Final Answer:
0.2578364 is not a rational number. Similarly, 1.2222 is a repeating decimal that can be written as 11/9, and 3.5 can be written as the fraction 7/2. On the other hand, 0.2578364 cannot be expressed as a fraction of integers since it doesn't follow a repeating pattern or terminate. Hence, it falls into the category of irrational numbers.
Step-by-step explanation:
Rational numbers are those that can be expressed as fractions where the numerator and denominator are both integers. 2/3, 1.2222, and 3.5 can all be represented as fractions: 2/3, 11/9, and 7/2 respectively. However, 0.2578364 is a decimal number that does not terminate or repeat, making it an irrational number.
The number 2/3 is a simple fraction that equals 0.666... when expressed as a decimal. Similarly, 1.2222 is a repeating decimal that can be written as 11/9, and 3.5 can be written as the fraction 7/2. On the other hand, 0.2578364 cannot be expressed as a fraction of integers since it doesn't follow a repeating pattern or terminate. Hence, it falls into the category of irrational numbers.
Understanding the distinction between rational and irrational numbers is essential in mathematics as it helps classify numbers based on their properties and representation. Rational numbers have a definitive pattern in their decimal representation, while irrational numbers lack such patterns, making them more complex to express in fractional form.