Problem 96 - Elementary Algebra

Question

Without doing the actual dividing, defend the statement, " \(\frac{1}{7}\) produces a repeating decimal." [Hint: Think about the possible remainders when dividing by 7.]

Step-by-Step Solution

Answer

Dividing 1 by 7 gives a repeating decimal because remainders repeat.

Step 1: Understand the Nature of Division

When dividing a number by another, the division can either terminate or repeat. A repeating decimal occurs when the remainders begin to repeat in a cycle.

Step 2: Consider the Possible Remainders

When you divide any number by 7, the remainder must be one of the integers from 0 to 6. This is because any number either divides evenly (remainder 0) or leaves a remainder that is less than the divisor.

Step 3: Analyze the Repeating Cycle of Remainders

Since there are only 7 possible remainders (0 through 6), during the long division of 1 by 7, eventually, these remainders will have to repeat. Once a remainder is repeated, the digits in the decimal will also start to repeat, leading to a repeating decimal.

Key Concepts

Remainder

Every time you perform a division, you often end up with a part that is left over. This leftover part is known as the remainder. For instance, if you divide 10 by 3, you get 3 whole parts of 3, and what's left is a remainder of 1. Understanding remainders is crucial when exploring repeating decimals because they determine the cycle of repetition.

When dividing a number like 1 by 7, we need to look at what remainders are possible. Since 7 is the divisor, the possible remainders can only be numbers from 0 to 6. This is because any remainder must be less than the divisor. If you get a remainder of 0, it means it divides evenly, but if the remainder is any other number, it sets the stage for potential repetition in the decimal result.

Division Cycle

A division cycle is the repeating pattern that occurs in the remainders when performing long division. When a division problem produces the same remainder more than once, a specific sequence of numbers will repeat in the quotient as well. This is what we call a repeating decimal.

For example, when dividing 1 by 7, once we cycle through all the possible remainders—from 1 up to 6—the next remainder will be the same as the one you've already encountered. This repetition indicates that the decimal portion will begin to repeat. This cycle is inevitable because there are only a limited number of possible remainders, specifically 7 in this case.

Long Division

Long division is a method used to divide numbers to find a quotient and a remainder. It involves repeated subtraction of the divisor from the dividend and brings down subsequent digits of the dividend to continue the division.

When dealing with fractions like \(\frac{1}{7}\), long division helps illustrate how the remainders recur, leading to a repeating decimal. As you proceed with long division:

• Write the dividend (the number to be divided) and the divisor (the number you are dividing by) clearly.
• Perform the division step by step, bringing down digits as needed.
• Observe the remainders at each step, noticing when they start repeating.

In the end, long division not only provides the quotient but also shows directly how and when the decimal starts repeating due to the recurring remainders.