The Incredible Niftiness of Zero.

Last week’s Occupy Math spoke a bit about Arabic numerals and declared that zero, a deep and subtle innovation, would be the topic of a future post. In this week’s post we look at zero, the empty set, and the ways these objects affect the way we do math. The starting point for appreciating zero is Roman numerals.

Roman numerals have no zero. They also can’t be used on numbers bigger than a million.

Roman numerals are still used all the time. The twenty-second Superbowl game is “Superbowl XXII”. A feature of Roman numerals is that they are not positional notation, but instead each numeral has a fixed value; you add all the numerals present to get the number represented — for instance, XXVIII is two tens, one five and three ones, for 28. Because it could be confusing to have too many of the same numeral in a sequence, they invented subtractive notation, putting the number to be subtracted first — IV is five minus one, for 4, instead of the hard-to-read IIII, for instance; IX is ten minus one, for 9, instead of VIIII; XL is fifty minus ten, for 40, instead of XXXX. As Roman civilization got more sophisticated, they started inventing really odd notational hacks. Fractions were a nightmare. So what do you do instead?

The solution is to have a ones place, a tens place, a hundreds place, and so on!

Instead of inventing new symbols or methods all the time, Arabic numerals (which were invented by the Hindus) divide the universe up into powers of ten: 1, 10, 100, 1000, 10000, etc. Any number can be written as zero to nine of each of the powers of ten that are no larger than the number itself. So 591 means one one, nine tens, and five one-hundreds. At this point it becomes possible to say things like “two to the forty-first power is 2199023255552” without inventing nine new symbols for numbers after “M” before you can write the answer. The practice of building numbers this way is called positional notation and it is one of those subtle ideas that opens up possibilities for any civilization that gets their hands on it. The availability of positional notation is one of the underpinnings of our current civilization and most of us notice it no more than a fish notices water. A lovely song that illustrates positional notation is New Math by Tom Lehrer.

What else can you do with zero?

Set theory is the foundation of modern mathematics. In a humorous error, an educational theorist heard this and, as a result, grade-schoolers in the United States were taught elementary set theory for some time. Occupy Math remembers this happening to him in grade school. To be very clear, the relationship between set theory and mathematics is similar to the relationship between quantum physics and cooking. Yes, quantum physics is the foundation of the natural laws that govern cooking. No, knowing quantum physics does not make you a better cook. Getting quantum physics to explain cooking is so hard that practice and consultation with other cooks are completely superior ways to become a better cook.

One of the really hard things when you’re first getting used to sets is the idea that there is a set with nothing in it. It’s called the empty set and, as you can see, it is important enough to have its own Wikipedia page. Let’s try a plain language exposition of the weirdness of the empty set.

Consider the set of human beings with eight functional arms.

Short of granting some kind of octopus or spider human rights, a fairly common reaction to this idea is “there is no such thing”, which is false. There is a set of human beings with eight functional arms – it’s just empty. Now this seems like a lawyer’s difference (and it is – math is a great major if you’re intending to go to law school), but until you understand that sets of impossible things exist, but are empty, it’s hard to reason logically about mathematical truth. There is a whole type of joke based on the empty set that only mathematicians laugh at.

The size of a set is the number of elements in the set – and zero is the number of elements in a set with nothing in it. Once you have zero, you can find the empty set. Let’s learn something new on our way to an example. The symbol Σ means “add up these things”. So, for example,

eq1

shows how to use Σ to say “add up the numbers from 1 to 10”. If we knew that the set S={1,2,3,4,5,6,7,8,9,10} then a different way to say the same thing would be:

eq2

Now suppose that E is the empty set. Then what happens when we add up all the members of that set? It’s tempting to say that’s not allowed, but if you understand the empty set, the summing no numbers at all comes out like this:

eq3

It’s not a coincidence that adding up no numbers gives you the special number, zero, that can be added to other numbers without changing them. Let’s change to multiplication. The symbol Π means “multiply these things” so, for example,

eq4

Another way to talk about this is the factorial.  The factorial of a positive whole number is what you get when you multiply the numbers from one to n together.   Factorial is denoted with an exclamation points so 5!=120, according to the formula above.  What happens when we multiply the members of the empty set? Well, one is the number you can multiply by other numbers without changing them, so:

eq5

My editor, after reading this, asked “how can multiplying zero give you one?”  Well, it can’t.  The thing is, multiplying zero together would be multiplying one number and with the empty set we are multiplying no numbers.  See how odd the empty set is?  Notice that when we are adding “nothing at all” is zero, but when we are multiplying “nothing at all” is one. This is because zero is the “identity” of addition — it changes nothing when you add it to a sum — and one is the identity of multiplication.

Lest you think Occupy Math has wandered off into the land of mathematical obsession, this shows up in applications.

The example above says that the product of the numbers one through five is 120. This can be written 5! or five factorial. The product of the whole numbers from 1 to n is “n factorial”, written n!, and it counts the number of orders in which n people or objects can be placed. So there are 120 different orders for five people to line up in. The factorial also appears a great deal in probability theory. So where does the empty product show up? It appears as 0!=1, a fact without which a lot of probability theory dies before it starts. Occupy Math wondered for years why zero factorial was one – he hopes this explanation saves another math major some mental irritation.

Occupy Math hopes this post has not wandered too deep into the ocean of math. The take-home message is that zero is important, a foundation of our civilization, and understanding its full impact is actually pretty tricky. We didn’t even look at the word with no letters in it. Computer scientists use this object, which is named λ, and called the “empty string for a number of things. Was this topic too abstract? Suggest another in your comments and tweets!

I hope to see you here again,

Daniel Ashlock,

University of Guelph,

Department of Mathematics and Statistics

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