What is a real-world metaphor for irrational numbers?

Question

In an effort to develop better number sense (and to create my own journey from fish to infinity), I have been going through Khan Academy's math material from the very beginning. So far, I have been able to develop strong intuitions and metaphors for most elementary mathematical ideas. I now see that arithmetic, division, negative numbers, fractions, decimals, factors, multiples, etc. all have clear grounding in reality; I have a much stronger grasp of the material.

So far, the only place my intuition has broken down is irrational numbers. It was surprising to me because it happened so suddenly, like the ground giving way. It was as if we saw this gaping hole in the ground, gingerly walked around it, and pretended it wasn't there. This bothers me.

Is there any straightforward metaphor for irrational numbers? Could you explain it to a child?

Answer 1

Here is a physical metaphor:

Draw a circle, and prepare two sticks:

• One with the length of the circle's diameter (2r)
• One with the length of the circle's circumference (2πr)

You cannot cut both sticks into pieces of the same length, no matter what length you choose.

In other words, you cannot measure both sticks using the same measurement-unit.

You can try meters, feet, inches, miles, or even invent your own measurement-unit.

You will never be able to accomplish this, because the ratio between the sticks is irrational.

Answer 2

I hear this once, not sure its origin:

The rationals viewed as points of light on the real number line are as the stars in the night sky.

Answer 3

To expand on my comment above, which as stated was somewhat incorrect, the usual idea as fractions as wedges of a circle can be expanded upon to include irrational numbers.

Now, to reexamine the rational case: normally when you explain $1/7$ to a child you picture a circle divided into seven equal wedges and $1/7$ corresponds to one wedge while $2/7$ corresponds to two wedges. A better way to think about this, in some respects atleast, is to notice that $1/7$ is the wedge that if you repeat it seven times it goes around the circle once. While $2/7$ is the wedge that if you repeat it seven times it goes around the circle twice. From this we can then say any rational number $r/s$ corresponds to the wedge that if you repeat it $s$ times goes around the circle $r$ times.

What happens then if you pick any arbitrary wedge? Well almost always no matter how many times you repeat it, it will never exactly go around the circle an integer many times. These wedges correspond to the irrationals.

Edit: The fact that each wedge we can choose almost always corresponds to an irrational is due to the cardinality of the rationals vs the irrationals. However, this doesn't provide any intuition of why we should expect this.

If we draw the wedges on the circle, by drawing lines from the centre to the circumference. We may eventually draw a line that seems to overlap a previous line. However, what we are drawing are really representation of an "infinitely thin" line. If I drew a line that was infinitely thin and then asked you to trace it with your own infinitely thin line, do you think you could do so? I doubt it, the more likely outcome is that if we zoom in close enough, eventually we will see that your line and my line are distinct. In some ways this intuition carries over to why almost all wedges will never exactly overlap. It would be just as difficult to draw a wedge that when repeated goes around exactly an integer many times.

Answer 4

There are two words involved here: ratio and reason or reasoning.

It seems that the word "irrational" in current mathematics is related just to "ratio" rather than "reason". A rational number is a number that can be written as a ratio of whole numbers while an irrational is just a number that it is not rational.

However, the first irrational that has been known is the length of the diagonal of a side $1$ square. This happened for the first time when non-rational numbers were unknown (even the negative integers were still at their birth aurora). And in such a circumstance it would seem that $\sqrt 2$ was something contrary to reason, i.e. “irrational”.

It is well known that there are many more irrational than rational numbers which would indicate that in politics often what counts most for the meaning of "irrational" is the old sense of $\sqrt2$. Can you get of this a real-word metaphor for irrational numbers?

Answer 5

Suppose we have a square whose side length is $1$, then according to Pythagoras theorem, its diagonal length is a number whose square is $1^2+1^2=2$.

It would indeed be catastrophic for geometry if there were no actual number that could describe the length of the diagonal of a square.

The Pythagoreans tried to make do with a notion of ‘actual number’ that could be described simply in terms of ratios of whole numbers; i.e. find a number $\frac{a}{b}$ such that $\left(\frac{a}{b} \right)^2=2$. But it can be proved that $\left(\frac{a}{b} \right)^2=2$ has no solution for integers $a$ and $b$, where we can take these integers to be positive. Thus, there is no such rational number that squares to be $2$.

Therefore, there should be other kinds of numbers that are not rationals and hence, irrational numbers were discovered.

Answer 6

If the "gap in the ground" of the irrational numbers is more noticeable and intimidating, then that's because you are understanding the situation accurately.

Let's go over definitions, first. You know what integers are, and how they fit on the number line. If you try to divide by integers, you get rational numbers; these too fit on the number line. You might think, then, that the rational numbers "fill up" the whole number line. Certainly, between any two rational numbers, you'll be able to find another rational number. Nevertheless, I intend to convince you that there are gaps between rational numbers, which we will fill with irrational numbers. If we want the number line to be complete, without any gaps, we have to add something.

The usual example is $\sqrt{2}$. By definition, $\sqrt{2}$ is this number $x$ that, when multiplied by itself, yields $2$. That is $x^2 = x\cdot x = 2$. Notably, there is no rational number that behaves this way (often, this is proven by infinite descent).

Let's look at this another way: pretend for the moment that all numbers are rational. We can divide the positive half of the number line into two pieces. One piece (call it $S$) is the set of all positive $x$'s for which $x^2 < 2$, and another is the set of all positive $x$'s for which $x^2 > 2$ (call it $T$). That is, we've created a Dedekind cut.

Clearly, every rational number is either in $S$ or in $T$, and any number in $T$ is bigger than every number from $S$. Intuitively, this means that these two sets cut the positive number line into two pieces. However, we haven't touched any rational numbers with our cut; there is no rational number on the border between $S$ and $T$. Our cut through the number line (which was supposed to be made completely out of rational numbers) has missed; it didn't hit any numbers.

What we can do, however, is look at the location of the cut on the number line, and declare that at this boundary is a number, which we call $\sqrt{2}$. If we do so with every gap in the number line, then we get the real numbers: the full, gap-less number line.

So, what are the irrational numbers? They're the numbers that fit between the rational numbers. Their defining property is that, like all other numbers, we can put them in order. For example, I could tell you that $1.41 < \sqrt{2} < 1.42$, or that $1.414 < \sqrt{2} < 1.415$, but $\sqrt{2}$ is not itself a rational number.

Answer 7

An interesting question.

You're not the first to fall into a hole when faced with the idea of irrational numbers. Here's part of the myth about the Greek discoverer:

https://en.wikipedia.org/wiki/Hippasus#Irrational_numbers

Irrational numbers may be the first thing you find in your mathematical journey that has no good counterpart in "ordinary life". You never need them to measure things as accurately as you wish, or to cut up pizza. They come up only when you want to pursue pure mathematics for its own sake. That effort started with the Greeks. Book X of Euclid's Elements begins with a definition of "incommensurable" (explained too in @BarackManos 's answer). Then in Proposition 10 Euclid proves that the side and the diagonal of a square are incommensurable. (The notes at that site say this proposition may not really belong to the original Elements.)

When you start to learn about infinite sets you'll discover that there are (in a sense that can be made precise) many more irrational numbers than rational ones.

Enjoy the journey.

Answer 8

Here a metaphor based on the irrationality of $\sqrt{2}$:

Every day two friends $A$ and $B$ enter a square field at one of its vertices, jog for a while, and leave by the same vertex. $A$ jogs along the boundary of the field (always in the same direction) whilst $B$ jogs along the diagonal joining the entry vertex and its opposite (changing the direction only when he/she reaches a vertex).

$A$ and $B$ never jog the same distance.

Answer 9

The "high school" definition is usually something like

A number is irrational if its decimal expansion neither terminates nor repeats

In other words, whenever a number is rational, you can express it with a decimal that either eventually stops (as in $1.234$) or repeats (as in $1.234234234234... = 1.\overline{234}$). However, we could come up with numbers where the decimal does something else. For example, $\pi = 3.14159...$ neither ends nor repeats, and $0.10100100010000100000...$ follows a non-repeating pattern.

These non-repeating decimals are numbers; you can easily order them in between rational numbers. For example, it's clear from comparing the decimal expansions that $3 < \pi < 3.\overline{3}$. However, we can't write these numbers as a fraction of integers.

It is troubling, however, that one must take it on faith that these objects are really "numbers" in the same sense as rational numbers.

Answer 10

I am a historian of science and I am intrigued by this question, because I am a non-specialist in mathematics but for many years I taught classes dealing with the Pythagoreans. One thing you might consider in making irrational numbers "real" is to imagine why they were deemed so revolutionary and important by the ancient Greeks. In other words, instead of looking for an illustration of their practical reality, look instead to why they were so darned interesting. One answer that hasn't yet been addressed is the contention that shapes are a better representation of reality than numbers. An irrational number is the only way, in the language of numbers, to represent any "real" distance" that cannot be expressed as a relationship between two whole numbers. And yet we all know that such distances are in fact real, i.e. the hypotenuse of a right triangle. That reality, by contrast, is quite easily expressed visually through geometry, which is one reason Euclid spent so much time laying out the rules of geometric shapes. Remember that he was fundamentally a philosopher. One could argue that geometric shapes (to which Platonic philosophers believed all reality could be reduced, i.e. the "forms") are a better representation of reality than numbers are.

By bringing up metaphors, you've put your finger on the fascinating problem of reality raised by irrational numbers. The "whoa" moment regarding irrational numbers usually comes when we rethink what's behind the theorem we all had to memorize in school. Irrational has a literal meaning (no ratio possible), but also it suggests the cognitive dissonance between numbers and shapes, and helps to explain why forms were perceived as fundamental among some of the ancient Greeks.

Answer 11

The reason irrational numbers are so hard to grasp is that the rational numbers are already dense: In an arbitrary small region around any rational number you'll find infinitely many other rational numbers. In particular, if you have a rational number, there is no "next higher" rational number. Therefore it intuitively seems there's no gap left in between for even more numbers.

But it turns out that there are such gaps, although you cannot label them with a single pair of rational numbers. This is unlike with the integers where you can clearly say "there's a gap between 0 and 1, where more numbers can be filled in."

To specify the gaps between the rational numbers, you have to specify not just one pair, but a whole sequence of pairs.

For example, consider the square root of two. It can be proven that there does not exist a rational number whose square is 2, and you surely already know that proof. But is this a fundamental problem (like the fact that there is no number $x$ such that $0x=1$) or just because we are missing some numbers?

One thing we can find for sure is a rational number whose square is less than $2$; for example, $1$ is such a number, as $1^2 = 1 < 2$. We can also find another rational number whose square is larger than $2$, for example, $\left(\frac32\right)^2 = \frac94 > 2$. Since for positive numbers, $x>y$ implies $x^2>y^2$, so if we assume that this should hold also for those "extra numbers" (which we term "irrational" because, well, they are not rational), we get for the (for now, hypothetical) number $x$ with $x^2=2$ that $1 < x < \frac32$.

But we can find a larger number than $1$ whose square is less than $2$. For example, $1 < (5/4)^2 = 25/16 < 2$. And equivalently, we can find another, smaller rational less than $\frac32$ whose square is larger than $1$.

Indeed, we can find a whole growing sequence of rational numbers whose squares are all smaller than $2$, but which eventually will grow above any rational number whose square is less than $2$. That is, any rational number that is above all the numbers in that sequence must have a square that is at least $2$. Similarly, we can find a falling sequence of rational numbers with squares larger than $2$ that eventually falls below any rational with square larger than $2$. Now it is obvious that there is no rational number that is larger than any number in the first sequence, and at the same time smaller than each number in the second sequence, as the square of such a rational number would be neither greater nor less than $2$, and we already know that a rational number with square exactly $2$ does not exist. Thus those two sequences describe a gap in the rational numbers; a gap tjhat is above all the number of the first sequence, but below the numbers in the second sequence.

As a cross check, we can do the same with $4$ in the place of $2$. Here we get a growing sequence of rational numbers with square below $4$ and a second sequence of rational numbers with square above $4$. Of course the first sequence consists of rational numbers below $2$, and the second sequence consists of numbers above $2$. And there is exactly one rational number enclosed by those sequences, namely $2$. And indeed, $2^2=4$.

Now if you look at the two sequences, you see that as you progress, the terms of those sequences get arbitrary close to each other, but the first always stays below the second. And one can show that whenever this condition is met, then there can be at most one rational number in between, but there might be none.

Now where there is none in between, those sequences identify a gap in the rational numbers; and we can now just fill in the gaps by declaring that each of the sequence pairs has exactly one number enclosed. In the case where this is no rational number, it is an irrational number (which just means, not rational).

Now of course one has to prove that this gives a consistent definition for those numbers, but one can do that, and indeed, this is one of the standard definitions of real numbers (though usually the two sequences are described as end points of nested intervals, but that is technical details).

So in short, the irrational numbers are the gaps in between the rational numbers, but because the rational numbers are already dense, you cannot specify that gap with a single pair of rational numbers, but you need to use two sequences of numbers "closing up" to each other in order to identify such a gap.

Answer 12

Imagine you can build chronometers. They all share a common mechanism such that one round takes the same unit of time. To fix ideas, say it is one minute. Now, you can draw any number of ticks (indices?), regularly marked around the dial's edge. It could be $60$ (marking seconds), but you can choose $17$, or any integer.

An irrational is a duration (expressed in your unit of time, here one minute) such that, no matter the number of rounds around the clock, you will never be able to design one chronometer (with regularly spaced ticks) for which the hand would stop exactly on one of the ticks, for this duration.

Answer 13

Irrational numbers do not exist in the "real world". For example, You cannot physically construct an exact square or a perfect circle. These are purely mathematical concepts. Take for example the corners of one face of a bcc diamond crystal. Although they form a "square", they never form a "perfect" square due to thermal fluctuations and quantum effects. So to say that the length of the diagonal is $\sqrt 2$ times the length of its side is never exactly true. Even if you take a statistical average, you will never reach a value of $\sqrt 2$. However, your average will always be an rational number. So in that sense, an irrational number is the limit toward which your statistical average (a rational number) approaches, but never reach, as you increase the number of observations of a measurement.

Answer 14

As you want a metaphor:

Consider you two numbers, say 6 and 8, and you want to related them by lining them up end to end to see where they meet up. Starting at 0, you lay out the six and it goes to 6. You lay out the eight and it goes to 8 so they don't match. You lay out a second six and it goes 12 (no match). You lay out the eight to 16 (no match). You lay out a third six to 18 (no match). You lay out a third eight to 24 (no match). You lat out a fourth six to 24 and you have a match.

4 sixes = 3 eights. six to eight is a ratio of three to four or 3/4. Such a nice ratio of whole numbers is called a rational number.

Try it with something not so nice. $5 \frac {16}{23}$ and $52 \frac {10}{21}$ for instance will be pretty tedious but they will eventually match up if we lay out 582958 of the $5 \frac {16}{23}$ and 2751 of the $52 \frac {10}{21}$ creating a ratio of 2751 to 582958 or the rational number $\frac{2751}{582958}$.

Now if we pick two arbitrary values, there is absolutely no reason on earth to assume that they ever will match up in any whole number of steps to attempt to cram them together.

If we try to match $1$ with $\sqrt{2}$ or $1$ with $\pi$ we'll never be able to. So the ratios of $\sqrt{2}$ to $1$ or $\pi$ to $1$ will never have any nice whole number ratios.

It's important to note that we can get as close to cramming to within any close but not complete margin as we like. It's also important to note any value we can express in terms of fractions has already been divided neatly into even parts so any two fractions (or whole numbers, or terminating decimals, are combinations thereof) will match up eventually. But if we pick values abtrarily by tossing an infinitely precise dart at a mile long yard stick there is no reason to believe the resulting value is a fraction of anything.

The question is how do we "really get" what such a value is if we have no method of expressing it. That is where it feels the "floor is pulled out from under us". But we shouldn't feel as the the floor was removed. We should feel instead that we had been building a floor out of cards flimsily relating whole numbers to whole numbers when there is no reason to believe all values relate to whole numbers. We should feel instead that we built a flimsy but very wide net upon which to walk, only to discover there is a solid granite bedrock foundation beneath it.

It should make us feel good. But we're humans so instead we feel as though the wind was knocked out of us.

Answer 15

Irrational numbers entering the 'real world' of point masses and weightless strings: the harmonic oscillator.

Hang a point mass on a (infinitesimally thin yet non-stretchable) string and observe the period of oscillation in the limit of small amplitudes.

Now create a string of the same material and of exactly twice the length. Attach a point mass also to this string and observe the period of oscillations in the limit of small amplitudes. You notice the period of this pendulum is about $1.4$ times longer than the period of the half-length pendulum.

Now let's determine the exact ratio between the two periods. You start both oscillators in sync by letting go both weights from an elevated position at exactly the same moment, and wait till both masses are again simultaneously at their highest point.

When the short oscillator has completed 7 periods, the long oscillator is a bit shy of 5 periods. So you wait a bit longer. And longer. And again longer... You never observe both oscillator to be in sync.

Answer 16

Here's a simpler metaphor. Imagine an infinite flat plane, a precise sign saying "here", a dart point down some distance away. You measure the distance in miles. It isnt an exact number of miles. You measure it in 1/2 miles. It isn't an exact number of 1/2 miles. You measure it in 1/3, 1/4, 1/5, 1/6, 1/7 miles it isn't an exact measure of any of those either. You keep doing smaller increments, 1/2,897,854 miles, 1/2,897,855 miles. Is there any reason to believe this point will eventually measure exactly into any of those.

Well, a typical student will say well there must eventually be some fuzzy edge where you can't measure any finer and how do you mean the dart is "there"? The edge of the tip? The center of the tip? Where do the quantuum particles that make the dart begin and where does the "not dart" end? Try not to slap or browbeat or bully the poor student but gently point out that this is math and in math we have infinite precision.

Is there any reason to believe this point is any whole number measure of $1/n$ miles for some whole number $n$?

The answer is no, there is no reason to assume that and if it isn't so for that point, that distance is an irrational number.

An irrational value is a value that can not be divided into a distinct whole number of fraction parts with whole number denominators.

Yes, it feels like the floor is being pulled out because it never occurred to the student that any values wouldn't be divisible like that, but when one thinks about it, there was never any reason to think that they should.

Answer 17

Here is an idea for a metaphor:

First, I wish to point out that what I find difficult about the irrational numbers is that they are uncountable. And so something like the rational numbers, $\mathbb{Q}$, do not seem unmanageable to me even if they are unioned with, say, all the various roots, e.g., $\sqrt{2}$ or $(5/6)^{3/4}$ (algebraic numbers) or some of the numbers that have special names, e.g., $\pi$ or $e$ (specific transcendental numbers).

Bearing all that in mind, how about as the metaphor for rationals: Colors that you can name; thus, as the metaphor for irrationals: colors that go nameless.

Individual colors can be named, but try as you might - and even given an infinite amount of time and an infinite amount of paint - you could not possibly list names for all of the different colors.

I like this metaphor because it allows you to recognize the rationals (say, by viewing them as a ratio of one color to another) and even to pick other, individual examples (simply by pointing to a given mixture and giving it a name). But, again, you cannot name all the colors that arise through [a suitably continuous approach to] mixing them together.

This metaphor is far from perfect, and I am sure that there are ways to pick it apart. But you can also push through some related notions (e.g., around density) that might make it worthwhile.

Answer 18

What metaphor do you have for "number"? (specifically, "real number"; e.g. positive real numbers are the kinds of numbers you use to measure distance in the Euclidean plane)

That's what an irrational number is. Nearly everything that is a number is an irrational number.

There are some rare numbers that happen to be ratios of integers and we call those special numbers "rational" numbers. The adjective "irrational" simply means that the described number doesn't happen to be one of those special ones.

Answer 19

Grab a sheet of grid paper and draw two axes.

Draw a line — neither horizontal nor vertical — passing through the origin. If this line passes through the corner of one of the little squares (the origin excluded), then its slope is rational. If not, then its slope is irrational.

In other words, the existence of irrational numbers means that we can "stand" at the origin and point a laser pointer in some special directions and not "hit" any corner of any "little square".

---

Two lines passing through the origin are depicted below. The blue line has slope $2$ and the pink line has slope $\pi$.

The blue line passes through $(1,2)$ and $(2,4)$, which are corners of "little squares". The laser beam "hits a target". This is no coincidence. After all, to draw the blue line, we simply connected $(0,0)$ and $(1,2)$. We hit the target intentionally.

If we do not want the laser beam to "hit" anything, we can make the line pass through $(1,\pi)$. Since $\pi$ is irrational, the laser beam will travel forever, always missing the corners. Irrationality tells us that there is an escape from the "tyranny" of the grid.

---

More rigorously, we have an equivalence relation on $\mathbb Z \times \mathbb Z \setminus \{0\}$

$$(a,b) \equiv (c, d) \iff \frac{a}{b} = \frac{c}{d}$$

The blue line depicts the equivalence class

$$\cdots \equiv (-3,-6) \equiv (-2,-4) \equiv (-1,-2) \equiv (1,2) \equiv (2,4) \equiv (3,6) \equiv \cdots$$

which corresponds to the rational number $\frac 12$.

Answer 20

Two circus performers are riding unicycles. The tires on the two unicycles are not the same size (different diameters). They are riding along, side by side, when they run over a freshly painted line, and so now they each make a dot on the ground at regular intervals. When they crossed the line, the painted parts of their tires were both on the ground at the same. But as long as they both stay abreast of each other (their distances since crossing the line are both the same), they will NEVER EVER have their paint spots on the ground at the same time again. Enough paint to fill the entire universe wouldn't keep those spots wet long enough to reach another moment of unity like they experienced back at the line. Surely, such absurdity is preposterous! How could you ever explain such a thing? Well, it turns out that when the smaller rider (having the smaller tire) cuts across the tire and perfectly flattens it out, it happens to be the EXACT height (diameter) of the other tire!

Answer 21

There seem to be a lot of answers that dance around this issue so I'll get directly to the point. The $\sqrt(2)$ examples and the circle example, and ranting about stars are trying to find a simple way to explain a profound mathematical concept, density of numbers. While barak manos' circle example seems to be the most liked for it's intuition, it's incomplete.

I would ask a very simple follow up question. Why is it not possible to construct the prescribed lengths? The answer is, in some sense, because the rationals are too coarse. If you were only allowed to measure things up to a $\frac{1}{16}$th (like a typical school ruler) there would be lots of things you couldn't measure (width of a human hair for example).

You might then say, "Ok, I'll get a better ruler, one that is finer". But then the question becomes how fine can you go? As it turns out, the answer is infinitely fine. But even if you had an infinitely fine ruler, you would still have irrationals between the infinitely fine rational points. This is what is meant by "dense". In the stars analogy, the irrationals are the empty space that fills in the gap between stars.

If you consider Omnomnomnom decimal expansion approach, you can get a feel for how big the "space" is. Consider 0.5 and 0.51, and ask how many decimals are there between those two numbers. Since there is no limit to the number of digits you can tack the back of the expansion (e.g. 0.502, 0.5002, etc...), there are infinitely many. Some of these are rational (each time you tack on just one more digit, it's still rational because the decimal terminates). But for each rational you add, an infinite number of irrationals come along for the ride.

Why does this happen? Consider adding .501. It terminates, and thus is rational. Now ask how many decimals are there that start with .501 and have more digits past the 1? The answer is again infinitely many. You might stop and say well about .5011, it's in that list and is rational? But we can turn the question around again and ask about decimals that start with .5011. This process never ends but each time you add 1 rational, an infinite number of irrationals come with it.

This gets to the other point that is missed, the notion of "next". If I told you an natural number $\mathcal{(N)}$ for example 7, and asked you what the next number was, the answer would be 7+1=8. For natural numbers next makes sense, but If I gave you an irrational number and asked the same question, the choice is not obvious (7.1, 7.01,...?). Now you may ask, what is the next rational for a given rational. It turns out this can be defined but it's not obvious how (look up Cantor counting).

EDIT:

Noah Schweber points out that the cantor ordering is not natural ordering of $\mathcal{Q}$, and this is correct. This starts to dig into a bigger problem of asking how do you tell when two sets of numbers have the same amount of numbers (called the cardnality) in them. But to resolve this, we'll need more mathematical machinery, specifically bijective maps between sets.

There is some insight to be gained in the idea that the density and the ordering of a set of numbers end up being closely related. The bigger picture that I was aiming for, is that when you reach the point where you are trying to discern how the rationals are different from the irrationals, you've actually found the surface of a very complex problem. Gaining some intuition about this is actually a very good step on your journey.

Answer 22

There is no real world metaphor

$\mathbb{R}$eal numbers are the union of rational an irrational numbers. The irrational numbers are the step to abstract world of unrealistically continuous space.

They are a contradiction to $\mathbb{Q}$uantum world and if we believe in it, then the irrational numbers indeed are a hole in the non $\mathbb{R}$eal world. But I like to think it more as a pixel grid, so that there is no holes.

There is concepts of the uncountably infinite sets and even more uncountable to that. Irrational numbers are kind of the stepping point between abstract mathematics and the real world mathematics. Too hard to explain for the children, without watering the concept. The real world has no perfect shapes.

Answer 23

Imagine that you have a sifter like this one, and that you are sifting through sand or soil. Let's call the internal lengths of the sifter $x$ and $y$, and let's assume that every particle which falls over the sifter represents a number from $0$ to $1$. The number $0.657$ for example, would be represented by a particle that falls a distance (along the x-axis) of $0.657*x$ from one rim, and a distance of $0.657*y$ (along the y-axis) from the other rim.

Each way of dividing the space $x*y$ into a grid pattern represents a certain numeral system, utilizing a limited number of decimals.

If a particle falls exactly on the grid, it represents a value that can be written using the numeral system and number of decimals represented by the grid.

Now, imagine that the spaces in the grid pattern are evenly distributed, and there are as many spaces along the $x$ axis as along the $y$ axis.

If for example there are $n$ spaces along each axis, the first perpendicular thread along the x-axis will appear at a distance of $\frac{x*1}{n}$ from the rim, the second at a distance of $\frac{x*2}{n}$ amd so on.
Now, only those particles that fall exactly on the grid will stuck, the rest will fall through. Let's say there are $5^3 - 1$ threads along each axis in the grid, dividing each axis into $5^3$ evendly distributed spaces.

This means that any particle which falls a distance from the rim (along the x-axis), consisting of a multiple of $\frac{x}{5^3}$ will get stuck on the grid.

Now, if you have a particle that falls a distance (along the x-axis) of $\frac{x*10}{5^3} + \frac{x}{5^4}$ from the rim of the sifter, it will fall through, unless you increase the number of spaces by $5$, so that you have $5^4$ spaces, and $5^4-1$ threads along each axis.

Now, imagine a particle that falls such a length from the rim of the sifter, along each axis, that it wont get stuck unless you have a grid pattern which is infinitely small. Such a particle represents an irrational number.

Answer 24

What does it mean for the ratio of the lengths of the sides of a rectangle to be rational, say $\frac{5}{3}$? It means that if the long side of the rectangle is divided into five equal parts, and if one counts out three of those parts, then the length of the resulting line segment equals the length of the short side of the rectangle. The short side can now be divided into three parts all equal to the parts of the long side. Hence the rectangle can be tiled as a $3\times5$ array of squares.

Conversely an $m\times n$ array of squares forms a rectangle whose ratio of sides is the rational number $\frac{n}{m}$. So to say that the ratio of sides of a rectangle is rational is the same as to say that the rectangle can be tiled as an array of squares. The question now is, can every rectangle be tiled as an array of squares? The answer isn't obvious. One might imagine that as long as we make the squares small enough, it can always be done.

Before answering this question, let's imagine that someone has told you that a rectangle can be tiled as an array of squares but hasn't told you what size square to use. How would you go about finding the size of square? To approach this, notice that if you manage to find a square that tiles a given rectangle, the same square will tile the shorter rectangle you get by chopping off a square section from the long end of the rectangle.

On the other hand, if the rectangle is, in fact, a square, then the rectangle is a square tiling of itself (using only one tile). Because of these two properties, we can find the square we want by chopping square sections off of the rectangle until the remainder rectangle is itself a square.

By reversing the process, this remainder square will tile the original rectangle, as the following image should make clear.

At this point, the idea that every rectangle can be tiled as an array of squares may seem much less plausible than it did previously. In order for a tiling with squares to exist, the remainder rectangle in the chopping-off process must eventually be a square. But it is not clear that this always has to happen. Why couldn't it be the case that, for certain rectangles, the chopping-off process continues forever, never resulting in a square? After all, for the remainder rectangle to be a square, its two sides have to be precisely equal. That the two sides should always be at least slightly unequal seems much more probable, from a random starting rectangle, than that they should ever be exactly the same.

These misgivings are all well-founded, but the true situation can actually be even worse than this. For certain starting rectangles, the sides of the remainder rectangle never even get close to approximate equality, much less exact equality. This happens, for example, when the sequence of remainder rectangles falls into a repeating pattern, which occurs when a remainder rectangle is geometrically similar to an earlier remainder rectangle in the sequence.

The simplest example of such a rectangle is the golden rectangle, which is defined by the property that chopping a square section off of the long end of the rectangle results in a rectangle that is similar to the starting rectangle. The ratio of the long side to the short side is known as the golden ratio and has decimal expansion $1.61803\ldots$. All remainder rectangles in the sequence have side lengths in this ratio, and hence none is ever close to being square. As a consequence, the golden rectangle cannot be tiled as a rectangular array of squares, and therefore the ratio of its side lengths is not rational.

The defining property of the golden ratio implies that its value is $(1+\sqrt{5})/2$. It turns out that similar numbers involving square roots, that is, numbers of the form $r+s\sqrt{d}$ where $d$ is a natural number that is not a perfect square and $r$ and $s$ are rational numbers, always fall into a repeating, but generally more complicated, pattern of remainder rectangles. None of these numbers are rational.

These examples are but the simplest way to see the phenomenon of irrationality. An interesting irrational number has decimal expansion $1.433127\ldots$. If the chopping-off process is carried out on a rectangle whose horizontal side has this length and whose vertical side has length $1$ then, after chopping a square off the long side, the vertical side becomes the long side; after chopping two squares off the long side, the horizontal side again becomes the long side; after chopping three squares off the long side, the vertical side becomes the long side; after chopping four squares off the long side, the horizontal side becomes the long side; and so on. Hence among the remainder rectangles are rectangles that become progressively longer and longer relative to their width. Assuming that this continues, it follows that the remainder rectangle is never a square and hence that that original rectangle does not have a whole-number ratio of sides. Actually proving that this pattern continues for this particular ratio (which is $I_0(2)/I_1(2)$, where the functions $I_n(z)$ are things called modified Bessel functions of the first kind) is considerably more work than in the case of the golden ratio.

A more familiar number that exhibits a similar, but more complicated pattern is $e$, which is therefore also irrational. The chopping-off pattern in this case is $[2,1,2,1,1,4,1,1,6,1,1,8,\ldots]$, where the the numbers represent how many squares get chopped off with each change in orientation of the long edge. Unlike the examples we have looked at so far, however, most irrational numbers exhibit a rather unpredictable chopping-off pattern. For example, $\pi$ has the pattern $[3,7,15,1,292,1,1,1,2,\ldots]$. In any case, it is only when the chopping-off process terminates that the side lengths have a whole-number ratio.

Incidentally, the chopping-off process is usually called the Euclidean algorithm, and the sequences of numbers representing squares chopped off with each change in orientation of the long side are called the coefficients of the continued fraction. Examples of continued fraction coefficients for various numbers, some of which were discussed in this post, are listed below. $$ \begin{aligned} 5/3&=[1,1,2]\\ 22/9&=[2,2,4]\\ 99/34&=[2,1,10,3]\\ (1+\sqrt{5})/2&=[1,1,1,\ldots]\\ \sqrt{2}&=[1,2,2,2,\ldots]\\ (6+\sqrt{10})/4&=[2,3,2,3,1,3,2,3,1,3,2,3,1,\ldots]\\ I_0(2)/I_1(2)&=[1,2,3,4,5,6,\ldots]\\ e&=[2,1,2,1,1,4,1,1,6,1,1,8,1,1,\ldots]\\ \sqrt[3]{2}&=[1,3,1,5,1,1,4,1,1,8,1,14,1,10,2,1,\ldots]\\ \pi&=[3,7,15,1,292,1,1,1,2,1,3,1,14,2\ldots] \end{aligned} $$