how real numbers has been made?
113789
Go to http://en.wikipedia.org/wiki/Real_numbers#History.
901768
This is actually the special case of the von Neumann's construction of natural numbers. This is also regarded as the standard construction of natural numbers in set theory.\n\nBefore we construct the set of natural numbers, we must first define succession, which was stated by Peano axioms as follows:\n[1] There is a natural number "0".\n[2] Every natural number a has a natural number successor, denoted by a+. \n[3] There is no natural number whose successor is 0; i.e. no such x exists such that x+ = "0".\n[4] If a ≠ b then a+ ≠ b+. In other words two different natural numbers have different successors. \n[5] If a property is possessed by "0" and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers. (This postulate ensures that the proof technique of mathematical induction is valid.) \n\nNote that the "0" in the said axiom is just a label and not the usual 0 that we know. In other words, the "0" here is just a starting point in which when successively applied by a successor function (+) will satisfy the Peano's axioms.\n\nNow, to construct the set of natural numbers, we do the stardard way. That is we define 0 to be { } and define the successor element, a+ to be a+ = a ∪ {a} for every set a. (Note: we read a+ to be the successor of a). Then we have the following:\n0 = { }\n1 = {0} = {{ }}\n2 = {0, 1} = {0, {0}} = {{ }, {{ }}}\n3 = {0, 1, 2} = {0, {0, {0, {0}}}} =  {{ }, {{ }}, {{ }, {{ }}}} \nand so on.\n\nTo generalize, we get\nn = {0,1,2,…,n−2,n−1} = {0,1,2,…,n−2} ∪ {n−1} = (n−1) ∪ {n−1}.\n\nFrom this construction, we see that there are exactly n elements set n. So, n ≤ m if and only if n is a subset of the set m.
71038
Looks like the first 4 answerers have a good imagination but know nothing about countable sets\n\nSpecifically:\n\n> if you take the real number system,\nit extends from -infinity to +infinity.\n\nLOL, how does that answer the question?\n\n> I think it's uncountable. Yeah, definitely.\n\nOh, are you sure? I mean, really, really sure?\nHow about a reason?\n\n> Uncountable. because between any two real numbers there are infinite number of real numbers.\n\nSo what? Infinite can be countable or not.\n\n> however if u think they are countable, give it a try\n\nLOL, really?\nLet me see you count all natural numbers then (which ARE countable)\n\n> The set of real numbers is uncountable as it contains all the positive natural numbers and the natural numbers are uncountable.\n\nThe natural numbers ARE countable by definition!!!!!\nDo you even know what countable means?!\n\n\n\nOk, let me give a serious answer now.\nThe set of real numbers is uncountable. It was proven by Cantor, in at least two ways.\nThe proofs are not simple enough to be copied here, so I'll just refer you to them.\n\nHis first argument: http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof\nHis diagonal argument (best known): http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument\n\nRead these pages and follow the proofs.
40697
imaginary numbers are used in physics and engineering to describe waves.  it does not mean that the physical thing you are describing is imaginary, but rather that the mathematics that describes it uses complex numbers.\n\ncomplex numbers are made up of 2 parts:  one part is called "real" and the other part is called "imaginary".  the imaginary part is just a real number multiplied by i, where i is defined as the square root of -1.  so a complex number looks like: 4 + 7i.\n\nif you know any basic math, you know that square root is only defined on the positive real numbers.  think about it - you cannot square anything and get a negative number, because the minus signs always cancel.  this is why the square root of -1 is called "imaginary" - the concept was derided for a long time by early mathematicians, until it was seen to be so useful that people finally accepted it.\n\nthe relationship with a wave comes from euler's equation:\n\ne^ix = cos x + i*sin x\n\nas you can see, e raised to an "imaginary" power can be written as the sum of 2 waves.  there are a lot of applications in physics and engineering where it is far more convenient to write the equation for a wave in terms of an exponential rather than in terms of sine and cosine.  in fact, in many advanced works you almost never see sin/cos and waves are often only described as a complex exponential.
511802
Our story begins with a primitive man, who needed a way to keep track of things. Eventually, he needed a way to know how many wives, children, pets, how much gold, etc he had. He discovered the natural numbers (1,2,3,4,...)\nNatural numbers were probably discovered independently by many of our early ancestors.\n\nFractions were introduced later to deal with division of stuff into different portions. Some weirdo also thought of negative numbers.\n\nFrom here, people began to add to the natural numbers. The Hindus invented 0 (zero) to represent nothingness, a topic which could consume an entire book. By that point we had integers.\n\nThe Greeks then discovered that they could arrive at numbers which could not be expressed as a ratio of two whole numbers. Using Pythagoras' Theorem, they stumbled upon irrational numbers. Now we have all of the real numbers.\n\nWhen mathematicians began to search for solutions to higher-degree equations, they invented a square root for -1 and created the complex numbers.\n\nCool, eh?
981541
No, it follows from 1+1 = 2\n\nWhich follows from 1+0 = 1\n\nBut that can be considered an amazing "fact".  In reality, it's an axiom - a definition.  One of the only true defintions that form the basis of mathematics.
1007041
The roman numeral for seven (VII) can be made by cutting the Roman numeral for twelve (XII) in half hortizontally.\n\n-unpredicktaboh-
1062811
The numbers 1, 2, etc are just symbols we give to a identify a quantity.  As for mathematical equations these have been calculated in order to work something out, and then tested repeatedly.  The rules are determined when, using the equation, the answer is always the same and/or predictable.  This is what all scientific theories are.  They are ways of measuring things, which are verifiable.  They have no 'reality' outside our minds, because they are used to measure the world as we perceive it, but they do help us to make sense of the reality that we perceive.
169470
The domain is the whole set R (all real numbers) since your function is defined for every x from R.\nSimply, you can put any real number to be x and you will have  real number as an answer.
120877
well i dont fully understand what you mean by structure but the real numbers are pretty much everything that is not imaginary so \n(-infinity,infinity) this includes rationals, irrationals, integers, and natural numbers.  Numbers that are not included are imaginary number (things that have an i attached to it), these numbers occur when you take the square root of -1 (sqrt(-1))\n\nhope that helps
909263
The square root of -1 cannot be a real number, because there's no number which, when multiplied by itself, gives -1.  So the square root of any negative number cannot be a real number, because for any number -n, it is equal to -1 x n.  You can take the square root of the positive n, but not the -1.  We call the square root of negative one by the letter "i", standing for "imaginary".  The square root of -16 would be +/-4i.  Zero is a real number, because it denotes the number of things when there aren't any, and it is a placeholder for the origin of a real line.  Negative numbers are also real numbers, as used on a number line, or as a measure of the balance in your bank account if you go into your line of credit.
545628
---THIS IS THE CORRECT ANSWER!!---\n\nIn algebra, the letter "i" relates to the term imaginary.  "i" actually is a shorthand for the square root of -1. Real numbers includes any number positive, negative, or zero, rational or irrational. Basically, any possible number you can think of on a number line. A non-real number would then be imaginary, a complex number whose square is a negative real number.
1059308
Using the standard "construction" of the natural numbers, they ARE NOT embedded in the real numbers.  The real numbers are like a completion of a splitting field of a class of polynomials over the natural numbers. (Existance of transcendental numbers, such at pi and e, show that the real numbers themselves are not the splitting field of the set of polynomials over the naturals.  See any good textbook on abstract algebra for definitions of splitting fields, transcendental numbers, ...  )  As such, the natural numbers and real numbers are indeed disjoint sets.  HOWEVER!  Consider the function f which takes 1_N ( 1 in the natural numbers ) to 1_R ( 1 in the real numbers ), 2_N to 2_R, ... etc.  \n\nf( m_N + n_N ) = m_R + n_R\nf( m_N * n_N ) = m_R * n_R\n\nThese properties show that the map f is a homomorphism.  Moreover, the only element which maps to 0_R is 0_N showing that the map is, indeed, an embedding.\n\nWhile the sets themselves may be disjoint, the set of natural numbers is "isomorphic" to what we consider the natural numbers sitting inside the reals.  As we typically consider such sets only "up to isomorphism", we can thus assume that the natural numbers do indeed sit inside the reals.
625700
if your talking about the origin of the numbers we use:\n\nthey are the Arabic Numerals that were used way back during the time of early civilizations and trade routes.\n\nthorough the trade routes, their numbers spread, and were widely used\n\nthey also used the zero, which was created by either the romans or the Greeks\n\nWe dont have any idea as to who invented them, though. Our best idea is scholars or educators.
195989
Just by inspection, if n=3 the expression is divisible by 7. A more thorough examination shows, though, that n = 3m works for all integers m>0, i.e. n=3,6,9,12,15,18,21,... will all be divisible by 7.\n\nUPDATE: n can not be any real number because the term "divisible" in this context has a precise mathematical meaning, i.e. when you divide by 7 you will get a natural number as the result. The use of the variable n usually suggests too that n belongs to the set of naturals.
286920
they're already in scientific notation!\nI think you mean decimal form..
19513
This seems to be a popular question. There is no real number that is the square root of -1.  However, it turns out that it is useful to develop as set of numbers that allows the square root of -1.    This is called "i" as is known as an imaginary number.  The set of numbers that includes i is called the set of complex numbers.