To write 1/3 as a repeating decimal, we need to understand the concept of division, decimals, and repeating decimals. This process can be broken down into several steps, including long division, identifying repeating patterns, and writing the answer in the correct form. Let’s explore each of these steps in detail.
Step 1: Long Division The first step in writing 1/3 as a repeating decimal is to perform long division. Long division is a method of dividing two numbers by breaking the problem down into smaller, more manageable steps. In this case, we want to divide 1 by 3, which gives us:
0.33333333...
3 | 1.00000000
To perform this division, we start by dividing 1 by 3. We can’t do this directly because 3 doesn’t go into 1, so we add a decimal point and a zero to the dividend (1) to make it 10. We then divide 3 into 10, which gives us 3 with a remainder of 1. We write down the 3 above the division line and bring down the next digit of the dividend, which is another zero.
3 | 10.000000 9 — 10
Next, we repeat the process by dividing 3 into 10 again. This time, we get 3 with a remainder of 1. We write down the 3 above the division line next to the first 3 and bring down the next digit of the dividend, which is another zero
3 | 10.000000 9 — 10 9 — 10
We repeat this process of dividing and bringing down until we get to the desired level of precision. In this case, we want to write the answer as a repeating decimal, so we need to keep going until we see a repeating pattern. We can see that the remainder is always 1, so the decimal representation of 1/3 is 0.33333333…
Step 2: Identifying Repeating Patterns In the above example, we saw that the decimal representation of 1/3 is 0.33333333… But how do we know that this pattern repeats? To answer this question, we need to understand the concept of repeating patterns.
A repeating pattern is a sequence of numbers that repeats over and over again. In the case of 1/3, we can see that the pattern 33333333… repeats indefinitely. To show this, we can use a bar notation to indicate the repeating pattern. In this case, we would write 0.3̅, where the bar over the 3 indicates that the pattern repeats indefinitely.
Step 3: Writing the Answer in the Correct Form Finally, we need to write the answer in the correct form. In this case, we can write 1/3 as 0.3̅, which means that the decimal representation of 1/3 is a repeating decimal with the repeating pattern 3.
It’s important to note that there are different types of repeating decimals, including those with one-digit repeating patterns (like 0.33333333…), those with two-digit repeating patterns (like 0.16666666…), and so on. In some cases, the repeating pattern may not start immediately after the decimal point, which can make the problem more challenging. However, the basic process of long division, identifying repeating patterns, and writing the answer in the correct form remains the same.
In conclusion, writing 1/3 as a repeating decimal requires an understanding of long division