๐ Understanding Terminating Decimals
A terminating decimal is a decimal number that has digits that do not go on forever. In other words, it has a finite number of digits. Converting fractions to terminating decimals is easier than you might think! Here's how:
๐ A Brief History
The concept of terminating decimals has evolved alongside our understanding of fractions and decimal notation. Early mathematicians in ancient civilizations like Egypt and Mesopotamia dealt with fractions, but the formalization of decimal notation as we know it today came much later, largely through the work of mathematicians in the Islamic world and later in Europe during the Renaissance. Simon Stevin, a Flemish mathematician, played a key role in popularizing decimal fractions in Europe in the late 16th century.
๐ Key Principles for Conversion
- ๐ Prime Factorization: The key to determining whether a fraction can be written as a terminating decimal lies in the prime factorization of its denominator.
- ๐ข Denominator's Factors: If the denominator's prime factors are only 2s and 5s, then the fraction can be converted to a terminating decimal.
- ๐ Simplification: Always simplify the fraction first. For example, $\frac{2}{4}$ should be simplified to $\frac{1}{2}$.
- โ Division Method: Divide the numerator by the denominator. If the division results in a decimal that ends, it's a terminating decimal.
- ๐ Multiply to Obtain Power of 10: Convert a simplified fraction to a terminating decimal by multiplying the numerator and denominator by a factor that results in the denominator being a power of 10 (10, 100, 1000, etc.).
๐ก Real-World Examples
Let's look at some examples:
| Fraction |
Simplified |
Denominator's Factors |
Terminating Decimal? |
Conversion |
| $\frac{1}{2}$ |
$\frac{1}{2}$ |
2 |
Yes |
$\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10} = 0.5$ |
| $\frac{3}{4}$ |
$\frac{3}{4}$ |
2 x 2 |
Yes |
$\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 0.75$ |
| $\frac{5}{8}$ |
$\frac{5}{8}$ |
2 x 2 x 2 |
Yes |
$\frac{5}{8} = \frac{5 \times 125}{8 \times 125} = \frac{625}{1000} = 0.625$ |
| $\frac{1}{5}$ |
$\frac{1}{5}$ |
5 |
Yes |
$\frac{1}{5} = \frac{1 \times 2}{5 \times 2} = \frac{2}{10} = 0.2$ |
| $\frac{7}{20}$ |
$\frac{7}{20}$ |
2 x 2 x 5 |
Yes |
$\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 0.35$ |
| $\frac{1}{3}$ |
$\frac{1}{3}$ |
3 |
No |
0.333... (repeating) |
| $\frac{2}{25}$ |
$\frac{2}{25}$ |
5 x 5 |
Yes |
$\frac{2}{25} = \frac{2 \times 4}{25 \times 4} = \frac{8}{100} = 0.08$ |
โ
Conclusion
Converting fractions to terminating decimals becomes straightforward once you understand the role of the denominator's prime factors. If the simplified fraction's denominator only contains the prime factors 2 and/or 5, the fraction can be expressed as a terminating decimal. Otherwise, it will be a repeating decimal.