October 6, 2025

What Is A Transcendental Number

Q: Are all irrational numbers transcendental?

No. All transcendental numbers are irrational, but not all irrational numbers are transcendental. For example, √2 is irrational but is an algebraic number because it's a root of x² - 2 = 0.

Q: Why are transcendental numbers important?

They prove that there are numbers that cannot be solved by algebraic methods, showing the depth of the number system. This has implications for problems like 'squaring the circle' in geometry, which was proven impossible due to π's transcendental nature.

Q: Can you write a transcendental number as a fraction?

No. By definition, all transcendental numbers are irrational, meaning they cannot be expressed as a simple fraction (a/b) where 'a' and 'b' are integers and 'b' is not zero.

Q: How do mathematicians prove a number is transcendental?

Proving a number is transcendental is generally very complex, involving advanced mathematical techniques. It demonstrates that no possible polynomial equation with integer coefficients can have that number as a root.

Discover what a transcendental number is, how it differs from other irrational numbers, and why numbers like pi and e are famous examples in mathematics.

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Definition of a Transcendental Number

A transcendental number is a real or complex number that is not an algebraic number. This means it cannot be a root of any non-zero polynomial equation with integer coefficients.

Algebraic vs. Transcendental

An algebraic number is a root of a polynomial equation with integer coefficients, such as x² - 2 = 0 (where x = √2). Transcendental numbers, however, cannot satisfy such an equation, making them fundamentally 'non-algebraic'.

Famous Examples: Pi (π) and e

The most well-known transcendental numbers are pi (π ≈ 3.14159), which describes the ratio of a circle's circumference to its diameter, and Euler's number (e ≈ 2.71828), the base of the natural logarithm.

Significance in Mathematics

The existence of transcendental numbers, first rigorously proven for 'e' by Charles Hermite and for 'π' by Ferdinand von Lindemann, expanded our understanding of the number system, demonstrating its richness beyond algebraic roots and influencing areas like geometry and number theory.

Frequently Asked Questions

Are all irrational numbers transcendental?

Why are transcendental numbers important?

Can you write a transcendental number as a fraction?

How do mathematicians prove a number is transcendental?