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Periodic Decimal Expansions

Periodic Decimal Expansions — Part I

E. L. Lady

unit IV

set 2.1

MATH 111

In this set we will look at repeating decimals from some new angles, seeing many new

things about them.

What is the quick way to multiply a number in decimal form by a power of ten?

Apply your rule by multiplying the following numbers by 1000:

.04315

.142857

.3571.

3.872

731.69

What is the quick way to divide numbers in decimal form by powers of 10? Apply your

rule by dividing the following numbers by 100:

135.48

96.7

1834

974.58

14.285714

17.09

Divide 1000 by 37. What is the remainder?

Divide 999 by 37. What is the remainder?

Calculate the decimal expansion of 1/37.

Divide 100,000 by 41. What is the remainder? Divide 99,999 by 41. What is the

remainder? Calculate the decimal expansion of 1/41.

What did you notice?

Can you ﬁnd a number like 9, 99, 999, 9999, etc. that is divisible by 15? How can you

be sure? How is the decimal expansion for 1/15 diﬀerent from the expansions of 1/37 and

1/41?

Periodic Decimal Expansions