The Irrational Ratio

π


Part of the peterjamesthomas.com Maths and Science archive


The Real Number pi – or \pi to use Greek typography, which I will do henceforth – is one of the most misunderstood in all of Mathematics. Mystery swirls around it, unlikely properties are claimed for it, false equalities are propounded for it and – most perniciously – characteristics are uniquely ascribed to it, which are actually properties of the vast majority of other Real Numbers as well. In the box below are presented just a selection of the mangled menagerie of misconceptions [1]:

Ten common misconceptions about \pi:

  1. \pi =\dfrac{22}{7}.
     
  2. \pi =\dfrac{355}{113}, or indeed any other Rational Number.
     
  3. \pi is infinite.
     
  4. \pi’s decimal expansion stops at some point.
     
  5. \pi’s decimal expansion has a pattern.
     
  6. If you use a different [Natural Number] base, then \pi is Rational.
     
  7. \pi has properties that are not shared by any other number.
     
  8. \pi cannot be represented exactly, only approximated.
     
  9. The decimal expansion of \pi contains The Complete Works of Shakespeare, the secrets of Alchemy and the ‘phone number of your future significant other.
     
  10. The value of \pi could be different in other Universes.

Here is the beginning of \pi in decimal notation (a notation that lies behind much of the confusion covered in the box above):

3.141\hspace{2mm}592\hspace{2mm}653\hspace{2mm}589\hspace{2mm}793\hspace{2mm}238\hspace{2mm}462  643\hspace{2mm}383\hspace{2mm}279\hspace{2mm}502\hspace{2mm}884\hspace{2mm}197\ldots

The ellipsis at the right-hand end indicates that the numbers keep on going forever; more on this later. At the time of writing, the record for the number of digits calculated stands at 22,459,157,718,361 [2], that’s twenty-two trillion digits. However, this feat – impressive as it is – has more to do with computer science than Mathematics and it is the Mathematics of \pi that we will be focussing on here.

In particular, I will hope to address all of the misconceptions listed in the box above and – more positively – to provide a clearer insight into the true meaning of \pi. As in previous pieces, I will need to develop some Mathematical apparatus along the way. Here this will include, the use of polygons to estimate \pi, the properties of Rational and Irrational Numbers and how these relate to decimal fractions, different ways to represent \pi and, finally, the distribution of digits in \pi‘s expansion. In a sense, the journey may be as interesting – and as illuminating – as the destination. Let’s start with a historical perspective.



 
Origin Story

Print Gallery by M. C. Escher, 1956

The modern day definition of \pi relates to its role in the period of the Exponential Function, which is 2\pi i. This is something that I covered in length across both Euler’s Number and The Equation. As explained in these articles, the Exponential Function is a much more fundamental part of Mathematics and, to an extent, we can think of \pi as a by-product of this.

However, no schoolchild is introduced to \pi this way and it was not how the ancients stumbled across the concept either. Instead \pi first came to prominence in considering circles. Buried in the mists of time, someone (or more likely some several people independently) noticed that – as in the image appearing at the top of this article [3] – if you fixed a point on a wheel and rolled it along until the point returned to the same place, laying out the “edge” of the circle on the ground as it were, then the distance travelled by the wheel was in fixed proportion to how “wide” the circle was. Wider wheels traveled further with one rotation, smaller ones travelled less far, but in each case the distance traveled was just a bit more than three times the “width” of the circle. Some sort of fundamental constant of nature seemed to emerge from the properties of circles.

Using more precise terminology, the “edge” of a circle is its circumference and its “width” is its diameter. The distance between the centre of a circle and its circumference is the same in all directions – that’s the definition of a circle – and this quantity, half of the diameter, gives us another important dimension called the radius. As ever, a picture paints a thousand words:

Elements of a Circle

If the various illustrated elements of our circle above have the magnitudes shown (so the diameter is d, the radius is r and the circumference is c), then we have:

d = 2r

What empirical evidence suggested to our forebears was that, for any circle, no matter what the size of its diameter (d) and circumference (c), we have:

\dfrac{c}{d} = \text{three and a bit}

So at some point in antiquity, the existence of a constant relating the dimensions of a circle was discovered. It is estimated that this occurred somewhere of the order of 4,000 years ago. The ancients also knew that this ratio was not precisely three, but a bit more. A Babylonian tablet of around this age shows the state of the art in estimation back then [4]:

\dfrac{c}{d}\approx\dfrac{25}{8} =3\frac{1}{8}

What the “three and a bit” was actually equal to, took some time to establish. This value was not initially referred to as either “pi” or “\pi”. The terminology was first coined by Mathematician William Jones as late as 1706. Jones’s idea was later popularised by our old friend Leonhard Euler. Taking their lead, we can now write:

\dfrac{c}{d} = \pi\hspace{5mm} \text{or} \hspace{5mm}\dfrac{c}{r} = 2\pi

For the sake of clarity, I will use this notation going forward, regardless of what people may have written historically.

Turning the clock back several centuries from Jones and Euler’s work in the 1700s, the 4,000 year old Babylonian estimate for \pi was improved upon 2,000 years later by another giant of Mathematics, Archimedes of Syracuse. Archimedes developed lower and upper bounds for the value of \pi as follows [5]:

\dfrac{223}{71} \hspace{2mm} \textless \hspace{2mm} \pi \hspace{2mm} \textless \hspace{2mm} \dfrac{22}{7}

It is probably via Archimedes showing that \pi was less than \frac{22}{7} that this fraction began to be mistaken for the actual value of \pi; we will return to \pi and fractions later. How did Archimedes come up with these inequalities? The answer is with polygons. In the next section, albeit with some more modern elements, I will try to explain Archimedes’s approach.



 
Pliable Polygons

Inscribed and Circumscribed

Let’s consider a circle with unit radius (and hence, by our definition above, with a circumference of 2\pi) and to make what we are doing a little more clear colour it red. We will inscribe a regular octagon inside it (i.e. draw an octagon inside of the circle with each of its vertices touching the circle) and circumscribe a regular octagon outside it (i.e. draw an octagon outside of the circle with the mid-points of each side touching the circle). In either case, if we draw lines from two adjacent vertices to the shared centre of both figures, these form an angle which is obviously one eighth of the whole circle, i.e. one eighth of 360^o [6]. This is shown in the above diagram.

The idea is to calculate the perimeter of each of the two octagons. The inscribed octagon’s perimeter will be an underestimate of the circle’s circumference (an underestimate for 2\pi) and the circumscribed octagon’s perimeter will be an overestimate for it. Before proceeding further, let’s cover the same brief refresher on trigonometry that I have shared in other articles:

Aside:

Consider a generic right-angled triangle as in the figure below:

Trigonometry

Here the bottom left-hand angle has a value of \theta, the hypotenuse has length c, the adjacent side has length a and the opposite side has a length of b. We then have the following definitions:

\begin{aligned}\sin\theta &= \dfrac{b}{c}\hspace{5mm}\Rightarrow\hspace{5mm} b = c\sin\theta \\ \\\cos\theta &= \dfrac{a}{c}\hspace{5mm}\Rightarrow\hspace{5mm} a = c\cos\theta \\ \\\tan\theta &= \dfrac{b}{a}\hspace{5mm}\Rightarrow\hspace{5mm} b = a\tan\theta\end{aligned}

Here we will use the definitions of \sin and \tan

With these equations lodged in our minds, let’s proceed. For both of our octagons, let’s draw a line from the mid-point of the topmost octagon side to the centre of the figures. This line is obviously perpendicular to the side in both cases. The line also bisects the angle we were discussing above and thus the angle subtended by this bisector is one sixteenth of 360^o. The preceding paragraph ably demonstrates why diagrams are so useful. Tale a look at the the following, it’s a lot easier to grasp:

Inscribed and circumscribed

In both cases we have a right-angled triangle with the bottom angle being equal to \frac{360^o}{16}. In the inscribed case, the hypotenuse of this triangle is of length one. In the circumscribed case the adjacent side of the triangle is of length one. Our aim is to find the length of the opposite side, x for the inscribed octagon and to find the length of the opposite side y for the circumscribed octagon. By our trigonometric relations, we have:

\begin{aligned}x&=\sin\left(\frac{360^o}{16}\right)\\\\y&=\tan\left(\frac{360^o}{16}\right)\end{aligned}

Now \frac{360^o}{16} is one of those angles whose trigonometric values end up being rather nice. We have:

\begin{aligned}\sin\left(\frac{360^o}{16}\right)&=\dfrac{1}{2}\sqrt{2-\sqrt{2}}\\\\\tan\left(\frac{360^o}{16}\right)&=\sqrt{2}-1\end{aligned}

If we recall that the circumference of the circle in question is 2\pi, then we need to consider half of the perimeters of each octagon in order to get our estimates for \pi. Both x and y are one eighth of these half perimeters. So if we want the length of the half perimeters for each octagon, we need to multiply each of x and y by eight. Doing this we get:

4\sqrt{2-\sqrt{2}} \leq \pi \leq 8(\sqrt{2}-1)

Evaluating these approximately we get:

3.061467459 \leq \pi \leq 3.313708499

As one is an upper bound for \pi and one a lower bound, we can even take their average and assert:

\pi\approx 3.187587979

This isn’t a very accurate approximation, but is at least in the right ballpark. It can be seen that, as we increase the number of sides of our polygons, we increase our accuracy. It was via such an approach that Archimedes was able to derive his estimates; he used a 96-gon.

Following in Archimedes’s footsteps, if we take just the formula we established above for the half perimeter of the inscribed octagon, we can generalise it to say that the perimeter of an inscribed n-gon would be:

n\sin\left(\dfrac{360^o}{2n}\right)

Acknowledging the limits of Excel as a tool for doing anything bar addition, we can form the following table, which starts with an octagon and then doubles the number of sides as we go down to the next entry:

n n\sin\left(\dfrac{360^o}{2n}\right)
2^3 3.\hspace{2mm}061\hspace{2mm}467\hspace{2mm}458\hspace{2mm}921
2^4 3.\hspace{2mm}121\hspace{2mm}445\hspace{2mm}152\hspace{2mm}258
2^5 3.\hspace{2mm}136\hspace{2mm}548\hspace{2mm}490\hspace{2mm}546
2^6 3.\hspace{2mm}140\hspace{2mm}331\hspace{2mm}156\hspace{2mm}955
2^7 3.\hspace{2mm}141\hspace{2mm}277\hspace{2mm}250\hspace{2mm}933
2^8 3.\hspace{2mm}141\hspace{2mm}513\hspace{2mm}801\hspace{2mm}144
2^9 3.\hspace{2mm}141\hspace{2mm}572\hspace{2mm}940\hspace{2mm}367
2^{10} 3.\hspace{2mm}141\hspace{2mm}587\hspace{2mm}725\hspace{2mm}277
2^{11} 3.\hspace{2mm}141\hspace{2mm}591\hspace{2mm}421\hspace{2mm}511
2^{12} 3.\hspace{2mm}141\hspace{2mm}592\hspace{2mm}345\hspace{2mm}570
2^{13} 3.\hspace{2mm}141\hspace{2mm}592\hspace{2mm}576\hspace{2mm}585
2^{14} 3.\hspace{2mm}141\hspace{2mm}592\hspace{2mm}634\hspace{2mm}339
2^{15} 3.\hspace{2mm}141\hspace{2mm}592\hspace{2mm}648\hspace{2mm}777
2^{16} 3.\hspace{2mm}141\hspace{2mm}592\hspace{2mm}652\hspace{2mm}387
2^{17} 3.\hspace{2mm}141\hspace{2mm}592\hspace{2mm}653\hspace{2mm}289
2^{18} 3.\hspace{2mm}141\hspace{2mm}592\hspace{2mm}653\hspace{2mm}515
2^{19} 3.\hspace{2mm}141\hspace{2mm}592\hspace{2mm}653\hspace{2mm}571
2^{20} 3.\hspace{2mm}141\hspace{2mm}592\hspace{2mm}653\hspace{2mm}585
\pi\approx 3.\hspace{2mm}141\hspace{2mm}592\hspace{2mm}653\hspace{2mm}589

So we can see that for an inscribed 1,048,576-gon [7] our estimate for \pi matches the first 12 figures of its decimal expansion. That would be an approximation that is accurate enough for virtually any real world usage.

Borrowing the concept of a limit that we introduced in Euler’s Number, we can say that [8]:

\displaystyle\lim_{n \to \infty}n\sin\left(\dfrac{360^o}{2n}\right)=\pi

Number 8 on our list of misconceptions about \pi was that it cannot be represented exactly. Well – at least to Mathematicians – the above formula is just such a precise representation of \pi. In a later section, we will meet some more of these. For now we will move on to consider whether or not the ratio we are discussing is a ratio or not. Confused? Hopefully the next section will make things clearer.



 
Repeating Rationals

Ouroboros

Elements of the beginning on this section have been adapted from the part of A Brief Taxonomy of Numbers relating to Rational Nunbers.

Misconceptions 1 and 2 at the start of this article related to \pi being equal to (as opposed to approximately equal to) some Rational Number, with both \frac{22}{7} and \frac{355}{113} being offered as candidates. Misconceptions 4 and 5 relate to the decimal expansion of \pi. There is a connection between these two areas and we will explore it in this section and the next. Let’s start with some definitions, just what is a Rational Number?

The set of Rational Numbers, denoted by \mathbb{Q}, consists of fractions both positive and negative, so numbers like:

\displaystyle\frac{1}{2}, \hspace{10mm} \displaystyle\frac{41}{7}, \hspace{10mm} \displaystyle-\frac{2}{3}, \hspace{10mm} \displaystyle\frac{35,931}{12,343},

and so on.

However, a loose definition of fractions would include that abomination, \frac{1}{0}, which as every schoolchild learns is the work of the devil and to be avoided at all costs. It would also include, for example:

\dfrac{1}{\sqrt{2}}\hspace{5mm}\text{and}\hspace{5mm}\dfrac{2\sqrt{13}}{83 - \sqrt[3]{19}i}

which is probably not what any of us had in mind when using the term.

A way to avoid such complications is to define \mathbb{Q} in terms of the numerators and denominators of its elements as follows:

\displaystyle\mathbb{Q} = \left \{\frac{a}{b} \hspace{2mm} \bigg | \hspace{2mm} a\in\mathbb{Z}, b\in\mathbb{N} \right \}

Where \mathbb{N} is the Natural Numbers, \{1,2,3,4,5,\ldots\} and \mathbb{Z} is the Integers, \{\ldots,-3,-2,-1,0,1,2,3,\ldots\}.

Aside:

Here we introduce some more notation. What we have above is a sort of recipe for creating the members of a set, namely the Rational Numbers. The bit directly after the first curly bracket and before the vertical bar shows the general pattern of set members. The bit after the vertical bar and before the second curly bracket provides restrictions on the general pattern. The vertical bar itself can be read as “such that”. For example, consider the set \{2n \hspace{2mm} | \hspace{2mm} n\in\mathbb{N}\}. This is the even Natural Numbers; the set of numbers of format 2n such that n is a Natural Number.

So our definition of the Rationals can similarly be read as “numbers consisting of a divided by b, such that a is an Integer and b is a Natural Number”.

With this definition under our belt, let’s also think a bit about decimal representations. We use decimals so frequently that maybe their meaning slips under the surface of consciousness. What does a decimal integer, say 1,234,567, actually mean? Well a moment’s thought yields the following:

\begin{array}{lrr}&1,000,000&\hspace{5mm}= 1\times 10^6\\&200,000&\hspace{5mm}= 2\times 10^5\\&30,000&\hspace{5mm}= 3\times 10^4\\&4,000&\hspace{5mm}= 4\times 10^3\\&500&\hspace{5mm}= 5\times 10^2\\&60&\hspace{5mm}= 6\times 10^1\\+&7&\hspace{5mm}= 7\times 10^0\\&\overline{\underline{1,234,567}}&\end{array}

So the position of each number indicates to which power of ten the number relates. Moving from right to left, position one (units) is the number times ten to the power zero (which equals one), position two (tens) is the number times ten to the power one, position three (hundreds) is the number times ten to the power two, and so on.

How about decimal fractions? Well, by extension from the above, 0.1234567 means:

\begin{array}{llr}&0.1&\hspace{5mm}= 1\times 10^{-1}\\&0.02&\hspace{5mm}= 2\times 10^{-2}\\&0.003&\hspace{5mm}= 3\times 10^{-3}\\&0.0004&\hspace{5mm}= 4\times 10^{-4}\\&0.00005&\hspace{5mm}= 5\times 10^{-5}\\&0.000006&\hspace{5mm}= 6\times 10^{-6}\\+&0.0000007&\hspace{5mm}= 7\times 10^{-7}\\&\overline{\underline{0.1234567}}&\end{array}

where we of course recall that: 10^{-n}=\frac{1}{10^n}, i.e. the reciprocal of 10^n.

We have the same idea, but this time we are looking at the number of tenths, the number of hundredths, the number of thousandths and so on.

My aim here is going to be to tie Rational Numbers and decimal representations together. Let’s start with some examples.

Most will be entirely au fait with equalities like:

\begin{aligned}\dfrac{1}{2}&=0.5\hspace{20mm}\dfrac{1}{4}&=0.25\\&&\\\dfrac{1}{5}&=0.2\hspace{20mm}\dfrac{1}{100}&=0.01\end{aligned}

Some may also be well-aware of more complex-looking relationships like:

\begin{aligned}\dfrac{1}{3}&=0.333333333333\ldots\\&\\\dfrac{1}{7}&=0.142857142857\ldots\end{aligned}

But how do we go about establishing these and what – if anything – do attributes of Rational Numbers tell us about their decimal representations? The answer lies in the pre-calculator disciple of long division. Let’s consider dividing 1 by 7 and see how we get on. We start of course with:

One divided by seven

Clearly, 7 does not go into 1, so we instead consider 10 divided by 7 (keeping track of the shift by a multiple of ten via using a decimal point), getting an answer of 1 with a remainder of 3 as follows:

One divided by seven

Once more, 7 does not go into 3, so we instead consider 30 divided by 7, getting an answer of 4 with a remainder of 2.

One divided by seven

We continue this way until we have a remainder of 5. 7 does not divide this, so we look at dividing 50 by 7 instead. This goes 7 times remainder 1. See below:

One divided by seven

At this point, we are back precisely where we started and – if we continue – we will simply repeat the sequence we have just established. It is pretty easy to see that:

\dfrac{1}{7}=0.142857142857142857\ldots

which is the result we stated above.

Thinking about this more generically, if we perform long division of an Integer a by a Natural Number b, then at any point the remainder must be a number r where 0\le r< b. If we continue the long division for enough time, we must either hit a remainder of 0, in which case the decimal expansion halts like:

\dfrac{5}{8}=0.625

or we hit a remainder that we have already seen and the sequence repeats itself indefinitely as in the \frac{1}{7} example above.

We can thus say that the decimal expansion of any Rational Number either halts, or goes on forever repeating a sequence [9].

What about the other way round, is a number whose decimal expansion either halts or goes on forever repeating a sequence a Rational Number?

Well the halting case is easy. Consider, for example, 0.03607675, we can of course write this as:

\dfrac{3,607,675}{100,000,000}

This is clearly a Rational Number and we are done.

Now let’s think about a decimal expansion that repeats a sequence forever. Let’s take the example of:

x=0.487901132487901132487901132\ldots

Here the repeating section, 487901132, is of length 9. Given this, we now multiply x by 10^9 to get:

10^9x=487,901,132.487901132487901132\ldots

Subtract the larger number from the smaller and we have:

10^9x-x=487,901,132

The forever repeating elements to the right of the decimal point just cancel. Rearranging we get:

x=\dfrac{487,901,132}{10^9-1}=\dfrac{487,901,132}{999,999,999}

This is clearly a Rational Number and so we have demonstrated that Rational Numbers and decimal numbers which either halt, or repeat a sequence forever are exactly the same thing.

So we have tied together Rational Numbers and at least certain types of decimals. An obvious question is “are all decimals Rational Numbers?”, we will explore this issue in the next section.



 
They appear to be Illogical Numbers Captain

But that's illogical

We are going to start this section, by finding two numbers (with decimal expansions) that are not Rationals. The first we will construct explicitly to not be Rational, the second is a pretty simple number that frequently appears in everything from algebra to geometry, to trigonometry, and which we will show cannot be Rational. Either of these approaches demonstrates the existence of a new set of numbers, the Irrationals [10]. Let’s start be being constructive.

We have determined that any decimal fraction that either halts or repeats itself must be a Rational Number. So if we want to find a non-Rational Number, then we need to consider a decimal expansion that neither terminates, nor repeats itself. The combination of these two properties means that such a number must have a decimal expansion that goes on forever and in which no continuously repeating set of digits ever occurs. This may sound like a daunting prospect, but it’s not really so bad. Let’s think about a number, x, that starts like this (with spacing to emphasise the construction):

x=0.1\hspace{2mm}01\hspace{2mm}001\hspace{2mm}0001\hspace{2mm}00001  000001\hspace{2mm}0000001\hspace{2mm}00\ldots

This clearly has a pattern, the decimal expansion is made by tagging on an ever increasing number of zeros, followed by a 1. However, equally clearly, no sequence of digits ever repeats. It is also evident that the decimal goes on forever. These two criteria taken together tell us that x cannot be a Rational Number. Instead it is our first example of an Irrational Number.

This construction may seem a little artificial to some readers (it is nonetheless entirely valid), so let’s spend some time on something that is hopefully more concrete, a very simple right-angled triangle where the two shorter sides are both of length one. Here is this rather non-esoteric object:

Right-angled Triangle

We can calculate the value of the mysterious question-mark, the length of the hypotenuse of the triangle using Pythagoras as follows:

?^2=1^2+1^2=2
 
?=\sqrt{2}

My claim is that in \sqrt{2} we have found another number which is not Rational.

This claim has some history. It was supposedly made by a member of the Pythagorean Sect, a certain Hippasus of Metapontum. It is rumoured that the Pythagorean Sect was less than happy with this result, feeling that it pointed to disharmony in their otherwise neat and tidy view of numbers. It has even been suggested that Hippasus may have paid with his life for advancing such a heretical idea. This story, apocryphal as it may be, is further proof – if it was ever needed – that being right is no guarantee that others will appreciate your position.

However posterity has taken the side of Hippasus and \sqrt{2} is indeed our second example of an Irrational Number. A sketched proof of this appeas in the box below:

Aside:

The content of this box has been adapted from a footnote to Chapter 4 of my book on Group Theory and Particle Physics, Glimpses of Symmetry.

There are many proofs that \sqrt{2} is irrational. The one with the longest history (dating back into antiquity) proceeds by a tried and tested Mathematical technique called reductio ad absurdum (Latin for reducing [an argument] to absurdity). If we want to prove a statement X is true, we start by assuming that X is instead false. We then proceed to show that this assumption leads to something that is patently false, or a contradiction of some sort. As this conclusion cannot be true we are forced to accept that X must indeed be true. Let’s apply this approach to the Rationality of \sqrt{2}.

If – contrary to our assertion – \sqrt{2} is a Rational Number, we can express it as the ratio between two Natural Numbers a and b [11]

\sqrt{2}=\dfrac{a}{b}\hspace{20mm}(1)

Further we can assume that at least one of a and b is odd. If both are even then:

a=2x
 
b=2y

For some x and y\in\mathbb{N}

Then:

\dfrac{a}{b}=\dfrac{2x}{2y}=\dfrac{x}{y}

We can keep going on this way until one of our numbers is no longer divisible by two (this is just the process of reducing fractions that many will recall from school).

Now if we rearrange our initial equation (1) we see that:

a^2 = 2b^2\hspace{20mm}(2)

This means that a^2 must be even (it equals some number multiplied by two, which is the definition of even).

What does this tell us about a? Well if we multiply an odd number by itself, we get another odd number:

(2n + 1)(2n +1) = 2n^2 + 4n + 1

Which we can write as:

2(n^2 + 2n) + 1

Which is clearly an odd number as the first term is even and we add one to it.

Because a has been shown to be even, this means that there is some Natural Number c such that:

a=2c

Substituting this in (2) we get:

(2c)^2 = 2b^2

Or:

4c^2 = 2b^2

So:

2c^2 = b^2

Which means that b is also even by the same argument we have just employed above.

So both a and b are even, which is a contradiction of our first step where we showed that we can assume that at least one of these is odd.

The contradiction means that our assertion that \sqrt{2} is a Rational Number is false. Which means that \sqrt{2} is not a Rational Number.

Based on our hard-won equivalence between Rational Numbers and decimal fractions that either repeat or terminate, we have now shown that the decimal expansion of \sqrt{2} goes on forever and never repeats itself. In fact it begins like this:

\sqrt{2}=1.414\hspace{2mm}213\hspace{2mm}562\hspace{2mm}373\hspace{2mm}095\hspace{2mm}048  801\hspace{2mm}688\hspace{2mm}724\hspace{2mm}209\hspace{2mm}698\ldots

As per misconception 7 in our initial box, it seems that humble \sqrt{2} has a property often thought of as applying only to members of an exclusive club containing \pi, e [12] and sometimes \phi [13].

In fact, if we consider Rational Numbers and Irrational Numbers, while both sets are infinite (you will never get to the end of them), it can be shown [14] that the size of the Irrationals in some sense exceeds that of the Rationals; the Irrationals have a size which is at a different level of infinity to the Rationals. Collectively, the Rationals and Irrationals are known as the Real Numbers. One way of thinking about the Real Numbers is anything you can write as a decimal, either terminating or non-terminating, repeating or non-repeating. Perhaps a better analogy is to think of a continuous line stretching both ways from a central 0, negative numbers to the left, positives to the right. If you pick any point at random on this line, its distance from 0 may be an Integer, it may be a Rational Number, but it is guaranteed to be a Real Number. That is the Reals can be thought of as every single number on an infinite line with no gaps whatsoever.

The result we mention above, that the Irrational subset of the Reals is infinite to a higher degree than the Rational subset, means that – at least in one sense – the vast majority of Real Numbers have decimal expansions that are non-terminating and never repeat. Such numbers are more the rule than the exception.

The observant reader may at this point have made a reasonable inference; they may have drawn the conclusion that, like our two numbers above, \pi is probably also Irrational. Congratulations to such people, you are right in your guess. In fact this result was proved in the 18th Century by Johann Heinrich Lambert. Neither Lambert’s proof, nor any of the more modern alternatives is entirely straightforward and the details are accordingly not repeated here [15].

The reason that the decimal expansion of \pi is thought to hold so many mysteries is precisely that \pi is an Irrational Number. Just like \sqrt{2}, its decimal expansion is non-terminating and non-repeating. However, as we have seen in this section, this is a wholly unremarkable property, one shared by the vast majority of Real Numbers. We will consider later if having a non-terminating and non-repeating decimal expansion is sufficient to guarantee a number includes the entire works of Shakespeare as per misconception number 9 (HINT: It’s not).

Perhaps here is a good point to tick off another two misconceptions, number 3, \pi is infinite, and number 6, \pi is Rational in some other number base. The first of these is probably as much down to an issue with terminology as anything else. The decimal expansion of \pi is indeed infinitely long by virtue of the fact that it is an Irrational number. However, \pi itself is anything but infinite. It’s decimal expansion starts 3.14\ldots this means that we must have:

3.1<\pi<3.2

3.1 and 3.2 are clearly finite numbers. If \pi sits between them, it too must be finite, indeed we could have just used its Integer part to state:

3<\pi<4

There are no infinite numbers between 3 and 4 of course.

Moving on to different number bases, above we elucidated what the number 1,234,567 meant in base 10 (aka decimals), namely:

\begin{array}{lrr}&1,000,000&\hspace{5mm}= 1\times 10^6\\&200,000&\hspace{5mm}= 2\times 10^5\\&30,000&\hspace{5mm}= 3\times 10^4\\&4,000&\hspace{5mm}= 4\times 10^3\\&500&\hspace{5mm}= 5\times 10^2\\&60&\hspace{5mm}= 6\times 10^1\\+&7&\hspace{5mm}= 7\times 10^0\\&\overline{\underline{1,234,567}}&\end{array}

We could however choose a number other than 10 as our base. For example hexadecimal numbers, or base 16 numbers, are frequently used in computing. These are related to powers of 16 and use the numerals:

0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

We have to add more “numbers” like A,B,\ldots,F as 10 in hexadecimal is equivalent to the decimal number 16. Then we need something to represent, for example, the hexadecimal number 10-1, which is 15 in decimal. F is the, essentially arbitrary, symbol that has been chosen to play this role.

So what is the meaning of the hexadecimal number D6D6D6 [16]? Well it is as follows, noting that D in hexadecimal is 13 in decimal:

\begin{array}{lrrr}&13,631,488&\hspace{5mm}= &13\times 16^5\\&393,216&\hspace{5mm}= &6\times 16^4\\&53,2484&\hspace{5mm}= &13\times 16^3\\&1,536&\hspace{5mm}= &6\times 16^2\\&208&\hspace{5mm}= &13\times 16^1\\+&6&\hspace{5mm}= &6\times 16^0\\&\overline{\underline{14,079,702}}&\end{array}

Another well-known number base, also associated with computing, is of course binary, you can read more about binary in the Data & Analytics Dictionary.

Typically number bases are Natural Numbers. We could – if we were masochistic enough – define numbers base \pi, in which case we would have 3.1415\ldots in decimal equals 10 base \pi. However, this way lies insanity. In any Natural Number base, \pi is still Irrational. In general number bases are different ways of representing the same number. So 10 base 16 is precisely the same number as 16 base 10. Things like Irrationality are properties of a number, not how it is represented. While we have not presented Lambert’s proof that \pi is Irrational, it does not depend on how \pi is represented, the argument would be just as valid in hexadecimal. To get a sense of this, take a look at the proof that \sqrt{2} is Irrational that is provided above. What elements of this would be invalidated in a different number base?

Before moving on to the next section, it is worth pointing out one way that \pi is actually different to \sqrt{2}, this is that \pi is a Transcendental Number, whereas \sqrt{2} is not. Mathematicians seem to love applying misleading terms to concepts, Transcendental Numbers have hardly anything at all to do with meditation. Instead they are numbers which do not satisfy any finite expression such as:

x^n+a_{n-1}x^{n-1}+\ldots+a_2x^2+a_1x+a_0=0

where the a_i are all Rational Numbers

Such an expression is known as a polynomial of degree n. Polynomials are important in many areas of Mathematics, but we will not have anything else to say about them here.

The proof that \pi is Transcendental was provided by German Mathematician Ferdinand von Lindemann in 1882 [17]. As an aside, this result was essentially a corollary of work that showed that e is Transcendental; the connections between \pi and e are manifold. This same result also settled the problem of squaring the circle that dates to antiquity; as \pi is Transcendental, this cannot be done.

Having addressed misconceptions 1, 2, 3, 4, 5, 6 and 7 so far. The next section will debunk misconception 8. The following one will address misconception 9, leaving us only misconception 10 to consider, which we will do in the final section.



 
Representation of the πεωπλε

Representations

This section presents a number of formulae for \pi, ancient and modern. With a couple of exceptions, I will just state these rather than derive them; this article would expand exponentially (in both length and complexity) if I provided the background necessary to prove all of these equalities.

Towards the end of a previous section, we discovered one way in which Mathematicians can precisely define \pi. This involved the concept of a limit. The same idea can generate many other formula precisely equal to \pi. Rather than being limits of a formula, some of these are the sum of infinite series (another concept covered in Euler’s Number).

The first of these was discovered by Indian Mathematician Madhava of Sangamagrama in the 1300s. It is as follows [18]:

\pi=4\bigg(1-\dfrac{1}{3}+\dfrac{1}{5}-\dfrac{1}{7}+\dfrac{1}{9}-\ldots\bigg)

Where again the ellipsis denotes that the sum goes on for ever. With the way the history of Mathematics goes, results being lost and then rediscovered, this expression used to be known as the Leibniz series after the great 17th Century German Mathematician Gottfried Leibniz. It is now often called the Madhava–Leibniz series.

We can use the summation notation that we also introduced in Euler’s Number to write this more concisely as:

\displaystyle\pi=4\sum_{n=1}^{\infty}\dfrac{(-1)^{(n-1)}}{2n-1}

Madhava didn’t rest on his laurels, he also formulated the following [19]:

\pi=\sqrt{12}\bigg(1-\dfrac{1}{3\times 3}+\dfrac{1}{5\times 3^2}-\dfrac{1}{7\times 3^3}+\ldots\bigg)

Which we can also write as:

\displaystyle \pi=\sqrt{12}\sum_{n=1}^{\infty}\dfrac{(-1)^{(n-1)}}{(2n-1)3^{(n-1)}}

An extremely famous formula for \pi was the result of a Mathematician solving a then outstanding problem to do with the sum of reciprocals of the squares of Natural Numbers, the so-called Basel Problem. The answer proved to be:

\dfrac{\pi^2}{6}=\dfrac{1}{1^2}+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+\dfrac{1}{5^2}+\cdots

Who was the person who provided this solution? None other that Leonhard Euler in 1734.

There are also many formulae for \pi employing infinite products, such as:

\pi=\dfrac{4}{1}\times\dfrac{4}{3}\times\dfrac{8}{3}\times\dfrac{8}{5}\times\dfrac{12}{5}\times\dfrac{12}{7}\cdots

We use a capital sigma – \Sigma – to denote summation, the capital version of another Greek letter is used for products. Can you guess which one from this equivalent expression?

\displaystyle\pi=\prod_{n=0}^{\infty}\bigg[\dfrac{16(n+1)^2}{({2n+1})(2n+3)}\bigg]

As stunning an example of self-reference as I have seen!

Another fruitful way of defining \pi; is via continued fractions. Instead of sums that go on for ever, these are fractions that go on for ever. Here is just one example:

\pi=\cfrac{4}{1+\cfrac{1^2}{3+\cfrac{2^2}{5+\cfrac{3^2}{7+\cfrac{4^2}{9+\cfrac{5^2}{11+\cdots}}}}}}

A more direct way to define \pi; is via Integral Calculus [20]. Such an approach leads us to the formula:

\pi=\displaystyle\int_{-1}^1\dfrac{1}{\sqrt{1-x^2}}\hspace{2mm}dx

Some of the details of this derivation are covered in the box below for those who may be interested:

Aside:

The equation for a circle of unit radius (or indeed any other radius) centered at the origin can be obtained via Pythagoras’s Theorem:

The Equation of a Circle

If we consider a generic point on such a circle, with x-coordinate a; and y-coordinate b;, then we can form the right-angled triangle shown above, which has as its hypotenuse a radius of the circle (recall that this is of length 1). Pythagoras then says that:

a^2+b^2=1

If we wanted to write this in terms of x and y, we of course write:

x^2+y^2=1

Rearranging this to express y in terms of x, we get:

y=\sqrt{1-x^2}

We can then calculate the length of the upper semi-circle running from (-1, 0) to (1, 0) using the generic formula for the length of a curve, y=f(x), running between x-coordinates a and b:

\displaystyle\int_a^b\sqrt{1+\bigg(\dfrac{dy}{dx}\bigg)^2}\hspace{2mm}dx\hspace{20mm}(1)

Here \dfrac{dy}{dx} is the derivative of y with respect to x, something we defined in Euler’s Number.

In our case, we have:

y=\sqrt{1-x^2}

and need to calculate its derivative. To do so, we employ something called the Chain Rule. Once more in Euler’s Number, we defined functions, we can rewrite the above by first definining two functions, g(x) and f(x), such that:

g(x)=1-x^2

and

f(x)=\sqrt{x}

These can be combined so we have:

y=f(g(x))

which means “do f(x) to the output of g(x)”.

The Chain Rule helps us with derivatives of combined functions and can be written as [21]:

[f(g(x))]^{\prime}=f^{\prime}(g(x))g^{\prime}(x)\hspace{20mm}(2)

Note that, as both g(x) and f(x) are in general simpler than f(g(x)), they may be easier to differentiate separately than the combined expression.

Applying the Chain Rule to our definitions of g(x) and f(x) above, we have:

g^{\prime}(x)=-2x

then, noting that \sqrt{x}=x^{\frac{1}{2}} and applying the rule for differentiating powers of x that we again met in Euler’s Number, we have:

f^{\prime}(x)=(x^{\frac{1}{2}})^{\prime}=\dfrac{1}{2}x^{(\frac{1}{2}-1)}=\dfrac{1}{2\sqrt{x}}

If we combine these together using the Chain Rule definition (2) above, we get:

\dfrac{dy}{dx}=\dfrac{1}{2\sqrt{1-x^2}}\times-2x=\dfrac{-x}{\sqrt{1-x^2}}

Plugging this into our formula for the length of a curve (1), we get:

\begin{aligned}1+\bigg(\dfrac{dy}{dx}\bigg)^2&=1+\bigg(\dfrac{-x}{\sqrt{1-x^2}}\bigg)^2\\&\\&=1+\dfrac{x^2}{1-x^2}\\&\\&=\dfrac{(1-x^2)+x^2}{1-x^2}\\&\\&=\dfrac{1}{1-x^2}\end{aligned}

Recalling that the x-coordinates run from -1 to 1, the length of the upper semi-circle (which is of course just \pi), is given by:

\pi=\displaystyle\int_{-1}^1\dfrac{1}{\sqrt{1-x^2}}\hspace{2mm}dx

Which is the equality that we wanted to prove.

However, my absolute favourite formula for \pi is one that is much easier to arrive at. It starts with Euler’s Identity and progresses via some very simple rearrangements as follows:

e^{i\pi}+1=0

Subtracting 1 from both sides, we get:

e^{i\pi}=-1

Take the Natural Logarithm of both sides, the equation becomes:

i\pi=\log{(-1)}

Divide both sides by i (which is of course equal to \sqrt{-1} [22]) to get:

\pi=\dfrac{\log{(-1)}}{i}

Multiplying both top and bottom of the right-hand side by i, we have:

\pi=\dfrac{i\log{(-1)}}{i^2}

Noting that of course i^2=-1, we finally have:

\pi=-i\log{(-1)}

We have covered a lot of ways of expressing the ratio of a circle’s circumference to its diameter in the course of this section. However, we have not as yet covered the most obvious one. Let me introduce it to you: \pi. That’s it, just \pi. \pi is a symbol that equates to a number, just the same way that 1 or 2,480,367 does. In the confusion caused by decimal expansions, infinite sums, continued fractions and definite integrals, we can sometimes lose track of the simple fact that all numbers are just symbols and that \pi is a perfectly sound representation of the number we have been discussing in this article.



 
Normal Again [23]

The Scottish Play

Here we get to our penultimate misconception about \pi, that it includes, somewhere in the depths of its infinite decimal expansion, any string of numbers that you could think of. This property is typically equated with some actual number, say your telephone number, or – by mapping numbers to letters [24] – some actual text, like the Declaration of Independence or – canonically – the Complete Works of Shakespeare.

We know that the decimal expansion of any Irrational Number is a stream of digits which neither stops nor settles down to repeating a sequence. Assertions like the ones above are based on the assumption that a stream of digits with these properties must contain any finite length sequence of digits that you can name. For example, if we pick a sequence:

878\hspace{2mm}533\hspace{2mm}370\hspace{2mm}852\hspace{2mm}108 628\hspace{2mm}152\hspace{2mm}204\hspace{2mm}656\hspace{2mm}679

(which is just made-up gibberish of course), it must appear somewhere in the never-ending stream of digits. This might be at the twenty-six trillionth position. It might be later than that. The assumption is that you just have to wait long enough for the chosen sequence to appear.

For the decimal expansion of a generic Irrational Number, this assumption is clearly false. As a counter-example, let’s consider the Irrational Number that we constructed as our first example of the breed above:

x=0.1\hspace{2mm}01\hspace{2mm}001\hspace{2mm}0001\hspace{2mm}00001  000001\hspace{2mm}0000001\hspace{2mm}00\ldots

Clearly the single digit 8 is never going to appear in this, let alone the rest of our 30 digit sequence above.

We obviously need to think about things in a little more depth. We will do this by first considering some Probability Theory. Though some of the concepts introduced here are meant to apply in all Natural Number bases, we’ll focus on just decimals.

Let’s assume that we have a back box which randomly spits out a chain of digits in the range 0 to 9. If the black box is a genuine random number generator, then we would expect that each of the ten digits, 0 to 9, let’s say 7 [25], appears on average a tenth of the time:

\text{Frequency of 7}=\dfrac{1}{\text{distinct digits}}=\dfrac{1}{10}

We would also expect that, if we considered consecutive pairs of digits in the range 00 to 99, let’s say 17 [26], these would appear on average one hundredth of the time:

\text{Frequency of 17}=\dfrac{1}{\text{distinct pairs}}=\dfrac{1}{10^2}

Generalising, any string of n digits would appear on average \frac{1}{10^n} times.

Let’s think about this for a moment. If a sequence of a certain length, say 30 like the one above, appears on average every \frac{1}{10^{30}}^{\text{th}} of the time, then this is the same as saying that \frac{10^{30}-1}{10^{30}}^{\text{th}} of the time it does not appear. Or we might say that, we have to wait for an average of \frac{10^{30}-1}{10^{30}}^{\text{th}} randomly generated digits to go by before we see our 30 digit sequence [27]. At one digit per second, that equates to:

999,999,999,999,999,999,999,999,999,999 seconds
 
or about 3.17\times 10^{22} years.

Our best estimate for the current age of the Universe is:

13.8\times 10^9 years

The average waiting time for our 30 digit sequence is more than two trillion times longer than this. It might take a while [28].

To return to the main point, the essentially uniform distribution of digits, pairs of digits, triplets, etc. that we describe above is the essence of a definition of a Normal Number [29]. Here is an example of a number that is Normal in base 10:

0.12345678910111213141516\ldots

That is the concatenation of all the Natural Numbers as a decimal fraction. This number, known as Champernowne’s Number, exhibits this random-like distribution of its digits. Its decimal expansion clearly goes on forever and equally no sequence ever ends up repeating indefinitely. Therefore Champernowne’s Number is an Irrational Number and – in addition – is Normal base 10. Translating this into less technical language, Champernowne’s Number does indeed contain the entire text of Macbeth somewhere in its decimal expansion. While this may seem remarkable, it also contains the same text somewhere else, but with all references to Macbeth replaced by your name, or a version in which Macbeth lives happily ever after [30].

It is however worth pointing out that, using the same numbers to text mapping that yielded Macbeth, the vast majority of this number would be total gibberish. When I say “vast majority”, I mean that – much as we found with our 30 digit number above – if you began counting at the Big Bang and took a second to read each digit starting from the decimal point, then you could well reach the present day before you stumble across:

“When shall we thre[…]” (18 characters into the first line)

Sadly, while we might be hoping for:

“[…]e meet again”

to appear next, it is overwhelmingly likely that the following text will be more like:

“D7ak shhhodDIUHKme#((j&&(8nlkLJlkj”

After a trillion Universe lifetimes, you might come across a sequence which completes the first line. After some more time goes by, you might find a series of numbers that gives you the first ten lines. Of course eventually you are guaranteed to find a sequence which yields the whole text, but the time taken to do so is most likely a number of seconds that makes TREE(3) [31] pale into insignificance.

Well this is all very interesting I hear you say, but this is not meant to be an article about Champernowne’s Number, it’s meant to be about \pi. The $64,000 question is surely, “is \pi Normal in base 10?” After all this build up, I appreciate that the answer may be rather anticlimactic, but – at the time of writing – we really don’t know. It is notoriously hard to determine whether a specific number is Normal or not.

Once more \pi is not exceptional in having this property. We don’t know whether any of \sqrt{2}, \sqrt{3}, e, or \log{2} are Normal. With the odd way that Mathematical proof works, it can be shown that the vast majority of Irrational Numbers are Normal Numbers, we just don’t know which ones. There are only a handful of numbers that we can say are Normal and most of these have been specifically constructed to have this property.

Trillions of digits of \pi have been calculated and a statistical analysis of these suggests that \pi may well be normal (similar studies indicate the same about \sqrt{2} and the rest), but “suggestive” and “may well be” don’t cut a lot of ice in Mathematics. Unfortunately, while we suspect that \pi may contain both the full text of Macbeth and all of the acceptance speeches for next year’s Oscars, we cannot definitely say that this is the case at present.

With these thoughts on the information that \pi may or may not include, this section draws to a close. In the final section, I’ll look to summarise what we have learnt and to address our last misconception, number 10.



 
Closing Thoughts 

Multiverse?

So, as advertised at the beginning of this article, the journey has been a long and winding one. At least from my perspective, some of the scenery along the way is just as interesting as the destination.

In terms of our list of misconceptions (cross-referenced in square brackets), we have explained that \pi can never be equal to any Rational Number, let alone either of \frac{22}{7} or \frac{355}{113} [1 and 2]. Trivially we noted that, as 3<\pi<4 it must be finite [3]. We were able to demonstrate that – like every other Irrational Number – \pi’s decimal expansion goes on forever [4] and never settles into a repeating sequence of digits [5]. We took a detour to establish that – again like every other Irrational Number – \pi remains Irrational in any Natural Number base [6] (though obviously \pi=10 in base \pi – not that any sane person would want to use this number system). As already referenced, through much of the above, we also showed that what holds for \pi also holds for most other Irrational Numbers, most obviously surds like \sqrt{2} [7]. We had an entire section devoted to precise representations of \pi, noting of course that the symbol \pi is just such a representation [8]. Most recently, we explained that the jury is still out on whether or not the decimal expansion of \pi contains your favourite piece of text or not, though indications are that maybe it does [9]. While ticking off these misconceptions has provided a framework, the Mathematical machinery we have developed and the results we have demonstrated or cited have, to my mind, been the real value of this article.

To close, let’s consider the final misconception, that the value of \pi could be different in other Universes, or that we could play a game of What If?, as in “what would the world be like if \pi=3?”

The concept of other Universes used to be the preserve of Science Fiction. Today, several respectable theories in Cosmology and Physics suggest that the idea may not be so far fetched. Some interpretations of Quantum Mechanics [32] admit a Multiverse as does a core element of the Standard Model of Cosmology [33]. Through millennia of investigation and theorising, we have come up with a series of quantities that define key parts of the reality that we observe. Things like the Fine Structure Constant; the rest mass and other properties of the Quarks, Leptons and Bosons that make up the Standard Model of Particle Physics; the strength of the Gravitational Force; and a number of other more esoteric constants.

In alternative Universes, there is nothing to say that the value of these constants may not be different. Maybe protons are heavier and Gravity is weaker. Indeed one idea in Cosmology [34] suggests that – in a manner analogous to Darwinian Natural Selection – we are only here to observe the grandeur of our particular Universe precisely because it is one in which these fundamental constants have just the right values. If the values of all of these numbers could vary, surely the same goes for \pi. The answer is a resounding “no”; let me try to explain why.

All of the constants I reference above are Physical constants, their values are derived by measurement. They are tied to our reality and in different Universes different measurements could be made. \pi is not a Physical constant, it is a Mathematical constant. In none of our work on \pi in this article has the charge of an electron been a key factor; the speed of light in a vacuum has never made an appearance in one of our equations. In general, while Mathematics is an uncannily useful tool for describing the Universe around us, one that “we neither understand nor deserve” [35], it is also entirely independent of physical reality. A sentient being in any Universe we could imagine could conceive of a set of points equidistant from a central one and arrive at a definition of a circle, thereby discovering \pi. To take a different definition, they could also determine the value of 2x, where x is the smallest positive number such that \text{cosine}(x)=0 and find that it begins 3.1415926\ldots

Despite the best efforts of the Indiana Legislature back in 1897, we will always have:

\pi=\displaystyle\int_{-1}^1\dfrac{1}{\sqrt{1-x^2}}\hspace{2mm}dx

or indeed any of the other many varied formulae for \pi that we saw in an earlier section. Can \pi ever be equal to anything except -i\log(-1)? Well, as I put it in a Quora response a while back, the answer is:

  • Not in some alternate time-line where Kirk is married to Spock
  • Not a long time ago in a galaxy far far away
  • Not in another dimension
  • Not in a parallel Universe with different values of the fundamental Physical constants
  • Not in a box / Not with a fox / Not in a house / Not with a mouse

Because, like the rest of Mathematics, the value of \pi emerges from a set of logical steps, rather than from measurement, it has the same value in every Universe and indeed in none. Concepts like \pi are in a sense supra-Universal. In a world subject to the oxymoron of constant change, I for one take some comfort in this indisputable and invariant fact.
 


 

<< Two Escapees Six Easy Pieces >>
Part of the peterjamesthomas.com Maths and Science archive.

 
Notes

 
[1]
 
If you want to experience these things first hand, try hanging out on in the pertinent part of Quora for an hour or so.
 
[2]
 
Peter Trueb, based on an algorithm by Alexander J. Yee.
 
[3]
 
Though this exhibit shows a half-turn, this is related to the confusion between diameters and radii when it comes to \pi.
 
[4]
 
The curly equals sign – \approx – means “approximately equal to”.
 
[5]
 
Archimedes actually did better than this. He came up with what we would now call an algorithm for generating estimates for \pi using an approach I am about to explain.
 
[6]
 
Against my normal practice, I am going to stick to degrees here rather than radians.
 
[7]
 
Answers on a postcard for the Greek name of this figure, the best reply wins a prize.
 
[8]
 
And of course that:

\displaystyle\lim_{n \to \infty}n\tan\left(\dfrac{360^o}{2n}\right)=\pi

This approaches \pi from above, whereas the other expression approaches it from below.

 
[9]
 
You can actually finesse the distinction by stating that 0.5\equiv 0.4999\ldots, an equality that probably merits an article of its own at some point.
 
[10]
 
And of course the Real Numbers, which consists of both the Rationals and Irrationals.
 
[11]
 
Here as the number we are considering is positive, we can use the Natural Numbers in the numerator as opposed to the Integers.
 
[12]
 
Euler’s Number.
 
[13]
 
The Golden Ratio.

\phi=\dfrac{1+\sqrt{5}}{2}
 
[14]
 
See the section of A Brief Taxonomy of Numbers referring to Real Numbers. This actually shows that the size of the set of Real Numbers, \mathbb{R}, is a greater type of infinity than that of the Rational Numbers, \mathbb{Q}. But, as the Irrational Numbers are just the set \mathbb{R}\backslash\mathbb{Q}, this result follows simply.
 
[15]
 
Wikipedia has a helpful page providing an overview of a number of proofs that \pi is Irrational.
 
[16]
 
D6D6D6 is the HTML code for they shade of grey I use to fill boxes in this article. It equates to (214,214,214) in RGB format, precisely because D6 in hexadecimal is equal to (13\times 16^1)+(6\times 16^0)= 208+6=214 in decimal.
 
[17]
 
Again the proof that \pi is Transcendental is non-trivial. Wikipedia provides a proof of the more general Lindemann–Weierstrass Theorem for those who are interested.
 
[18]
 
Madhava actually presented a more general result, this series is a special case of it.
 
[19]
 
This second series starts to give an accurate estimate of \pi after fewer terms than the Madhava–Leibniz series.
 
[20]
 
Again Euler’s Number provided an introduction to the other kind of Calculus, Differential Calculus. An introduction to Integral Calculus will have to wait for another article.
 
[21]
 
Using f^{\prime}(x) to denote the derivative of f(x)
 
[22]
 
For a whistle-stop introduction to i, see the relevant section of A Brief Taxonomy of Numbers. For a more leisurely review, see Chapter 7 of Glimpses of Symmetry.
 
[23]
 
Yes that is a Buffy reference:

Institutionalise

“Normal Again” – Season 6, Episode 17.

See also: 25 Indispensable Business Terms

 
[24]
 
This might be done by paring up consecutive digits and mapping them as follows:

\begin{array}{cl}00&=A\\01&=B\\02&=C\\&\vdots\\24&=Y\\25&=Z\\26&=\text{space}\\27&=\text{comma}\\&\vdots\end{array}

Or you could be lazy and just use the existing ASCII mapping, which requires three digits, e.g. 065= A.

The mechanism doesn’t really matter of course, just that such mappings exist.

 
[25]
 
Or any other digit, such as 0 or 6.
 
[26]
 
Or any other pair of digits, such as 04 or 92.
 
[27]
 
Of course it might be much less or much more, we are just talking about averages here.
 
[28]
 
This might be British understatement.
 
[29]
 
The full definition should insist that this applies in any Natural Number base.
 
[30]
 
I am unsure about the copyright implications of this. Perhaps not an issue for older works, but surely Stephenie Meyer is going to sue as the entire Twilight Saga, but with more believable characters, will appear in the same number somewhere.
 
[31]
 
A very big number indeed – for an introduction see TREE(3) is a big number, I mean really big by Josh Kerr.
 
[32]
 
The Many Worlds Interpretation.
 
[33]
 
Eternal Inflation.
 
[34]
 
The Anthropic Principle.
 
[35]
 
Part of a quote by Nobel Laureate, E.P. Wigner, which I have cited in full before on this site.

Text & Images: © Peter James Thomas 2018.
Published under a Creative Commons Attribution 4.0 International License.