# Real numbers in math

What are real numbers for a person who doesnt know ANYTHING about math, and had to explain them what real numbers are.

Are real numbers only rational and irrational? if so then do we have to say what are rational and irrational

and please it cant have a lot of math involved in it (even if it sounds silly). Like say I am in uni or maybe you as a graduate of a university you want to explain to your niece, your little brother that is what I want like I know the good hard theorems but he wont get it.

So i want to simply break it down for someone who wants to know what a real number actually is

so this is what i have so far

So basically say you are a non math person and this is my explanation would it be valid. So a real number is basically any number that you can think of, let it be decimal numbers, rational numbers which are just fractions, irrational numbers. Real numbers can range anywhere from $\infty$ to $-\infty$ where $\infty$ is the highest number you can think of in your imagination and $-\infty$ is the lowest number you can think of in your imagination. Like numbers really dont end say the biggest number is $1000000000000000000000000000000000000000$ but there is a number bigger than that like $1000000000000000000000000000000000000000$, or the lowest number you can think of is $-1000000000000000000000000000000000000000$ what about $-2000000000000000000000000000000000000000$, etc real numbers are never ending

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Somebody will certainly give you a better write up later, but here's the answer to your questions: the real numbers are all rational and irrational numbers. A number is rational if it can be expressed as the quotient of two integers ($p/q$). So $5$, $2.5$, and $1.333333...$ are rational, because they can be expressed as $5/1$, $5/2$, and $4/3$ respectively. A number is irrational if it cannot be expressed as the quotient of two integers. $\pi$, $e$, $\sqrt{2}$ are all numbers that are irrational. –  George V. Williams Feb 10 '13 at 3:32
@George: saying that a real number is either a rational or an irrational number seems merely to beg the question of what is an irrational number. (If anyone were to ask me that, my immediate response would be that an irrational number is a real number which is not rational!) –  Pete L. Clark Feb 10 '13 at 3:34
Rather, your answer seems to be implicitly predicated on decimal expansions. I think making that explicit is probably a good way to go to explain real numbers to the layman. –  Pete L. Clark Feb 10 '13 at 3:35
yeah but i cant say all that because that is too "Mathy" if you know what I am trying to say, like say I am in uni or maybe you as a graduate of a university you want to explain to your niece, your little brother that is what I want like I know the good hard theorems but he wont get it. –  MathGeek Feb 10 '13 at 3:39
@MathGeek: I think you need to clarify who your target audience is. Decimal expansions are taught to "everyone" in the sense that one learns about them in elementary and junior high school. If you're talking to a five year old who doesn't know about fractions and decimals, then I think the only honest answer to "What are real numbers?" is "You'll learn about them when you get older." –  Pete L. Clark Feb 10 '13 at 3:43

If you can write it as a decimal, even if the decimal goes on forever, it's a real number.

If you can write it as a fraction of any two numbers, integers or decimals, or as a power of a positive number (or as a whole power of a negative number), it's a real number.

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yeah but i cant say all that because that is too "Mathy" if you know what I am trying to say, like say I am in uni or maybe you as a graduate of a university you want to explain to your niece, your little brother that is what I want like I know the good hard theorems but he wont get it. –  MathGeek Feb 10 '13 at 3:37
Just say my final line: "If you can write it as a decimal, even if the decimal goes on forever, it's a real number." Alternatively, it could be something that is returned by a standard calculator. –  SSumner Feb 10 '13 at 3:39
Make what explicit? The concept of decimal expansions? –  SSumner Feb 10 '13 at 3:41
Isn't that basically what I said? –  SSumner Feb 10 '13 at 3:47
You said it in the last sentence, yes. I think if you took that sentence and expanded it a little (not too much), you'd get a good explanation. The quoted paragraph doesn't say the word "decimal" but puts decimal expansions in parentheses, so I don't think it's a good explanation. –  Pete L. Clark Feb 10 '13 at 3:49

If you're explaining to a non-math person, the easiest way to describe a real number is to say:

"Pick any number, and it's real." (And say after that there are some exceptions that you only come across only in "higher" math.)

This is the way I explain it to people who aren't mathematically mature enough to understand what a complex number is (that is, they don't know square roots yet). Someone who can take a square root (or name a complex number) is probably math-literate enough to understand a more rigorous explanation.

Will this explanation make mathematicians cringe? Yes. Will it get the point across to a layperson? Also yes. Sometimes specificity needs to be sacrificed to explain a difficult concept to a beginner.

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So basically say you are a non math person and this is my explanation would it be valid. –  MathGeek Feb 10 '13 at 4:10
So a real number is basically any number that you can think of, let it be decimal numbers, rational numbers which are just fractions, irrational numbers. Real numbers can range anywhere from $\infty$ to $-\infty$ –  MathGeek Feb 10 '13 at 4:12
where $\infty$ is the highest number you can think of in your imagination and $-\infty$ is the lowest number you can think of in your imagination. Like numbers really dont end say the biggest number is 1000000000000000000000000000000000000000 but there is a number bigger than that like 1000000000000000000000000000000000000000, or the lowest number you can think of is -1000000000000000000000000000000000000000 what about -2000000000000000000000000000000000000000, etc real numbers are never ending –  MathGeek Feb 10 '13 at 4:14
is there anything else i could add to this? –  MathGeek Feb 10 '13 at 4:15
Regarding your first comment: Essentially, as long as you include the part in parenthesis. Second through fourth comment: What you've said is basically right, but I'd leave out the part about "...say the biggest/lowest number is..." There is no biggest or lowest real number--they are never ending (as you noted). –  anorton Feb 10 '13 at 4:21

I'd at least take a stab at a rough "story" that mimics (in a very loose sense) the construction of the real numbers in math classes:

Say you start with 1. I imagine most people will let you just start there. For a long time you hang around with 1, and 1 is great, but after a while you wonder if maybe there's better stuff out there. So you you think about what would happen if you got another 1 and put them together, and call this $+$. And you look at $1+1$ and think that's pretty cool, and wonder what would happen if you did that again. So now we have $1+1+1$. And at this point it becomes pretty clear to you that 1. you can keep doing this and 2. writing things like this is really clumsy. So you decide to give $1+1$ the shorter name of $2$ and $1+1+1$ the shorter name of $3$. And in this way you get a number that you call $4$, then $5$, and so on. But after a while, maybe around a number you've decided to call $2000$ or so, this loses its appeal and you decide you kind of get what's happening here: you're just adding $1$ over and over. It was a pretty natural choice, so you call these numbers the natural numbers.

But wait a minute, you have these numbers, but adding two of them makes a different number. You should make something that does nothing when its added to numbers. So call this new number $0$. And now $0 +$ any number is just that number again. But after a while you wonder if you can go in the other direction. Can you add something to a number to get $0$? OK, make that happen too. For a given natural number $n$ you define $-n$ to be the number that when added to $n$ gives 0 in the end. So all told you've expanded your number system to include $0, 1, -1, 2, -2, \ldots$ with no end in sight. This is pretty good. But you want to distinguish this new bunch of numbers from the natural numbers, so you call this bigger collection the integers.

So you spend some time adding integers together. $46 + 11 = 57$, you say, and $8000 + (-12000) = -4000$. And $4 + 4 + 4 + 4 = 16$. But eventually even this loses its luster. And writing down the addition of a bunch of numbers is annoying, so you decide to simplify things by renaming $4 + 4 + 4 + 4$ as $4 \cdot 4$, and you call this multiplication. And this simplifies things a lot, and gives you something new to fool around with. But then you get nostalgic for the $1$ you started with, and you wonder if there's a way to multiply a number by something and get $1$ again. But you're dismayed to discover that none of your numbers work for this (except $1$ and $-1$, for some reason). So you define a big new bunch of numbers to do this. $1/2, 1/3, 1/4, -1/5, -1/83, \ldots$. All of those. And now you have new numbers enough that anything can get back to $1$ by multiplication (except for $0$, which remains immovable).

But then you remember how fun it was to multiply the integers, because you've been doing this for a while now and are getting nostalgic all the time, and decide to try multiplying those new numbers by integers. So you get $47/6, 35/2, -4/19, \ldots$. And you decide to call all numbers of these forms rational because hey, it was a pretty logical trip to get here. And you notice that all of the numbers you've looked at so far are rational. $0$ is the same as $0/1$; $8$ is the same as $8/1$. And so on.

And for a very, very, very long time you stick with the rational numbers, and everything seems okay, until one day it isn't. Because you're trying to find a number that multiplies by itself to give $2$ - the other day you did this with $4$ and were pleased to find that $2$ worked - and you can't. You try for a while and get nowhere, and then you try for even longer and get nowhere, and then through some great effort you find that none of your numbers will work. So you have to define new ones. Ones like $\sqrt{2}$. Because apparently these things can't be expressed in the forms you were working with. They're not rational, and you find them kind of counterintuitive anyway, so you call them irrational. Eventually you find more of these by mucking around in geometry and probability and you give them weird symbols like $\pi$ and $e$. And you finally decide that yeah, this is probably more numbers than anybody needs, and you're done. So you call everything you've done the real numbers, because that's what it's been. Real.

And then you realize that none of your numbers, when multiplied by itself, gives -1...

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My idea about non-mathy explanation of (positive) real numbers - "it's length of line segment if unit segment is given".

Because geometry is more natural than decimal numbers and their expansions, or limits of rational number sequences or something like that. It nicely captures such qualities of real numbers as connectedness and ordering (by $<$).

And negative numbers follow if I choose direction on the line. (Real numbers as one-dimensional vectors, actually.) Addition then is geometrical addition of vectors.

Multiplication then becomes vector with length equal to area of corresponding rectangle and orientation in positive side if vectors are oriented equally and in negative side if they are oriented in opposite ways. (This part is potentially a micro-mindfuck, but not enough for me to reject this idea. The same mindfuck as the first time when learning multiplication of numbers with different signs.)

( tom gave link to much better alternative to area of rectangles. Representing the multiplication of two numbers on the real line )

It's completely legitimate way to start from axioms of Euclidean geometry and construct real numbers geometrically.

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could you explain a little more on this perspective if you dont mind? –  MathGeek Feb 10 '13 at 4:18
@MathGeek Expanded. –  zaarcis Feb 10 '13 at 4:52
Mentioned unit segment too. –  zaarcis Feb 10 '13 at 5:01
Multiplication can be explained without a notion of area. see math.stackexchange.com/questions/139340/… –  tom Feb 10 '13 at 23:30
Because you explain number as length. But multiplication as area and area is number and number is length. So you get in circle. –  tom Feb 10 '13 at 23:34

I would consider explaining it via the real number line. The benefit of that is simply that it is visual (and therefore, your brother has something to look at) and it is easily understood. How I would explain it:

1. Start off with a line, mark the 0,1,2,3,-1,-2,-3. Say that these sequences go on and on forever in both directions. These are the naturals and integers.
2. Make him look at the spaces in between. For example, in the Interval $[0,1]$. Mark 1/2, 1/4, 1/8 and tell him that in between all of these, we can put one in the middle again. These are effectively the rationals.
3. Last but not least, tell him that in between every two rationals, there are other numbers, numbers that will never be hit by any rational number. This might be harder to explain. I would prefer to do the step from fractions to decimal expansions somewhere in between rationals and irrationals. Tell him that all rationals keep repeating themselves. Obviously we can define other numbers that have a non-repeating pattern. These are the irrationals.

I guess this might be easier understood than trying to clarify the dimensions of the reals, as they do not really charakterize the reals.

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Roughly speaking, if you could write it only with numbers, dots and bars, even if you take a large amount of time, it is a real number.

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dots and bars ? –  Elements in Space Feb 10 '13 at 4:41
if you have one dot, and one vertical bar; you can represent a number that is not real –  Elements in Space Feb 10 '13 at 4:43
only with numbers. –  Parth Kohli Feb 10 '13 at 6:17

All numbers you ever have heard of, like $7$, $-51$, $10^6$, ${5\over 9}$, $2.4137$, $\sqrt3$, $\pi$, are real numbers. Mathematicians have invented various means to talk about them in a unifying way. In the end they just say: "Let $x$ be any real number."

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Even as a schoolchild I had heard of numbers like $\infty$ and $\sqrt{-1}$. So this does not seem to be a good answer. –  Pete L. Clark Feb 10 '13 at 20:32

Real numbers are numbers that are on the real number line,

$x\in\mathbb R \iff -\infty<x<\infty$

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