Rational number of 0.89 in the simplest form.
The Correct Answer and Explanation is:
Sure!
Let’s first convert 0.89 into a rational number in its simplest form:
Step 1: Write 0.89 as a fraction
Since 0.89 has two decimal places, it can be written as: 0.89=891000.89 = \frac{89}{100}0.89=10089
Step 2: Simplify the fraction
Check if the numerator and denominator have any common factors.
The numerator is 89 and the denominator is 100.
- 89 is a prime number.
- 100 factors into 22×522^2 \times 5^222×52.
- 89 and 100 have no common factors except 1.
Thus, the fraction 89100\frac{89}{100}10089 is already in its simplest form.
Final answer: 0.89=891000.89 = \frac{89}{100}0.89=10089
Detailed Explanation
A rational number is any number that can be expressed as a fraction of two integers, where the denominator is not zero. The decimal 0.89 is a terminating decimal, which means that it can be converted into a fraction easily. To do this, observe that the decimal 0.89 has two digits after the decimal point. This means that 0.89 can be rewritten as 89 divided by 100. Thus, 0.89 is equal to the fraction 89100\frac{89}{100}10089.
The next step is to simplify this fraction, if possible. Simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the numerator is 89 and the denominator is 100. The number 89 is a prime number, which means its only divisors are 1 and itself. The number 100 is not divisible by 89, and they share no common factors other than 1. Therefore, the GCD of 89 and 100 is 1.
Since the GCD is 1, the fraction 89100\frac{89}{100}10089 is already in its simplest form. No further simplification is possible. The rational number representation of 0.89 in simplest form is therefore 89100\frac{89}{100}10089.
This process demonstrates that every terminating decimal can be expressed as a rational number. In this case, 0.89 is a rational number because it can be written as the fraction 89100\frac{89}{100}10089, fulfilling the definition of a rational number.
