Arithmetic Musings #8    Division

Division belongs to Contraction on the third level of the Arithmetic operators.

The contraction process at each level comes to boundaries or thresholds which have led to new concepts.

We will get to the 4^th level another day.

Humanity has used several (at least 4) different symbols to represent division throughout mathematical history.

The fraction bar or dash, — , was used in 12^th Century in Morroco by the Arab mathematician, Al Hassar.  It was brought into Europe by Italian Fibonacci around 1200.  The numerator or dividend was written over the bar, and the denominator or divisor was written under the bar.  So four fifths is shown as:

4
—
5

A miniature picture of this, ÷, (called an obelus [named after a ‘sharp stick’ used for cutting things] was introduced in Switzerland by a Professor Rahn around 1660.  Gutenberg had developed the movable type printing for the Bible in the 15^th Century.  Soon, even books on mathematics were being printed rather being hand-copied by scribes.  Now division could be printed in one line, not three (as above):  numerator ÷ denominator!

A few decades later in Germany, Leibnitz even removed the little bar in the middle and simply used the colon, : so three quarters would be 3:4.

In Britain in the 19th century, printers introduced the slash, / (solidus) to represent division, again on one line not three.

Powerful

You will notice that the operators become more powerful at each level.

Subtracting 5 is a matter of condensing the repeated act of counting down five steps into a single act: -5.

Similarly division is a process of repeated subtraction until there is nothing left, or until the remainder is too small to do it again.

We should introduce some terms:
DIVIDEND = the amount to be shared or divided, say a basketful of eggs;
DIVISOR = the number of recipients of an equal share;
QUOTIENT = the equal share (the number of eggs) that each recipient receives;
REMAINDER = the amount left over – not enough to give every recipient another egg.

For example - sharing a dozen eggs between different numbers of people:

You will note the relationship between them is:

DIVIDEND = (DIVISOR x QUOTIENT) + REMAINDER.

There is the idea of equality or fairness or sameness in the sharing hiding behind this process.

We have come to a boundary – the unshared remainder.  Instead of sharing the whole of something, can we share a part?  Can we break or cut a whole into smaller (but equal) parts?

Here is born the idea of FRACTIONS, related to the word ‘fractured’ (broken).

Let’s make it pictorial:

Here are 3 apples to be shared equally between 4 plates/people.

First, cut one apple into 4 parts – quarters.

And put one quarter on each plate.

After all three apples have been quartered and shared, each plate will have 3 quarters of an apple.  Even if one apple is bigger that the others, each plate will get the same amount.

Even with three different things, the process still works.  Here the Dividend is three pieces of food to be shared between four people:  The knife is dividing agent or operator.

Becomes

Numerically, we can show 3 shared between 4 in this way:

3

4

Or more simply

3
—
4

Twist the knife a little and you get 3/4  written in a single line, not three lines.

The symbols that we use can have interesting stories - can we find them - or - imagine them?

RB

More in the coming weeks