Other Ways to Write Fractions

The ancient Egyptians wrote fractions in a way that was very unusual from our point of view.

When we write fractions, we write them with both a numerator and a denominator. That is, a fraction like four-fifths is written as:


where four is the numerator, and five is the denominator: so the denomination of the fraction is such that it's a fraction in units of one-fifth, dividing one into five equal parts, and the number of those units is four.

The ancient Egyptians did have a symbol for the fraction two-thirds in addition to the fraction one-third.

But this was their only exception to the general rule that any fraction would be written as the sum of several fractions, each of which had an implied numerator of one. So only the denominator was written.

Thus, where we would write 3/5, they would write 1/2 plus 1/10. Where we would write 4/5, they would write 1/2 plus 1/5 plus 1/10. Where we would write 5/12, they would write 1/3 plus 1/12.

I can't recommend this system for general use, but it may well have seemed appropriate to people who made only limited use of arithmetic, and who didn't have sophisticated concepts about numbers. The notation 1/2 plus 1/10, instead of 3/5, does make it more immediately clear how big the fraction involved is; exactly 1/10th more than a half!

Ways to write fractions that may look a bit odd to many of us didn't die out with the ancient Egyptians.

A recent textbook on arithmetic for office purposes which I once saw showed problems in which people worked with numbers that looked like this:

.3 ---

It isn't too hard to figure out what this kind of number means. It's a decimal mixed fraction, where you take .3, or 3/10, and add 1/70, not 1/7, to it.

One place where this kind of notation might well be used is with amounts of money that are expressed in dollars, cents, and fractional parts of a cent... so a stock might be valued at

$3.12 ---

a share. And you will see amounts like that for some of the stocks at lower values in the stock quotes page of a newspaper, although these days perhaps only when looking at old newspapers on microfilm.

Going further back, the Liber Abaci of Fibonacci (the book that popularized the use of Arabic numerals instead of Roman numerals, which is also remembered for an example problem which had the Fibonacci series as its answer) used a notation for fractions that would, at first sight, seem quite mysterious. Thus, the fraction 2 37/240 could have been written as:

  1   1   1
------------- 2
 12   2  10

and someone today, looking at a fraction so expressed, would have no way to really guess what its value was.

However, unlike the fractions of the ancient Egyptians, this notation for fractions is really not so bad. Once you understand the code, it can actually make some types of arithmetic with fractions easier.

One problem with the fractions as written in the book was that they were being written from right to left. In modern notation, the Fibonacci form of 2 37/240 would become:

          1 -----
     1 -----------
2  ----------------

or two and one and one and one-twelfth halves tenths.

What on Earth would such a way of noting fractions be good for?

Well, suppose one had a column of figures to add like this: 5 1/7, 3 17/42, 2 6/35, 1 9/28.

Today, to add those fractions together, one would try to find a common denominator. In this case, the common denominator would be 420, so the fractions would be changed to...

5 60/420, 3 170/420, 2 72/420, 1 135/420

Using the Fibonacci system, though, one would handle these fractions in this way instead:

5  1/7  -> 5  1     /7 
3 17/42 -> 3 (2 5/6)/7
2  6/35 -> 2 (1 1/5)/7
1  9/28 -> 1 (2 1/4)/7

Basically, one is making use of the fact that all the denominators, 7, 42, 35, and 28, are multiples of seven, so one is doing arithmetic using smaller numbers by doing arithmetic on the sevenths.

So one adds the three fractions 5/6, 1/5, and 1/4. Finding the common denominator and doing the addition on such simple fractions is less likely to lead to a mistake in arithmetic, and so we add 50/60, 12/60, and 15/60, and get 1 17/60.

The 1 gets carried over into the whole sevenths, and so we have seven of them, and so another one gets carried into the units.

Therefore, our answer is 12 17/420.