What is the decimal expansion of the following fraction? This question often arises in mathematics, particularly when dealing with fractions that are not easily converted to whole numbers. In this article, we will explore the process of finding the decimal expansion of a fraction, discuss some common methods, and provide examples to illustrate the concepts involved.
The decimal expansion of a fraction is the process of expressing the fraction as a decimal number. This involves dividing the numerator by the denominator and continuing the division until a repeating pattern emerges or the decimal terminates. The decimal expansion can be finite or infinite, depending on the fraction.
To find the decimal expansion of a fraction, we can use the long division method. This method involves dividing the numerator by the denominator, writing down the quotient, and then multiplying the remainder by 10 and dividing again. This process is repeated until a repeating pattern is observed or the remainder becomes zero.
Let’s consider an example to demonstrate this process. Suppose we want to find the decimal expansion of the fraction 1/3.
Step 1: Set up the long division problem.
“`
1 | 3
“`
Step 2: Divide the numerator by the denominator.
“`
0.3
1 | 3
-3
—
0
“`
Step 3: Multiply the remainder by 10 and divide again.
“`
0.03
1 | 3
-3
—
0
“`
Step 4: Repeat the process until a repeating pattern is observed or the remainder becomes zero.
“`
0.0333…
1 | 3
-3
—
0
“`
In this example, the decimal expansion of 1/3 is 0.333… (with the 3 repeating indefinitely). This indicates that 1/3 is an irrational number, meaning it cannot be expressed as a finite decimal or a repeating fraction.
Some fractions have a finite decimal expansion, which means that the decimal terminates after a certain number of digits. For instance, the fraction 1/4 has a decimal expansion of 0.25, as the division process stops after two digits.
In conclusion, finding the decimal expansion of a fraction involves dividing the numerator by the denominator using the long division method. The resulting decimal can be finite or infinite, depending on the nature of the fraction. By understanding the process and applying it to various examples, we can gain a better grasp of decimal expansions and their significance in mathematics.
