Visualizing the square root of 2

Question

A junior high school student I am tutoring asked me a question that stumped me - I was wondering if anyone could shed some light on it here.

We were talking about how the square root of 2 is an irrational number, and that means you can't write that value as the ratio of two integers. The decimal form of this value goes on forever, without repeating. And then we tried visualizing what it means when a decimal number "goes on forever."

I tried explaining it as imagining drawing a line, but you keep adding smaller and smaller pieces to its length. The pieces end up being so small, you effectively get a finite length. It was here that I could hear my own explanation breaking down. I realized I truly don't know how to visualize this concept.

The diagram I was using to describe this line was the diagonal of a square with a side length of 1. The length of this diagonal is the square root of 2, and clearly, it has a finite length (it fits inside the box, after all). Yet looking at that length in decimal form, apparently it isn't really finite. Sure those additional values that keep getting added to it as you go out from the right of the decimal point keep getting smaller and smaller, but they each have substance, adding to the overall length of the line.

What am I missing here? Or, is there a better way to explain this concept?

Answer 1

Try explaining it with a number line:

Divide the interval between 1 and 2 into ten equal parts and number them 0-9. Note that these are tenths, and so numbers of the form 1.4... fall into division four.

Now divide the interval between 1.4 and 1.5 into ten equal parts and note that these are hundredths, so numbers of the form 1.41... fall into division one.

Continuing this way demonstrates that at each step, adding more digits gives us a smaller and smaller interval that the number can be in and so increases accuracy rather than just making the number grow.

Answer 2

I would start off with a much simpler example. Suppose we wanted to locate $1/3$ on a number line, but the number line is such that every interval is divided into tenths, not thirds (e.g., like a metric ruler). How would you locate such a number? Well, you would start off by moving $3/10$ units from $0$, so now you're at $0.3$. Then between $3/10$ and $4/10$, the ruler is again divided into ten equal sub-intervals, namely $0.31, 0.32, 0.33, \ldots, 0.39, 0.40$. We again choose the third tick mark, so now we're at $0.33$. By now we should see what to do next: between $0.33$ and $0.34$, there are ten more intervals, and we choose the third tick mark, which is $0.333$. This process never ends. We have gone from $0.3$ to $0.33$ to $0.333$ to $0.3333$, and so on. The position we are at after an infinite number of steps is exactly $1/3$.

Now you might say, "why does the ruler have to be divided into tenths? If the ruler had simply been divided into exact thirds, we would not need such an elaborate, infinite process." And that's correct. But the point is that the division of the ruler is analogous to the decimal representation of a number, where each successive digit can only be obtained by moving from one tick mark to the next. Otherwise, if we wanted to "measure" a number like $\sqrt{2}$ on a number line, we could just get ourselves a "special ruler" where there are just two tick marks: one at $0$ and the other one at exactly $\sqrt{2}$. But that doesn't illustrate that the decimal expansion of $\sqrt{2}$ is $1.41421356237309504880168872421\ldots.$ Each real number is identified with a unique point on the number line. But if we want to regard such numbers as being some kind of decimal expansion, then we must resort to using the ruler divided into successive tenths.

Answer 3

Here is another way of approximating the square root of two by rational numbers which doesn't depend on the decimal system.

Suppose $p^2-2q^2=\pm 1$ so that $\left(\cfrac pq\right)^2=2\pm\cfrac 1{q^2}$, then the larger we can make $q$ the closer $\cfrac pq$ is to $\sqrt 2$.

Consider now $(p+2q)^2-2(p+q)^2=p^2+4pq+4q^2-2p^2-4pq-2q^2=2q^2-p^2=\mp 1$ so that $\cfrac {p+2q}{p+q}$ is a better approximation.

From this we obtain the approximations $$\frac 11, \frac 32, \frac 75, \frac {17}{12}, \frac {41}{29},\frac {99}{70} \dots$$

The Wikipedia entry gives also that if $r$ is an approximation, $\frac r2+\frac 1r$ is a better one, which picks out $1, \frac 32, \frac {17}{12}, \frac {577}{408} \dots$ which converges very quickly, picking out a subsequence of the previous one..

This takes $$\frac pq \text{ to } \frac {p^2+2q^2}{2pq}$$

It also gives a geometric proof which is quite visual and may help - the problem with these kinds of proofs is that they often work by some form of descent, and therefore terminate, so don't give a sense of never-ending.

Answer 4

1/3 of a pie is clearly finite. The only infinite part is the number of digits required to express absolute accuracy (i.e. you can't) when written in Base 10. In Base 3, it'd be 0.1. It is finite, but it also has absolute accuracy.

Answer 5

As usual with visualizing things, a graph does help - here is a graph of y=$\sqrt x$
(y=$x^\frac {1}{2}$ if you prefer ):

As $x$ gets bigger, $\sqrt x$ gets bigger, but by smaller and smaller amounts - example values:

$\sqrt 1=1$, $\sqrt 2=1.414$~, $\sqrt 3=1.732$~, $\sqrt 4=2$, $\sqrt 4=2$, $\sqrt 5=2.236$~

Also, of course nothing appears below $y$=0 or to the left of $x$= as it impossible to get the square root of a negative

Answer 6

The first thing to note is that a decimal representation is just a bunch of fractions; how many whole tenths, how many whole hundredths etc.

Now an irrational number is one that is never a whole number of these fractions, however small you make the fractions.

As someone pointed out, you can demonstrate numbers with recurring decimal digits like 1/3; it is no whole 1s, 3 whole tenths, 3 whole hundredths, etc. Because this is recursive, the student should get the idea that it's never going to match the fractions based on 10s, and thus recurs without stopping (which is a more graspable phrase than 'infinitely').

Thus if the problem is that the decimal representation goes on without stopping, you have an example without going as far as irrational numbers.

By contrast, of course, an irrational number is one which goes on indefinitely regardless of the base of the number, and that's rather harder to explain.

Answer 7

The infinite series of the sum of the values of decimal places is converging, and therefore the value is finite.

Answer 8

Here's one way to visualize an infinite series adding up to a finite value:

A flea is 1 metre away from a wall. The flea jumps half way to the wall (1/2). Then it jumps half way again (1/4). And again (1/8). It keeps going forever.

After a large number of jumps, it gets really close to the wall. Pick any distance, no matter how small, and the flea gets closer to the wall than that!

This example shows us that: 1/2 + 1/4 + 1/8 + 1/16 + ... = 1

This infinite series adds up to a finite value. The same is true for the decimal expansion of the square root of 2. It's an infinite series that represents a finite value.

Answer 9

Think of a meterstick. Suppose you mark a point on it.

The meterstick is divided up into centimeters: you can narrow down where the mark is by checking which two centimeter marks it lies between.

Each centimeter is split into ten millimeters. You can further narrow down where the mark is within the centimeter by noting which pair of millimeter marks it lies between.

We could divvy each millimeter up into a thousand micrometers. We could further narrow down where the mark is within the millimeter by working out which pair of micrometer marks it lies between.

And so forth. If you have a scheme for continually subdividing your meterstick (e.g. you keep breaking it up into tenths), then your marked point lies in a unique subdivision (or it lies on a mark); this tells us the decimal digits of its decimal expansion.

Conversely, any way of (consistently) specifying which subdivision the point lies in, no matter how fine we make the subdivisions, is sufficient information to uniquely determine exactly where the mark lies on the meterstick.

Answer 10

Many of the above answers are quite excellent, but lack a certain experiencing of both the never ending nature of the square root of 2 and its irrationality.

Take for example, 1/3. It's a rational number and it repeats forever. The most common method of representing that is 0.3 with a bar over the 3. The same can be done for other repeating fractions, such as 1/7, which repeats after 0.142857. Both 1/3 and 1/7 "go on forever", but neither of those just keeps on changing the way the square root of 2 does.

This is actually the key to an irrational number -- it doesn't repeat. If it did, it would be rational.

The best way -- I think -- to comprehend the value is to compute it using something such as the Babylonian Method for computing square roots, as described in this article -- Babylonian Method. That will very quickly demonstrate that the square root of 2 just goes on and on and on.