The Summer Hill Journal Of New Physics

May 2009

Is Zero A Number?

Nothing highlights the problems we have created for ourselves with zero more than this simple question "Is Zero A Number?". Why? Because the answer is both Yes and No - the "two faces" of zero. Nature does not treat zero as a number - it treats it as "nothing" or the absence of a number, but human beings have invented "number zero" and introduced it into everyday use.

For example, "zero time" 0:00:00 is a natural concept, but the "zero hour" between 23:00 and 1:00 is a man-made construct.

Similarly the expression 0⁰ (zero raised to the zeroth power) contains both an empty quantity and the man-made construct "to the zeroth power". These are two quite different types of zeros.

Many Flavours Of Zero

We deal with many types of zeros:

• 0-dimension zeros, nulls or "nothing"
• 1-dimensional zeros, such as the common view of zero as a number
• 2-dimensional zeros (e.g. x, y co-ordinates)
• zero as a number between -1 and 1
• zero as an hour between 23 and 1
• zero as a minute between 23:59 and 00:01
• zero as the opposite of one, as in binary
• zeros that are just made up to fit on a number line
• integer zeros and floating-point zeros
• and many more

Note that several of these have rules that are quite different and incompatible with others. Some are quantities ( such as apples, money or energy ), some are ordinal values ( zeros that are made up to fit on a number line, such as the time 00:00 ).

Our "Rules" For Zero

When mathematicians formalised our conventional view of zero, as commonly used in mathemathics, they chose the wrong one. This can be demonstrated simply, as follows:

• If zero is a number, as in the conventional view, then 1/0 = Infinity.
• If 1/0 = Infinity then we can divide a line into an infinite number of zero-sized points.
• If we can divide a line into an infinite number of points then we should also be able to construct a line from zero-sized points. Starting at 0, simply stack up zero-sized points until you get to 1.

Try building a line using only zero-sized points and you soon realise it cannot be done. No matter how many points you add, even an infinite number, you never move from 0.

What's Wrong?

The view of zero commonly used in maths and physics is a one-dimensional "real" number, situated between -1 and 1 on the number line. Physics actually uses a 0-dimension version: "nothing" or "null". The mathematics of Physics not only supports this, but demands it. It is the only "flavour" of zero that meets the rules required by the mathematics of physics.

The differences between a 0-dimensional and a 1-dimensional zero are few but subtle. In the case of a 0-dimensional "null" there is no value, so mathematical operations can be "incomplete" - there are no results. This lack of result is itself a null, so the result is zero. This does not change addition, subtraction or multiplication, but the result of division by zero is zero, not infinity or "undefined".

Regards,

AJ Corcoran

Apr 2009

June 2009