So the integers are: 0, 1, -1, 2, -2, 3, -3, There are infinitely many rational numbers, but they do not form a continuous line. The continuous line of numbers is called the real number line. It includes all the previous numbers we have mentioned, but also numbers like sqrt 2 , pi and e , which are not rational. Some real numbers - such as sqrt 2 - are the roots of polynomials with integer coefficients. These are known as algebraic numbers. Irrational numbers are any real numbers that are not rational.
So 0 is not an irrational number. Some in fact most irrational numbers are not algebraic, that is they are not the roots of polynomials with integer coefficients.
It is when we extend the concept of number to cover the enumeration of things that do not exist that we find that zero has utility. At that point we move from the concrete to the abstract, as counting things that are not there requires numbers to represent ideas, not things. In terms of a definition, there is one - the set that contains the sequences that correspond to the process of determining the cardinality of non-empty sets i.
Each sequence has a label, being the last element of each sequence. These labels are what we recognise as Natural Numbers. This is however a definition of dubious utility when it comes to the problem of teaching children what zero is.
There is a reason why so few cultures independently devised the concept of zero, and that's that most ancient, pre-technological cultures without currency or finance had no need for an abstract concept where you had to count the things that were not there. The question is: Is 0 a Natural Number?
John Foster says that Natural numbers "are the ones we use to count things that are there. That is actually the definithion for Counting Numbers 1, 2, Hence the word count in counting. Hence the word "count. Hence the word "whole. Yes, there are two names for the same thing, Natural Numbers, or Counting Numbers. This is common in mathematics, we often have more than one name, or more than one symbol to stand for the same concept. Usually which one we use is decided by the context.
This is not tautological, unlike expressions like 'Whole Numbers are non-negative integers that are uncut, undivided, and not in pieces 0, 1, 2, 3…. It's interesting who sat in the various camps regarding this question is zero a natural number?
It appears that the Formalists say yes, and the Intuitionists and Platonists say no. Convenience is driven by context. They have different algebras, that make sense in their own contexts, although there are mappings between the sets, that also make sense in certain contexts. I contend that the Natural Numbers are those that children use once they move beyond "one, two, many".
In their context there is no need for zero. It mystifies me why some people feel so strongly that this is not a valid position. I contend the answers are Yes, Yes and No. It isn't tautology if a writer's or speaker's objective is to make certain that he or she is very clear to the reader or listener. The mere FACT that this is post 30 should be enough for the readers of this post to conclude that zero being a natural number IS a matter of opinion. I realize that certain people have very strong opinions one way or another.
It seems that they want to influence others with those opinions. I have my own opinions but I will not try to influence anyone with them. There are books that include zero and books that do not include it. There are teachers that include zero and teachers that do not include it.
I happen to give a student my opinion. That student does his math work based on my opinion. The teacher teaches based on a different opinion. Where do you think that would leave the student? You may notice in my previous post 26 that I never stated that zero was or wasn't included. Instead, my answer was that it is a matter of opinion. It will continue to be a matter of opinion until it is proven well enough one way or the other to become fact.
If the question was, "Is 0 a whole number or counting number;" then I doubt there would be any disagreements and think this thread would be much shorter. Especially if you are referring to that set often. Therefore, I find it more practical to define the set of the natural numbers to include 0, i. Seems like the blog doesn't like unicode. I also see I made a typo, the positive integers were obviously supposed to have been the integers greater than or equal to 1, not strictly greater than 1.
I consider a natural number as a value to something that you can see or is present. I can see 1 apple, 2 grapes, 10 trees.
I consider a whole number as a value attached to counting the number of the same things I can see. If I see no apples, then I see 0 apples. Anyway, I edited it and I think I have what you originally intended including the corrected typo. What a fascinating discussion! And what a great website, zac; I've just stumbled on it and added it to my RSS feed reader.
Arguably it is a matter of opinion on one level, but I'm with John Foster on this. The term "natural" strongly suggests a sense of intuition. Hence Euler, et al. I must say that I am unaware of England making such a distinction between 0 being included among the natural numbers.
I teach mathematics in England and on the journey into the world of rational versus irrational numbers my older students take a brief tour into the world of "natural" numbers; we discuss the abstract and philosophical notions and implications of non-integers, negatives, and zero.
For example, you cannot have half a piece of paper or half a chair. You can remove pieces, but it remains what it is until it is no longer what it was. I hope that makes sense. In other words, fractions exist to define relative comparisons or measures, whereas natural numbers define the actual quantity of usable items. So in the same way we talk about how unnatural the concept of zero actually is. It is quite natural to talk about three books or one calculator, but it makes no sense to talk about zero anythings.
If zero were natural then an infinite number of them would occupy some space. The room where I am typing this comment right now contains zero elephants with one written on its back, zero elephants with two written on its back, and so on.
There are an infinite number of zero elphants with N written on its back and yet there is space for me to be here. There is nothing natural about zero! Thanks for your reply, the esteemed "Euler"! You raise some great points. This bit gave me pause for thought - "you cannot have half a piece of paper or half a chair", since functionally the situation is somewhat different.
If I rip a piece of paper in half, I can still use the individual pieces of paper, but half a chair is as useless as no elephants with N written on them!
Your philosophy of numbers course sounds very stimulating. Thanks for the input about conventions in your part of England.
It's interesting that these things are not even necessarily standard across one whole country, let alone universally. I wrote about your Project Euler here. But they enjoy the opportunities to think outside of the basic curriculum diet. You're right about the degree to which you can remove pieces of paper and still describe it as a piece of paper compared to removing pieces or parts of a chair, but I still think that the idea has some merit, albeit tentative.
If you asked for a piece of paper and I gave you a fragment of paper measuring 1 mm by 1 mm then you would think I was crazy. It might contain the same matierial as paper but it would not function as paper. The phrase "one piece of paper" refers to a usable and practical measure of paper. Admittedly the point at which it is no longer describable as a piece of paper is somewhat subjective, but you would never describe it as half a piece of paper unless you were comparing it with, say, a piece of A4 paper which had been torn in half.
In which case you are comparing its size, not really describing it as half a piece of paper in terms of its function. Simialrly with the chair, if I continue to remove parts of it then at some point it ceases to be describable as a chair. Even if I took a chainsaw to through the centre of it then you might look at one "half" and say, "That's half a chair.
Technically what you're looking at is no longer a chair. It does not function as a chair any longer. But I recognise that even here with all this philosophising I am skating on very thin ice, and I wouldn't be foolish enough to defend my points with any authority. I simply don't possess it. The bottom line in the discussion of "Is 0 a natural number? You might have noticed at Project Euler, where our problems often venture into the realm of Number Theory, that we are careful to define the set of whole numbers not including zero, as the set of positive integers.
The problem with O in mathematics is that it is used to symbolise nothing, no-thing, and yet, mostly, it refers to unity, a whole or united thing. For human beings, no-thing is an abstract concept, meaning it has no-thing to do with our real experience of life, and in effect no-thing has to be imagined as a total blank, say the paper that something is written on, but of course this paper is a whole thing and it is only our focus on the writing that makes it a blank, no-thing, background.
No-thing is this background to focus, and once, space, the heavens, were seen as a black background of no-thingness against which the stars appeared as things. Nowadays this no-thingness is thought to be filled with fields, sequential influences and almost-things, and the no-thing is in doubt. At the other end of the scale we have atoms, quantum particles, strings and the something that they appear from, but if we focus on the strings and ignore the fields etc.
However, if I have an apple and someone steals it, takes it away, then I have no apple, no-thing. If I recover my apple I have a unified thing and if I cut it into sections it is a divided thing, and maths is based on this unified principle even as it ignores its own reality. This is the division of unity into things, or the many things, like a lot of apples, that create a unified concept. What is lost in maths today is the concept of a unified background, the unity that things appear from within or the unity that is being enumerated as things, and the sooner maths re-invents itself into a concept of unity and sees its no-thing for what it is, the better for everyone.
No-thing exists in my human experience when something is taken away, but what I experience before this event is a unified concept that can be divided into things or the things that can represent another unified concept. O as part of the numerical symbol for ten, a hundred and so on only describes a decimal form of mathematical unity, and the modern decimal system is based on this unified concept of ten things.
O shows that the integers in a column have been unified as the 1 in the next column, it represents a unity of the ten in this previous column, and so on with and etc. This says that no-thing exists in the unified column and this ability to ignore reality, the paper that maths is written on or the human being that first divided things in a numbered or quantified way, is forgotten today. Reintroduce the concept of unity as the background that maths is built on and which it uses all the time, and maths could make sense to everyone, but of course abstract thinkers will probably choose to rely on the unwritten rule of preference that created the zero.
Their abstract way of thinking depends on it. They will take away our humanly unified reality and leave us with an abstracted no-thing again. These "conventions" mentioned above are just that: conventions. There are times when it is easier to consider 0 a natural number and times when it is not. However, the asking whether 0 is a natural number is equivalent to asking whether you want to call the empty set "0" or "1". When the natural numbers are constructed rigorously in set theory, we have.
Thus, these sets either correspond to 0, 1, 2, 3, It seems more natural to me to call the empty set 0, since it contains nothing. Besides, 0 is the most "natural" number of all. Simply go into a room and think of all the things in the world. Then determine how many of each thing are in that room.
For example, there might be 1 desk, 1 chair, 1 bed Thus, of all the things in the world, there will be more things, 0 of which are in the room, than things 1 or 2 or Lucas Mentch comment, above, describes O very well in his last paragraph. There is reality, what is there, that which can be enumerated as 1 2 3 etc. Whether 0 is a natural number or not seems to be answered by this. For me, what is real, nature, represents what is natural, what is unreal, or produced only to suit the demands of abstract thinking, is non-natural.
Thinking, as a mathematical exercise is abstracted from reality and its connection to the natural world is representative of nature or of a fantasy that thought has dreamed up to suit it's own devices. The pattern of decimal numbering demand sets, what I was taught to call columns of ten as a child, units, hundreds, thousands, etc.
These columns form an abstract non-natural pattern in human brains and people become confused between what is natural, real, and non-natural, abstract, when the patterns set by these columns seem more real to their way of thinking than what they actually observe. In reply to a student, you don't really need to know if 0 is a natural number or not. You only need to know what the one who marks your homework thinks it is.
If you read the blogs you can see that the naming of numbers is more or less down to confusion and personal preference. Your marks are given by personal preference so I would say what you were taught to say, collect the sweetie, and then remember that many rules of maths are preferential.
Maths is not an exact science, it is a preferential one. It's really confusing everyone has a good point. But as I analize it based on your notions 0 is not a natural number if it represent on its own. It can only be a natural number when added to other numbers just like in 10,20, As Anton Pech points out, the problem is two different concepts, one symbol.
Nothing and Unity. Maths requires a second symbol, but it seems to prefer the confusion that comes from one symbol doing two jobs. Hitler did much the same when he made staff appointments. Two people, one unclear job, and both vying with each other, trying to win his preferential favors.
Maths is an authoritarian system, unclear preferences are unwittingly passed on by teachers who decide what 0 is, and confusion reigns until you see rules of preference for what they are.
Until then, confusion can over-rule peace of mind and conflict's mindset becomes superior to commonsense. Yes Philip I agree with what you said that, 'unclear preferences are unwttingly passed on by teachers who decide what zero is'. So, do you know who was the first one to say that '0' is a natural number? Maybe it was just also over-heard as natural instead of neutral.
However, he doesn't make the distinction between its use as a unifying symbol and a symbolic nothing. Like most mathematicians, he recognises its usefullness and its growth in several cultures, but doesn't draw the distinctions of use and application that may have existed in other mathematical systems.
Given the linguistic variations of the several inventors of zero, I doubt that a simple misunderstandng of a concept took place as you suggest, and as I see it, neutral is as far from unity as it is from nothing.
Nuetrality springs from neuter, and sexlessness, as neither masculine nor feminine, often appears in nature, even if human beings mostly judge this condition to be unnatural.
For me, authoritarianism rules by confusion, and by the acceptance of a social appearance that is similar to that described in the children's story of the Kings New Clothes. Maths spreads confusion through its preferences, and yet it is easy to see its naked reality once you stop trying to fit into an illogical system. Guess you could say that unclear preferences neutralise people's brains and stop imagination and inventive thinking from working together as equal partners.
This topic is very important for some teachers because they don't know if the 0 is or isn't in the natural numbers; in our country in the primare school, the teachers say that 0 isn't in the naturals, but not because they think it but they follow the government programs. Hi Luisa and Richard, The Mayan concept of 0 may be a little closer to home for you than the confusion spread by the western system, but you'll have to rediscover the beauty of the Mayan mindset to understand their maths.
Difficult after the dichotomies of rationality have invaded everything and everyone by pretending to be logical. I have always seen in books and written in my own, Prealgebra that natural numbers start with 1, and whole numbers start at 0. But when I was in grad school at Berkeley, the great Julia Robinson seemed to include 0 in the natural numbers in a theorem.
I asked, "are you considering 0 as a natural number? Hi Dan. I love that - answers to the great questions of math by force of personality and power of assessment! Whole numbers are unified ones, unity exists, and some one has noticed it, so where does no-thing come from? Perhaps it's a financial abstraction of maths, and only relevant in as much as the monetary or possessive state of having and not having can dominate westernized minds with calculated conceptualizations of some thing.
Preference started out as the order of courtiers and dignitaries, all trying to please a monarch, and being pleased by a ruler as they were shown favor. They were chosen ones, but the force of personality that chose them could vary from the benign to the insane. So, it was the position of monarch, the first one, that chose its favorite ones, and this assessment could be based on anything from promise to the needs of the state.
Monarchs preferred those who kept them in power. Preference arranged the system, position mattered, power might, and generally did whatever it wanted to do, and poor common folk had nothing to compare with it, so they strived to be some one, too.
I guess, when maths notices that natural numbers come out of unity, just as they form it, then the system could find balance again. But of course, it will mean letting go of the first position, the prime mover, the original one, and noticing that this was actually a unified everything before it diversified. The king will be dead, there will be no authority, but someone will shout long live the king and raise a new pretender to the throne because they prefer things that way.
This self-assessment seems to be pretence and it just becomes easier to accept the qualification of the system that promotes a preferential state of mind than to acknowledge and trust your unified self. I always explain to my students that the 'natural numbers' are the numbers that arose naturally when 'somebody' began to count. Not having something sheep, rock, cave Look at the stars from any point in the universe.. As has been stated above, 0 was invented as a place holder long after counting came about NATURALLY so I remain firmly in the "0 is not" camp and I will continue to tell this to my students let's hope that this does not upset their first exposure to set theory!
Looking at the stars also shows us the heavens, what we have learned to call space today, and we are looking at a unified concept. When counting the stars we often ignore this heavenly space, or see it as an infinite nothing when abstraction takes over.
As Einstein pointed out, trying to measure space doesn't work because measured frames of reference are always surrounded by space as well as containing it, so a way of thinking that focuses only on the stars has to limit its own machinations and it has to ignore what might be called the unity that includes the heavens. Infinity is a relatively recent concept in the west, and yet various groups of people have used it, under other names and for thousands of years, to demonstrate the limited capability of the abstract thinking that focuses on its own things by ignoring reality.
However, as with all languages, trying to turn their symbolism back into significance is fraught with problems. Abstraction disappears into its own orifices, and yet its symbols still seem to represent something.
The paradoxes of set theory show this anal search at work, and as fascinating as it might be to arrange meaningless symbols into sets that cannot belong to their own set, all that this actually demonstrates is the limitation of the rules applied. A number of square shapes can become a square and, linguistically, it is of its own set, but I can also accept that a number of circles cannot become a circle and this fact isn't paradoxical if I recall that the heavens always surround the stars, just as they do squares.
But to think this way, I have to accept that unity exists before I turn it into an abstraction, and that the sophistication of infinity, like nothing, or even set theory, is an unnatural offspring of the misunderstood and long ignored, unifying zero.
I'm not sure that I really follow this deep philosophical take; hey I'll bet Einstein would vote "not 0" but Hawking would probably go the other way. I'm way up there voting for the unity of the universe on the other hand. Also let's not forget the real masters of counting - the primes..
Yup, I'm sure the Primes would vote "not 0". The problem with your masters of the counting universe, the primes, Optimus and his friends, is that they follow a fundamentalist theory, where one is excluded and where like-minded adherents are included.
And like all fundamentalist ideas, its easier to follow rules that have been laid down by prime movers than to realize that one is essential, even to a prime, and that one can think for itself. However, if all ones are unique, and essentially primary, then the set of unity is also a unique one, as a paradox that doesn't exclude anything, and where nothing cannot exist.
What value does infinity's endless search for an unreachable goal bring to a unified equation, apart from the denial of its existence? Guess the difficulty comes from fundamentalist theoreticians not accepting that one can represent anything if it follows the rules of unity, they seek like-minded followers, and we are back to authoritarian theories and the masters of the universe once more.
I visited this page about a year ago And today, while revisiting, I feel like sharing some simple thoughts as a humble, ordinary guy. This comment of mine takes the total number of comments to Comments numbered 1 to No comment 'number 0' And now if I may digress slightly and only because 'nothing' has been mentioned quite often in the posts here!
In my humble opinion and with all due respect to Hawking no number of big bang, multiverse, string, or superstring theories can really answer that one! Rich: I've come across books that have a "Chapter 0", which explain basics before getting into the heavy content The debate about what particular set to call the natural numbers is silly.
Robinson is correct - 0 is a natural number. Modern logical formulations of computation start by generating the Naturals with the Peano axioms, with 0 being the first element.
Note that you can call the first element anything you want, say Fred. The "numbers" in the set inductively generated from the Peano axioms correspond to the number of times you apply the successor function to the initial element. That is why 0 is a Natural - it is what you get by applying the successor function no times. Much of the confusion comes from how to "index" the set of Naturals. Do you start at 0 or start at 1? That's not so clear. Sometimes starting at 0 is convenient, sometimes at 1.
Languages like APL gave you a choice of 0-origin or 1-origin indexing for vectors. Star magnitudes start at The observations that children don't know about 0 can only be made by someone that is not a parent. Ask any child how many candies are left in the bowl after Dad eats all of them, and they will say 0.
In Canada, all children also know about negative numbers from an early age, since we have them for temperatures in the winter. Guess that answers the question. If Dad leaves nothing for the kids then 0 must be natural.
Teaching kids what's natural is what being a parent is all about, isn't it? Or is it that what's become normal in an individual, family, society or nation can seem to be natural until you start to think about normality and discover another way of doing things? Teachers used to say, "Nature abhors a vacuum," but perhaps nature wasn't natural and maybe what they, Dads and other parent figures did, by leaving nothing for the kids, was.
For a selfish way of thinking, something or nothing is normal. For a natural way of thinking, something or something else is normal. Guess it comes down to Dad in the end. Is eating all the candies normal or is there a natural way to teach the kids? Really, now Different definitions of 'natural numbers' are used in math, all equally valid see post 30 and several others.
As children came into play re poster 62 in frigid Canada , I have a five-year-old who can add nonzero, one-digit positive integers has notion of counting and adding. He also knows that the '0' on the elevator button means 'ground floor' and the '-1' means 'garage', but has yet to grok that '0' can also denote 'nothing', 'none', or 'empty' and be used "to count". Personally, I think of 'natural numbers' as the same as 'counting numbers' or 'positive integers'.
But I have nothing but admiration for Peano's theorems, which start with "there is a zero" a non-negative integer, though I don't think he called it 'natural'.
One reservation: Preparers of university entrance exams should be careful to avoid 'natural numbers', and takers of such exams should be prepared to challenge any disqualified answers! Good point about the 0 representing the ground floor on an elevator, although in Germany it is generally an E for Erdegeschoss, ground floor, and from what you say, your 5 year old sees 0 as a symbol for the ground floor, which for him is a tangible something or he's hanging in a void.
I recall having a detailed discussion about 0 with a teacher not long after being a 5 year-old, well, I was actually on the losing side of an argument because teachers didn't discuss things with children in the early s, or maybe it was only the teachers I came across that seemed to prefer parrots.
My side of the discussion was asking whether 0 came at the start of the sequence or at the end of it and why, but what this memory indicates is I couldn't make sense of what I was being told and that the teacher couldn't explain it in a way that my young, and for me, very natural mind could understand. As I see it, 0 becomes a thing of preference, like the parrots, and in this case, a confused thing of preference where two or more of them exist as a conflict in the same symbol, each of which are taught with authority and neither of which make sense to a child until someone has taken all the candies for themselves.
I guess if Peno had given a clearer explanation of what he meant by zero, nothing or unity, or even the ground floor, then the value of the symbolic 0 could have been sorted out long ago, as could the symbols used in elevators. So, back to looking for the candies. Father, why hast thou forsaken me! Whereas, 2n-1, where n is a "natural" number, will generate all the positive odd numbers. Note that the converse is not true: all numbers of this form are prime.
On another note I think that any arguments which appeal to essential building blocks, and zero being a necessary component, need to consider the importance of In any system of counting which begins with the notion of incremental components, we would be unable to incorporate one of the fundamental rules of arithmetic: subtraction, without So is -1 among the natural numbers?
Or do we simply see subtraction as an extension of arithmetic by including the "reflections" of the natural numbers? In which case, what purpose does zero serve in this domain? Arguably, 0 appears from the clash of equal and opposite quantities, but as we continue extending the system we soon have all rational quantities. The line needs to be drawn somewhere. Personally I can see no benefit in including zero as the "first" natural number.
However, I do concede that with the advent of computing, it is often helpful to talk about the "zeroth" term, u 0. But equally, when working with iterative sequences it sometimes helps to begin with u -1 or u Those familiar with solving Pell equations will often make use of these terms. The bottom line, and it has been said here several times already, there now exists sufficient uncertainty in the proper meaning that is is sensible to either use the phrases: "positive integer", "non-negative integer", depending on your intention.
Alternatively include some proviso in your definition. Today is 31 December , the last day of the year. The cycle of the seasons has been completed once again, and a new cycle will begin as the earth follows its path around the sun.
That path can be described as 0, a circle, and however the year is divided, each division is part of the whole 0 cycle. Similarly, the sun rises as the earth turns, and its turning also describes a 0, a circular motion that defines a day as the passage of the risen sun to sunset and back to sunrise. In both these natural instances, 0 can be said to be a unified 1 or the completion of a cyclical movement that is now designated as 1 symbolic and naturally significant 0.
The end of a year does not result in nothing any more than the end of a day does, they both bring a new cycle, a new 0, that will be unified until the current natural course of events says that this cycle is no longer sustainable. What will happen then will also be described by 0s, circles and cycles, and new 0s, cycles will be formed as the movements demanded by gravity take over once more.
The complexity of mathematical thinking ignores its own conception and relies on the rules that it prefers to justify its existence, but the 0 of maths had a natural birth and perhaps many of its current preferences are only the result of an unnatural system of education.
I once wondered whether 0 is a natural number and went through the whole textbook to find the answer, but it is not stated. Concerning whether 0 is a natural number; I feel that there is a difference between natural numbers and counting numbers.
Counting numbers are numbers used in counting real objects. If students are asked to count their fingers, none of them will start from 0. However, 0 is used to express the absence of a real object. Natural numbers are defined as the non-zero negative integers. In each case the elements should be listed for students to observe the inclusion of 0 in the set of natural numbers.
So, we now have counting numbers for natural, material fingers and natural numbers for imaginative, abstract things.
And by this definition, we can start to feel that the fantasies of abstraction are more natural than the countable fingers of the material world. Additionally, 0 is the absence of both of them, and yet it is also the unified cycle of life that is naturally imaginative and material as a human body and mind.
And of course we must list the elements for students to make sure that they include 0 in the abstract set of natural numbers, to ensure that they understand as we do. The fact is that 0 unity is material and imaginary and that 0 nothing can be imagined and manifest by the actions of selfish people. So, why do we use a single symbol for opposing concepts? What is a number? Modern mathematicians have not explained it accurately.
Ancient Indian Vedic mathematicians had precise answer to this question! Each number is a condition. This is Ancient Indian logic! How and why a number is imagined as condition? Both counts are technically differrent! I mean, modern 'one to one count' and ancient Indian "one less states of one to one count" differs! While Ancient Indians used equal step increases by one they also used 'number' as 'condition'. Second one has 'one' before it, which condition is one!
Third one has 'two ones' before it, which condition is two. What is differrence? The rational numbers are those numbers which can be expressed as a ratio between two integers. All the integers are included in the rational numbers, since any integer z can be written as the ratio z 1. All decimals which terminate are rational numbers since 8. Decimals which have a repeating pattern after some point are also rationals: for example,.
The set of rational numbers is closed under all four basic operations, that is, given any two rational numbers, their sum, difference, product, and quotient is also a rational number as long as we don't divide by 0. An irrational number is a number that cannot be written as a ratio or fraction. In decimal form, it never ends or repeats. The ancient Greeks discovered that not all numbers are rational; there are equations that cannot be solved using ratios of integers.
What number times itself equals 2? But you'll never hit exactly by squaring a fraction or terminating decimal. The square root of 2 is an irrational number, meaning its decimal equivalent goes on forever, with no repeating pattern:. Other famous irrational numbers are the golden ratio , a number with great importance to biology:. Irrational numbers can be further subdivided into algebraic numbers, which are the solutions of some polynomial equation like 2 and the golden ratio , and transcendental numbers, which are not the solutions of any polynomial equation.
The real numbers is the set of numbers containing all of the rational numbers and all of the irrational numbers.
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