The actual smallest number is not a number at all, but a concept. The number system (...0,1,2...) is not a fixed system, so there is no endpoint. Thus, there is no smallest 'number', but for arguments sake, negative infinity can be considered the smallest number. However, if one were to get technical, 'infinity' does not represent a number but the concept that the system never ends. Hope I've confounded you. * Lets think about a real object (some quantity of matter) to represent our number. Lets say we have 50 units of matter, which keeps on decreasing in size. 50 down to 40, 30, 20, 19,18,17,10, 3,2,1, 0.8, 0.7, 0.6 etc. * Basically as small as you can get without hitting zero. This is 1 over infinity (which is said to approach 0 but never reach it) * Same as 1 over a very small number (eg 1/infinity) approaches a very very big number (infinity) , since 1/(1/infinity) cancels to = infinity. * 0 is not generally considered to be a number since we cannot equate anything to it.('cos zero means nothing at all not even a number (no matter or anything neither tangible or imaginary to represent)). * The danger of allowing anyone to answers these questions is just this: you get untrue statements like the above '0 is not generally considered to be a number since we cannot equate anything to it'. * Any small number that is added by -1 or subtract by 1, that is the smaller number than the given number. So, there is no such things as smallest number. * First, it depends what you mean by "number." The smallest natural number is zero. If you take "smallest" to mean smallest in absolute value, then the smallest real number is 0 as well, however there is no least real number. By the least element in an ordered set we mean the element that is not strictly greater than any element of the set. The least element can also be called the "smallest" element, but in the case of the real numbers, this is misleading because negative numbers with large absolute values should not be considered small. A "small number" almost always means a number very close to 0, in which case zero is the

* The smallest number is the "-" so which means that the number can also become the -99999999999999999..............99999999 any thing like that. or any thing like the number like -89797776565654356787879 but any thing with a bar on any negative number is he smallest number. * It depends what you are talking about: -if you are talking about LENGTH:

the smallest possible distance is the 'Planck Length' of 1.616252 x 10-35 meters. It the 'quantum of length', the smallest measurement of length with any meaning.

-if you are talking about TIME:

the smallest possible time is 'Planck Time' of 10-43 seconds. This is the 'quantum of time', the smallest measurement of time that has any meaning. No smaller division of either has any meaning. (Both named after Max Planck who first formulated the quantum theory in 1900 and won the Nobel Prize for Physics in 1918.)

* You can still talk about the number 10-10000 in the context of pure mathematics. You can even talk about the number 1/(10^(10^(10^(10^(10^10^10))))). This number still exists even though there is no length or time that small. * The smallest number is Zero. This whole discussion is rather silly. * The Question that was actually asked was: "What is the smallest Number?" not "What is the smallest Number actually used for?" . It has been proven by Quantum Mechanics that in the real world there is actually no such thing as "Zero" because there is a smallest possible length which cannot be sub-divided any further: it is an actual number, a finite distance. Actually "Zero" (ie 0.0000...0)) is not a Number: it is an abstract concept. The Collins English Dictionary defines "Zero" as: "The symbol 0, indicating an absence of quantity or magnitude; nought." * That is true. Zero is an abstract concept. All numbers are abstract concepts, so there is no problem there. In pure mathematics, everything is an abstract concept. Mathematics is not concerned with the real world. It is hard to argue that zero is not a number, because it obviously is. * In the book Naive Set Theory by Pual Halmos, it says: "How much is two? How, more generally are we to define numbers? ... How would a mathematician define a meter? ... by a more or less arbitrary convention an object is selected and its length is called a meter... motivated by the considerations above, we have earlier defined 2 as some particular set with (intuitively speaking) exactly two elements... * Lets assume 7 has already been defined. How should we define eight? Where, in other words can we find a set consisting of exactly eight elements? We can find seven elements in the number (set) seven. What should we use as the eight? A reasonable answer is the number (set) 7 itself... We are now ready to define the natural numbers. In defining zero to be a set with zero elements we have no choice; we must write 0 = [the empty set]. If every natural number is to be equal to the set of its predecessors, we have no choice in defining 1, 2, or three either; we must write 1 = {0} ... 2 = {0,1} ... 3 = {0,1,2} etc. " * Evidently zero is a natural number. * If we don't like 0 for an answer, and we accept the fact that negative numbers are not smaller numbers, and we accept that infinite is not a number but a concept, then the smallest number would be one divided by the biggest number. The biggest number with a name is Graham's number. One divided by Graham's number would be the smallest number. 1/g64 Or you could take One divided by Graham's number squared. * The smallest named number is 0.00000000000000000000001 one septillionth * I would think that the smallest (not least) natural number is 0. If we wanted to use positive numbers we should just use reciprocals of very large number like googolplex. Where N-minex=10-N, the numbers googolminex (the reciprocal of googolplex). But then again you could keep adding more -minex's to the name of the # to make it smaller. I think googolminex is the smallest named number but in theory the smallest number I think is an infinith. * Anything that goes away from zero gets bigger, so the closer you get to zero the smaller it is. Zero isn't the smallest number, because zero is nothing. * So the closest is near to 1/Graham's number to the power of Graham's number, Graham's number of times.