# Nothing

# BLOG: Heidelberg Laureate Forum

I try to find a varied range of topics to write about for this blog, but sometimes you run out of inspiration and find you have nothing to write about. Luckily, this has happened to me today – since nothing, also known as the number zero, is a crucial concept in mathematics and a vital ingredient of many mathematical theories and systems. Here are some of the ways in which nothing is really something.

#### 0. It comes before everything else

While most of us usually start counting from one (since until you’ve got one of something, there’s no real need to count it), mathematicians and computer scientists often start counting from zero. Sometimes called zero-indexing, doing this makes counting and indexing easier in computer programs, and is a more natural way to think about many mathematical series and patterns.

For example, the zeroth power of a number is always 1, and when thinking about number bases it’s natural to start counting with ones – in decimal, we have ones, tens, hundreds and thousands, and in binary it’s ones, twos, fours, eights and so on. The zeroth derivative of a function is just the function itself, and any sensible list of derivatives would normally include it as the first entry.

It’s most often used in numbering lists of things – you wouldn’t say that the set {a,b,c} contains 2 elements, because you started counting at zero! It’s got three elements, but a is the 0th element, b is the 1st and c is the 2nd.

Numbering the first element zero is sometimes also employed when an extra, more important and fundamental idea is discovered after the list is already established – like the zeroth law of thermodynamics, which was formulated in 1935, decades after the others.

So if you’re counting things and a mathematician or computer scientist is involved, make sure to check whether they’re zero-indexing (since, as the famous quote goes, “There are two hard problems in computer science: cache invalidation, naming things, and off-by-one errors.”) Or, use it as a handy excuse if you yourself have mis-counted something!

#### 1. It’s crucial for writing numbers

When writing numbers, we use a place-value number system – that is, the position of a digit within the number determines how much that digit is worth. For example, the digit 4 in the number 24 represents 4 ones, but in the number 496 it’s worth 400. In a system like this, zeroes are crucial, as if we didn’t have a symbol to indicate ‘no tens’, the number 205 would be indistinguishable from 25.

Also sometimes called positional notation, this system is the one we’re all used to working with, and it’s a great way to write down numbers. It allows for easy multiplication and addition of numbers (column-by-column), and combined with a decimal point we can represent any number, whole or fractional, using some combination of digits.

Prior to the invention and widespread use of positional notation, numbers were represented in less intuitive ways. For example, Roman numerals use symbols like I to represent 1, V for 5 and C for 100, and these letters will always have these values wherever they appear (except in the case a symbol is placed before a larger number, in which case its value is negative: IV is 5 – 1 = 4). Computing sums and products using this notation is not particularly intuitive – if you’ve ever tried it, you mostly find it’s easier to convert the numbers to decimal first! It’s also not necessary to have a symbol for zero in this kind of system, and the Ancient Greeks didn’t have a symbol for zero at all.

But this doesn’t mean zero wasn’t used or known about in ancient times – place-value systems have been around in some form for a very long time, and in any positional system, you need a zero, or at least some way of indicating it. The Ancient Babylonians used base 60 numbers (sexagesimal), Mesoamericans used base 20 (vigesimal), and early mathematics from many other cultures used a variety of number systems including base 10, and sometimes even binary (base 2). It was common to leave a gap between numbers to indicate a zero, and when specific symbols for zero were introduced they included a pair of slashes, a large dot, and later a circle shape, which became the basis of the symbol 0.

#### 2. It’s the identity element for addition

Groups are a fundamental concept in maths, and group structures crop up everywhere from arithmetic to symmetry to matrices. Briefly, groups are sets of objects on which there’s a well-defined combining operation – if you take two objects from the set, and combine them, you get another object from the set. If you read my post from last year on groups (or you know about them anyway) you might recall that every group has an identity element – one which when combined with any other object doesn’t do anything: it leaves the object the same.

One group which you’ve almost certainly already worked with extensively (maybe without realising it!) is the group of whole numbers under addition. It definitely has a group structure – if you add together two whole numbers, unless you’re doing it really wrong, you always get another whole number. In this group, the identity element – the unique important special one that you need for the group to make sense and work properly – is zero. Adding zero to something doesn’t change it!

This works in finite groups as well – if you think of the numbers 1-12 in clock arithmetic (so if you count past 12 you go back to the beginning), the zero here is the number 12, and the set is often considered to be the numbers 0-11. If you add 0 hours to the time (or 12 hours), you don’t change anything.

While the zero might not seem like the most interesting element of a group, the fact that we have it is crucial to the way addition works, and to the structure of the group.

#### 3. It’s the answer to Euler’s identity

If you ask a mathematician to tell you their favourite maths equation, quite often the answer will be Euler’s identity. Sometimes also called Euler’s formula, it’s an equation which relates some of the most interesting concepts in mathematics, and it’s most commonly seen in this form:

This simple relation connects five things mathematicians really like:

- π, the circle constant
- e, the base of natural logarithms
- i, the imaginary number
- 0, the identity for addition
- 1, the identity for multiplication

Written this way, it also includes all of the important mathematical operations – addition, multiplication (in there between the i and the π) and exponentiation (raising to a power). The formula is fundamental to trigonometry, and shows how all of these mathematical ideas are connected. Isn’t it cool? And zero is the answer to the equation, which cements its place as definitely one of the best concepts in maths.

#### 4. It’s a crucial component of binary numbers

Binary, which is the basis of all computer systems and many other types of thinking, couldn’t happen without zeroes. For all the reasons outlined in item 1 above, it’s needed to define the absence of a power of two in the number’s expansion – but in the case of binary, it’s even more fundamental. The intrinsic beauty of stripping information down to fundamental ones and zeroes is meaningless without zeroes – they’re literally half of the story.

1 and 0 are also used in logic – as true and false in truth tables, representing the truth value of a statement, and encapsulating the yin and yang, each meaningless without the other. And statements with the truth value 0 are increasingly popular these days! 🙂

#### 5. Mathematicians are always looking for zeroes

One final argument in favour of nothing: mathematicians are obsessed with finding zeroes. In particular, zeroes, also called roots, of functions. For example, if I have a function f(x), I might ask ‘for which values of x does this function equal zero?’

This might seem like a strange question, but if you want to study a function and understand what it looks like and how it behaves, the zeroes are crucial. If the function is continuous and can be plotted as a line on the page, the zeroes will tell you where the function crosses the axis, and how it’s positioned in space. If you’re prepared to do a little calculus and differentiate the function, the zeroes of its first derivative will tell you where it changes direction and where its behaviour changes. Mapping out the zeroes gives a picture of what’s interesting about the function.

In fact, looking for zeroes could potentially be a lucrative hobby – if you can say for sure where all the zeroes of the Riemann Zeta function are, it’ll net you a cool million dollars! As I wrote previously, the Riemann Hypothesis is one of the greatest outstanding problems in mathematics. Despite many decades of mathematicians working on the problem, we still don’t have a definite answer to the question of exactly where all the zeroes are. If you ask anyone working on it, they’ll certainly tell you how important zero is to their work in mathematics!

So, if you’re considering how important zero is, you might think nothing of it – but for mathematicians, it’s second to none.

Even in the year MMXX we still live with problems of the roman numerals notation and with the marriage of the arabic numerals into the latin alphabet.

This is the problem:

The digits used in the arabic numerals, namely 1 2 3 4 5 6 7 8 and 9 clearly differ from any latin letter but the digit 0 looks very similar to the latin letter O. While this is not really a problem in handwritten text, it is a problem in text for computer porgrams, because computer programs do not have the context information of a human reader and are not satisfied with an O if they expect an 0.

Der Ton der einen Hand 🆚 Das Gewicht von keinem Rucksack

Ein bekanntes Koan (Zen-Denkrätsel) lautet:

Ein gleichwärtiger Denkspruch für eine Mathematiker müsste wohl lauten:

“You know the sound of two hands clapping; tell me, what is the sound of one hand?”.

🆚

You know the weight of a backpack on your back. Tell me what the weight of no backpack feels like.

Katie Steckles wrote (20. Feb 2020):

>

[…] if I have a function f(x) […]>

[…] If the function is continuous and can be plotted as a line on the page […]… and if the curvature radius of the corrsponding

“line on the page”is nowhere zero, then …>

If you’re prepared to do a little calculus and differentiate the function, the zeroes of its first derivative will tell you where it changes directionIf by

“the direction”of a line (on a page), at a particular point, we mean the entire set of lines (on the same page) which “just touch (but do not intersect)” the line under consideration, at the particular point under consideration, andif by the line, at a particular point,

“changing directions”we mean the line having finite curvature radius at that point, then …… that line may change direction at points, too, where the first derivative of the corresponding function is not zero.

(And the zeros of second derivative of the function may indeed tell, where the corresponding line

does notchange direction.)Katie Steckles wrote (20. Feb 2020):

>

[…] if I have a function f(x) […]>

[…] If the function is continuous and can be plotted as a line on the page […]… and if the curvature radius of the corrsponding

“line on the page”is nowhere zero, then …>

If you’re prepared to do a little calculus and differentiate the function, the zeroes of its first derivative will tell you where it changes directionIf by

“the direction”of a line (on a page), at a particular point, we mean the entire set of lines (on the same page) which “just touch (but do not intersect)” the line under consideration, at the particular point under consideration, andif by the line, at a particular point,

“changing directions”we mean the line having finite curvature radius at that point, then …… that line may change direction at points, too, where the first derivative of the corresponding function is not zero.

(And the zeros of second derivative of the function may indeed tell, where the corresponding line

does notchange direction.)