Interpretation of 0.999.....

by Soo Jin Lee and Jaehong Shin

Primary Learning Outcomes

Help students appreciate the value of 0.999... as the same one of 1.

Objective : Understading infinite circulating decimals or Preparation for the idea of 'limit' in precalculus context

Procedures/Activities

The purpose of following activities is to help student's appreciation of a infinite decimal 0.999... by interpretating it in three diffenent ways using calculator. The activities also make students to think that it could be possible to represent infinite series as a finite bounded value and they naturally start to learn the concept of limits.

DISCUSSION

0.999..... = ???

Let students define the meaning of 0.999.... and discuss each other about their own expectation value.

Interpretation 1 : Converting Fractions to Decimals

Students can easily get the value of 1/3 by using a calculator, that is, 1/3 = 0.3333.... and by sum of three 1/3 students can get the values of 1 on both sides like above table.

Interpretation 2 : Using equation.

Let x=0.999....,then 10x= 9.999....

10x-x = 9x = 9.999.... - 0.999.... = 9

Therefore, x = 1 = 0.999....

By above two different interpretations, students will easily understand why 1.0 is equal to 0.999..... Actually, above contents would be taught at middle school levels, However, those concepts can play a good role in introducing the notion of 'limit' by suggesting third interpretation about 0.9999.....

Interpretation 3 : The sum of infinite series (Expansion)

0.999.... = 9/10 + 9/100 + 9/1000 + ..... = ???

Students already know that the answer is 1 by previous two interpretations. That makes students to think that it could be possible to represent infinite series as a finite bounded value and they naturally start to learn the concept of limits.

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