### The Number 0

#### Does the number 0 actually exist?

The number 0 would imply that the absolute nothing (what is that?) would exist. This has both philosophical (0=death) and mathematical consequences. In Mathematics the zero defines infinity (something/0=infinitity while division by 0 is not allowed!) and causes,as ultimate consequence, that everything is equal to everything (does infinity then also exist): is this death (everything equal to everything) and is death then also the number 0? Is there nothing after death (the zero)

Does the number 0 actually exist? Without getting too philosophical about the meaning of `exist’: yes. Mathematically it exists (was introduced) as a neutral element for addition; the defining property of 0 is that 0+a=a for all numbers a. The 0 and the negative numbers were introduced/invented in order to be able to add and subtract numbers without problems or exceptions: after a while it becomes a bother when 10-9 does have an answer and 9-10 does not.

It is a misconception that zero would define infinity: the only thing we expect from 0 is what was mentioned above. When we combine multiplication and addition this leads to the conclusion that the product of 0 with any other number is equal to 0. This means that there is ni number, a, such that 0×a=1. For every number b not equal to 0 there is a number, denoted b-1, whose product with b is equal to 1. `Division by b’ is defined as `multiplication by b-1’ and so there is no definitionfor division by 0.

The statement `something/0=infinity’ contains, for a mathematician, three undefined things `something’, `division by 0’ and `infinity’. If `something’ is taken to mean a number different from 0 then division by 0 remains undefined and the equality becomes menaingless.

There are many questions involving infinity and we will come back to that later in this series.

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Read all of KP Hart’s math questions here! About Dutch Mathematician KP Hart: In the beginning of this year the Dutch government opened a website, The Dutch Science Agenda, where everyone could post questions that they thought were of scientific interest. This was an attempt to involve the whole country in determining what the Dutch science agenda should be in the coming years.

I looked through the questions and searched for terms like `mathematics’, `infinity’ … to see what mathematical questions there were and I noticed various questions that already have answers (and have had for a long time). On a whim I decided to post answers to those questions, in Dutch. For your edification I will translate these posts into English.