0.1 Repeating as a Rational Number
In mathematics, a rational number is a number that can be expressed as the ratio of two integers, i.e., a fraction. One might wonder, can a repeating decimal like 0.1 be expressed as a rational number? The answer is yes.
The Repeating Decimal 0.1
The decimal 0.1 is a repeating decimal, where the digit 1 repeats indefinitely. This type of decimal is also known as a periodic decimal. The repeating pattern in this case is the single digit 1.
Expressing 0.1 as a Fraction
To express 0.1 as a rational number, we need to find a fraction that is equal to 0.1. Let's call this fraction a/b, where a and b are integers.
We can set up an equation based on the fact that 0.1 is equal to the fraction a/b:
0.1 = a/b
To get rid of the decimal, we can multiply both sides of the equation by 10:
1 = 10a/b
Now, we can see that b must be equal to 10, since the right-hand side of the equation is an integer. Therefore, we have:
a = 1
So, the fraction that is equal to 0.1 is:
0.1 = 1/10
Rationality of 0.1
We have shown that 0.1 can be expressed as a fraction, 1/10, which means it is a rational number. This is not surprising, since all repeating decimals can be expressed as rational numbers.
In fact, a repeating decimal can be converted to a fraction by multiplying both sides of the equation by a power of 10 that is equal to the number of repeating digits. In this case, we multiplied by 10, since there is only one repeating digit.
Conclusion
In conclusion, the repeating decimal 0.1 can be expressed as a rational number, specifically the fraction 1/10. This demonstrates that all repeating decimals can be expressed as rational numbers, and reinforces the idea that rational numbers are a fundamental part of mathematics.
