This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses nonterminating divisions. By the end of the module students should understand the meaning of a nonterminating division and be able to recognize a nonterminating number by its notation.
Section overview
- Nonterminating Divisions
- Denoting Nonterminating Quotients
Nonterminating divisions
Let's consider two divisions:
-
-
Terminating divisions
Previously, we have considered divisions like
example 1 , which is an example of a terminating division. A
terminating division is a division in which the quotient terminates after several divisions (the
remainder is zero ).
Exact divisions
The quotient in this problem terminates in the tenths position. Terminating divisions are also called
exact divisions .
Nonterminating division
The division in
example 2 is an example of a nonterminating division. A
non-terminating division is a division that, regardless of how far we carry it out,
always has a remainder .
Repeating decimal
We can see that the pattern in the brace is repeated endlessly. Such a decimal quotient is called a
repeating decimal .
Denoting nonterminating quotients
We use three dots at the end of a number to indicate that a pattern repeats itself endlessly.
Another way, aside from using three dots, of denoting an endlessly repeating pattern is to write a bar ( ¯ ) above the repeating sequence of digits.
The bar indicates the repeated pattern of 3.
Repeating patterns in a division can be discovered in two ways:
- As the division process progresses, should the remainder ever be the same as the dividend, it can be concluded that the division is nonterminating and that the pattern in the quotient repeats. This fact is illustrated in
[link] of
[link] .
- As the division process progresses, should the "product, difference" pattern ever repeat two consecutive times, it can be concluded that the division is nonterminating and that the pattern in the quotient repeats. This fact is illustrated in
[link] and 4 of
[link] .
Sample set a
Carry out each division until the repeating pattern can be determined.
When the remainder is identical to the dividend, the division is nonterminating. This implies that the pattern in the quotient repeats.
The repeating block is 703.
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We see that this “product, difference”pattern repeats. We can conclude that the division is nonterminating and that the quotient repeats.
The repeating block is 1.
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Divide 2 by 11 and round to 3 decimal places.
Since we wish to round the quotient to three decimal places, we'll carry out the division so that the quotient has four decimal places.
The number .1818 rounded to three decimal places is .182. Thus, correct to three decimal places,
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Divide 1 by 6.
We see that this “product, difference” pattern repeats. We can conclude that the division is nonterminating and that the quotient repeats at the 6.
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Practice set a
Carry out the following divisions until the repeating pattern can be determined.
Exercises
For the following 20 problems, carry out each division until the repeating pattern is determined. If a repeating pattern is not apparent, round the quotient to three decimal places.
Calculator problems
For the following 10 problems, use a calculator to perform each division.
Exercise for review