“He’d always felt he had a right to exist as a wizard in the same way that you couldn’t do proper maths without the number 0, which wasn’t a number at all but, if it went away, would leave a lot of larger numbers looking bloody stupid.” (Terry Pratchett, Interesting times)
Zero is a very strange and interesting number with a lot of amazing properties and, for many reasons, the most significant number of all. From its origins to the paradoxes that it creates, we will explore the facts that make 0 so special.
The first numeric systems such as the ones used in ancient Egypt or Greece didn’t have a zero. Quantities were expressed using combinations of symbols or letters, which made it very difficult to write big numbers and even harder to perform also simple calculations.
The firsts to develop a positional numerical system were the Sumerians in 3000 BC; the value of a symbol was given by the relative position of that symbol in relation to the others. This system was later further developed by the Babylonians in 2000 BC, who created a mark to indicate that a number was missing from one position (units, decimals, hundreds and so on).
Brahmagupta, an Indian mathematician and astronomer, was the first to formulate the concept of zero as a number around 650 AD. He also gave rules for its use, explaining how to use it in additions with positive and negative numbers (“The sum of two positives is positive, of two negatives negative; of a positive and a negative the sum is their difference; if they are equal it is zero. The sum of a negative and zero is negative, that of a positive and zero positive, and that of two zeros zero”), in subtractions (“A negative minus zero is negative, a positive minus zero positive; zero minus zero is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added”) in multiplications (“The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero”) and divisions, although his description differs from the one we have today (“A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is also negative. A negative or a positive divided by zero has that zero as its divisor, or zero divided by a negative or a positive has that negative or positive as its divisor”). What differs from nowadays conception of division by zero is that the result is now considered undefinable… we will see later what this means.
But it still took a few centuries for zero to arrive in Europe. It first reached the Middle East, where Arabian mathematicians based their numbers upon the Indian system. Finally, thanks to the conquest of Spain by the Moors, the zero reached Europe, where mathematicians and philosophers such as Fibonacci, Descartes, Newton and Leibniz further developed its understanding and conceptualisation.
HOW CAN WE DESCRIBE ZERO?
The first thing that makes us understand that zero is a special number is the fact that it’s not easy to describe it the same way we describe every other number: is zero positive or negative? Is it even or odd? These simple questions, in fact, don’t have a straightforward and intuitive answer.
Let’s start from the first issue: positive or negative?
Positive and negative are the two big categories in which we divide real numbers. But, to be completely correct, we should divide them into three categories: positive, negative and zero. Zero, in fact, is neither positive nor negative and the demonstration for that is very simple. Let’s suppose that we want to demonstrate that zero is positive using this simple equation:
0 x -3 =
= +0 x -3 =
We end up with a negative zero. For this reason, we say that zero is neutral.
The other question is: is zero even or odd?
Zero is an even number. In fact, it follows the properties of all even numbers such as: it’s divisible by 2, it’s adjacent to odd numbers on both sides, and a set of 0 objects can be divided into two equal sets.
However, people find it hard to classify it, and reaction times are much higher compared to those for other numbers. In an experiment conducted by Dehaene in the 1990s, numbers and number words were flashed on a monitor, and the subjects had to press a button for odd and another for even. A computer registered the reaction times for each answer. The results of this experiment showed that people took on average 10% more time to classify the number 0, regardless from age, nationality, linguistic background. The only discriminating factor was mathematical expertise.
In mathematics zero is a tricky number. We know that if we multiply any number by zero we have zero as an answer, but what happen if we try to divide a number by zero? Can we do it?
First of all, let’s think about what a division is. Let’s say that we have the number 10 and we want to divide it by 2. This basically means subtracting 2 from 10 until we have 0:
10 – 2 = 8
8 – 2 = 6
6 – 2 = 4
4 – 2 = 2
2 – 2 = 0
We did this 5 times, and our answer for 10/2 is therefore 5. But what happens if we want to divide 10 by 0? We have:
10 – 0 = 10
10 – 0 = 10
10 – 0 = 10
…. And so on. Thinking about it we can believe that the division by zero give us infinity. However, infinity isn’t a number, and we can’t treat it like if it was. So infinity is not the correct answer. As we mentioned previously, division by zero is undefined, or in other words it doesn’t make sense.
Let’s now think about 00. This gives problems because:
X0 = 1
0X = 0 where X = any number
What happen when we have 0 on both sides? There isn’t a definitive answer to this: some people argue that the result should be 0, others that it should be 1. Once again, the answer is that 00 is undefined; we can’t give a value for it because the limit varies depending on the number considered.
If you want to know more about the mathematical problems with number 0 check this video: