Understanding Rational Numbers: What is 1/3 and Why It's Not Transcendental

In the realm of mathematics, numbers are classified into various categories based on their properties. The number 1/3 is a prime example of a rational number, which has distinct differences from irrational and transcendental numbers. This article aims to explain what 1/3 is, why it's a rational number, and why it cannot be a transcendental number.

What is a Rational Number?

A rational number is any number that can be represented as a fraction of two integers. This includes all integers, fractions, terminating decimals, and repeating decimals. In the case of the fraction 1/3, it is a rational number because it can be expressed as the quotient of two integers, 1 and 3.

Expressing 1/3 as a Fraction

The number 1/3 is a simple yet important example of a rational number. It can be written as the fraction 1/3 or expressed as a decimal, which is a repeating decimal, 0.333… This repeating decimal indicates that the digit 3 repeats indefinitely, making it a rational number.

What is an Irrational Number?

An irrational number, on the other hand, is any number that cannot be expressed as a simple fraction. Examples of irrational numbers include the square root of 2 (denoted as sqrt{2}) and the mathematical constant pi (denoted as pi). These numbers have non-repeating, non-terminating decimal expansions, which means they cannot be written as a ratio of two integers.

What is a Transcendental Number?

A transcendental number is an irrational number that is not a root of any non-zero polynomial equation with rational coefficients. Among the many transcendental numbers, the most well-known is the number pi (pi). Pi is a transcendental number because it cannot be a root of any non-zero polynomial equation with rational coefficients.

Why 1/3 is Not a Transcendental Number

The number 1/3 is not a transcendental number because it is a root of a polynomial equation with rational coefficients. Specifically, 1/3 is a root of the linear polynomial equation:

3x - 1 0

This is the simplest form of a polynomial equation, where the coefficients are 3 and -1, both of which are integers. Since 1/3 is a root of this equation, it is not a transcendental number, as it satisfies the requirement that it is a root of a polynomial equation with integer coefficients.

Moreover, any rational number x (expressed as a fraction a/b) is not a transcendental number because it is a root of a linear polynomial equation bx - a 0. For example, for the rational number 1/3, the equation 3x - 1 0 demonstrates that 1/3 is indeed a root of this equation.

Infinitely Many Rational Numbers with No Finite Decimal Representation

Like 1/3, there are infinitely many other rational numbers that have no finite decimal representation. These numbers can all be written as fractions or expressed as terminating or repeating decimals. Examples include 2/3 (0.666…), 1/4 (0.25), and 1/11 (0.0909…).

Understanding the distinctions between rational, irrational, and transcendental numbers is crucial for a deeper comprehension of mathematics. While numbers like pi and e are fascinating for their transcendental nature, numbers like 1/3, while seemingly simple, are fundamental to many areas of mathematics and science.

Conclusion

In summary, the number 1/3 is a rational number—it can be expressed as a fraction and has a repeating decimal form. It is not a transcendental number because it is a root of a polynomial equation with integer coefficients. This distinction is important for understanding the categorization of numbers in mathematics.