When dealing with numbers, we often encounter two main categories: rational and irrational numbers. Rational numbers are those that can be expressed as a fraction of two integers, while irrational numbers cannot. The distinction between rational and irrational numbers is essential in understanding the behavior and properties of numbers.

In this article, we will specifically focus on the number 0.87 and determine whether it is rational or irrational. We will explore the definitions and properties of rational and irrational numbers, apply mathematical concepts, and provide a clear explanation to answer the question: is 0.87 rational or irrational?

Types of Numbers

### Rational Numbers

Rational numbers are numbers that can be expressed as a quotient of two integers (whole numbers), expressed in the form a/b, where a and b are integers and b is not equal to zero. Rational numbers include integers (whole numbers), fractions (such as 1/2 or -3/4), and terminating decimals (decimals that end after a finite number of digits, such as 0.5 or -0.125).

The set of rational numbers is denoted by Q. Rational numbers are dense on the number line, meaning that between any two rational numbers, there exists another rational number. They are also closed under the operations of addition, subtraction, multiplication, and division (except division by zero).

### Irrational Numbers

Irrational numbers, on the other hand, are numbers that cannot be expressed as a fraction of two integers. They are non-terminating, non-repeating decimals that go on forever. Examples of irrational numbers include π (pi), the square root of 2, and the golden ratio (φ). Irrational numbers are not dense on the number line, meaning that there are gaps between irrational numbers where no other rational or irrational numbers exist.

The set of irrational numbers is denoted by Q’. Irrational numbers are important in mathematics and science, and they are used to represent quantities that cannot be expressed exactly as a fraction, such as the ratio of a circle’s circumference to its diameter.

Determining Rationality

To determine whether a number is rational or irrational, we can use various mathematical tests. One common test is to check if the number can be expressed as a fraction of two integers. If it can, then the number is rational. If it cannot, then the number is irrational.

Another test for rationality is to examine the decimal representation of the number. If the decimal representation terminates or repeats after a finite number of digits, then the number is rational. If the decimal representation does not terminate or repeat, then the number is irrational.

Is 0.87 Rational or Irrational?

Now, let’s apply these tests to the number 0.87.

### Decimal Representation

0.87 is a terminating decimal, as it ends after two digits. According to our test, a terminating decimal is rational.

### Fraction Representation

We can also express 0.87 as a fraction by dividing the numerator (87) by the denominator (100):

“`
0.87 = 87/100
“`

87 and 100 are both integers, and 100 is not equal to zero. Therefore, 0.87 can be expressed as a fraction of two integers, making it a rational number.

Conclusion

Based on the tests and analysis presented, we can conclude that the number 0.87 is rational. It can be expressed as a fraction of two integers (87/100) and has a terminating decimal representation (0.87). Therefore, 0.87 belongs to the set of rational numbers, denoted by Q.

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