.33333 As A Fraction: Simplifying The Decimal
Table of Contents :
To understand the decimal number .33333 and how to express it as a fraction, we first need to delve into the basics of decimal to fraction conversion. The decimal .33333 can also be expressed as a repeating decimal, which means it is often represented as 0.333..., where the 3 repeats indefinitely.
Understanding Decimals and Fractions
Decimals are an extension of the number system that represents fractions in a base 10 format. They are useful in mathematics to express numbers that are not whole. Meanwhile, fractions represent a part of a whole, and they are expressed as a numerator (the top number) over a denominator (the bottom number).
Step 1: Expressing .33333 as a Fraction
To convert .33333 into a fraction, follow these steps:
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Identify the Decimal Place: The number .33333 has 5 decimal places. This means that it can be rewritten as: [ \frac{33333}{100000} ] This is because moving the decimal place five times to the right gives us 33333, and we place the number over 100000 (which is 10 raised to the power of 5).
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Simplifying the Fraction: Now we need to simplify the fraction (\frac{33333}{100000}) if possible. To simplify, we look for the Greatest Common Divisor (GCD) of the numerator and denominator.
Step 2: Finding the GCD
Finding the GCD of 33333 and 100000 involves using the Euclidean algorithm or prime factorization. However, in this case, it can be noted that:
- 33333 can be divided by 3 to give 11111 (since 3 is a factor of 33333).
- 100000 can be factored into (10^5 = 2^5 \times 5^5).
The prime factorization shows that there are no common factors between 33333 (which factors down to (3 \times 11111)) and 100000 (which factors down to (2^5 \times 5^5)). Thus:
[ \text{GCD}(33333, 100000) = 1 ]
Since they have no common factors other than 1, the fraction is already in its simplest form.
Step 3: Resulting Fraction
Thus, the decimal 0.33333 can be simplified and expressed as the fraction: [ \frac{33333}{100000} ]
However, to represent the repeating part, .333... can be simplified further.
Step 4: Expressing Repeating Decimals
To express 0.333... more commonly, we recognize that it is equal to: [ \frac{1}{3} ]
This conclusion comes from the fact that if you multiply both sides by 3: [ 3 \times 0.333... = 0.999... ] And we know that (0.999... = 1). Hence: [ 0.333... = \frac{1}{3} ]
Key Insights
- The decimal .33333 can be converted to the fraction (\frac{33333}{100000}), which is already simplified due to the lack of common factors.
- The repeating decimal form 0.333... is equivalent to the fraction (\frac{1}{3}).
Table: Key Conversions and Their Simplifications
<table> <tr> <th>Decimal</th> <th>Fraction</th> <th>GCD</th> <th>Simplified Fraction</th> </tr> <tr> <td>0.33333</td> <td>33333/100000</td> <td>1</td> <td>33333/100000</td> </tr> <tr> <td>0.333...</td> <td>1/3</td> <td>1</td> <td>1/3</td> </tr> </table>
Conclusion
Understanding how to express decimals as fractions is an essential skill in mathematics. The decimal .33333 demonstrates how repeating decimals can be efficiently converted into fractional forms, reinforcing the relationship between different number representations.
So, the next time you encounter a decimal that seems complicated, remember the steps to convert it into a fraction, and you'll find it much easier to handle!