Number 6 Fair View,
The Garth Chambers,
Dexter University Chapel,
Dexter.
NORREX
DX1 1AY

My Dearest niece Josephina,
thank you for your recent letter, which I enjoyed reading enormously. In answer to your question, yes I agree with you. Without doubt the most fascinating number of them all is zero which was discovered independently in India, the Middle East and China, but the concept came late to Europe.

Late in the twelfth century, a wealthy entrepreneur, Joseph King of Oxenford, who was a friend of many learned men of the day, in conversation asked where he should educate his son and heir at the highest of levels. The reply he recorded in his journal was interesting.

“Should it suffice that he needst learn addition and subtraction then any of the great western centres of learning are adequate. But should he needst learn multiplication and division he must go east. There is talk of Pisa but nothing is certain there. To be certain of mastery of the arts he must go to Persia at least.”

This may seem surprising today, but the explanation was the European number system in common usage was Roman and had no zero, so multiplication and division were nearly impossible whereas the number system used in the east was Arabic and used a positional system i.e. it used a zero.

For example 5 could be 5, 50, 500, 0.5, 0.005 &c. So only ten symbols were used to represent all possible numbers, but the tenth symbol zero was crucial to place the 5, in the correct place which made multiplication and division easy.

The Roman system needed an unlimited number of symbols to do that and hence it couldn’t.

I have been so fascinated by this concept that years ago I wrote a small book called ‘This Book is About Nothing’, of which I enclose a copy.

As to you question, ‘Is zero odd, even, both or neither?’ You do ask interesting questions, my dear.

The question has no validity if one considers ℕ the set of natural numbers which start at one and are {1, 2, 3, 4...} the counting numbers all children learn, for zero is not a member of ℕ.

However, if one considers ℤ the set of integers we discussed last month which zero is defined as a member of and which includes all positive numbers, zero and all negative numbers your question indeed has validity.

If one starts at say seven and counts down in twos one gets the double open ended sequence {...7,5,3,1,-1,- 3,-5,-7...} which is of course the odd numbers, all of which if divided by two leave a remainder of one.

With a positive number, say 7 one gets 7 = 2 x 3 + 1 the 1 is the remainder when 7 is divided by 2.

With a negative number, say -7 one gets -7 = 2 x -4 + 1 the 1 is the remainder when -7 is divided by 2.

Similarly if one started at six one gets {...6,4,2,0,-2,-4,-6...} which gives you the even numbers and zero is amongst them, all of which if divided by two leave a remainder of zero.

With a positive number, say 6 one gets 6 = 2 x 3 + 0 the 0 is the remainder when 6 is divided by 2.

With a negative number, say -6 one gets -6 = 2 x -3 + 0 the 0 is the remainder when -6 is divided by 2.

And finally with zero 0 one gets 0 = 2 x 0 + 0 the last 0 is the remainder when 0 is divided by 2.

Have your mother drop you off for dinner and a chat, my love, at my chambers any evening other than a Tuesday when I have a tutorial with a student. I’d be delighted to discuss such matters with any five year old who asks such delightful questions, and I’ll see if I can beat you at chess this time.

We shall certainly discuss ℚ, and ℝ and possibly even ℂ! Which I sure will fascinate you.

Here’s a puzzle for you. Why does none sometimes take a plural verb and sometimes a singular one? I know mathematics and not grammar is what you are interested in, but unfortunately they do overlap sometimes.

And finally something for you to investigate before we meet. Hexadecimal counting uses the digits, 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. How many numbers can one represent using just four digits? Don’t forget 0000.

My very best to my dear sister, your father and yourself,
Your loving Uncle Peter

P Halthorp

Professor P. Halthorp

P.S. I was reminded of a very old joke the other day. I think you’ll like it.

Only one in a thousand people understand binary, the other seven don’t.