 The Evolution of the

R E A L  N U M B E R S

6

# UNIT FRACTIONS

A -- One -- is the source of a number of anything. We count units.

A unit fraction is a fraction whose numerator is 1.

 12 , 13 , 14 , 15 ,  etc.
 Here is how we count 15 's:

"One fifth, two fifths,three fifths,"

and so on.

Every fraction is thus a number of unit fractions. In the fraction 35 , the unit is 15 .  And there are 3 of them.
 35 = 15 + 15 + 15 .

The denominator of a fraction names the unit  The numerator tells their number -- how many.

 Example.   In the fraction 56 , what number is the unit, and

how many of them are there?

 Answer.   The unit is 16 .  And there are 5 of them.
 56 = 5 × 16 = 16 + 16 + 16 + 16 + 16 .

Problem 1.   Which numbers are the decimal unit fractions?

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.1, .01, .001, .0001, etc.

 Problem 2.   Let 13 be the unit, and count to 2 13 .
 " 13 , 23 , 33 or 1.  1 13 ,  1 23 ,  2.  2 13 . "

Again:  Each fraction is a number of unit fractions.

 35 =  3 × 15 = 15 + 15 + 15 .

Problem 3.

 a) 49 =    4 × 19 b) 58 =  5 × 18
 c)   2 35 =  13 × 15 d)   .35  =  35 ×  .01

Problem 4.   Write each of the following as a sum of unit fractions.

 a) 23 = 13 + 13 b) 38 = 18 + 18 + 18
 c)   1 14 = 54 = 14 + 14 + 14 + 14 + 14

Problem 5.

 a)   What number is the largest unit fraction? 12

b)   What number is the smallest?   There is none.

Problem 6.   1 is equal to

 a) 33 b) 44 c) 55 d) 1010 e) 1616 f) 100100
 g)   5 × 15 h)   9 × 19 i)   24 × 1 24

We see:

1 is a multiple of every unit fraction.
We can write 1 with any denominator.

 Problem 7.   What is the first multiple of 1 32 that will be

greater than 1?

 33 32

For more on unit fractions, see Lesson 21 of Arithmetic. Next Topic:  Rational Numbers

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