Each unit fraction is a part of number 1. Half of 1, a third, a fourth, and so on.
Here is how we count 's. "One fifth, two fifths, three fifths," and so on.
Every fraction is thus a number of unit fractions.
In the fraction |
3 5 |
, the unit is |
1 5 |
. And there are 3 of them. |
The denominator of a fraction names the unit The numerator tells their number -- how many.
Example 1. In the fraction , what number is the unit, and how many of them are there?
Answer. The unit is |
1 6 |
. And there are 5 of them. |
5 6 |
= 5 × |
1 6 |
= |
1 6 |
+ |
1 6 |
+ |
1 6 |
+ |
1 6 |
+ |
1 6 |
. |
Example 2. Let |
1 3 |
be the unit -- and count to 2 |
1 3 |
. |
Again, every fraction is a sum -- a number -- of unit fractions.
3 8 |
= 3 × |
1 8 |
= |
1 8 |
+ |
1 8 |
+ |
1 8 |
. |
The symbols for all the numbers of arithmetic stand for a sum of units.
2 eighths + 3 eighths are 5 eighths. The unit we are adding is .
This illustrates the following principle:
We can only add or subtract things that have the same name,
which we call the unit. In the Example above , the name of what we are adding is "Eighths."
We will see this in Lesson 25. The denominator of a fraction has no other function but to name the unit.
Example 4. 1 is how many fifths?
Answer. 1 = |
5 5 |
("Five fifths.") |

1 5 |
is contained in 1 five times. |
Similarly,
And so on. We may express 1 as a fraction with any denominator.
Example 5. Add, and express the sum as an improper fraction:
Answer. |
5 9 |
+ 1 = |
5 9 |
+ |
9 9 |
= |
14 9 |
. |
It was necessary to express 1 as so many ninths because whatever we add must have the same name.

|