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.
