The Number √2: An Irrational and Real Number

Understanding the square root of two, denoted as √2, is pivotal in both mathematics and its applications. This article explores whether √2 is indeed a number, particularly focusing on its status as an irrational and real number.

What is √2?

Yes, √2 is indeed a number. Specifically, it is an irrational number, which means it cannot be expressed as a simple fraction of two integers. The decimal representation of √2 is approximately 1.41421356 and it extends infinitely without repeating. This unique property makes √2 an interesting subject in mathematical discourse.

Geometric Significance

The square root of two has significant geometric meaning. It represents the length of the diagonal of a square with sides of length 1. Ancient Greeks were the first to recognize that not all numbers could be expressed as ratios of integers, marking √2 as one of the earliest irrational numbers discovered.

Between Natural Numbers

While natural numbers include integers starting from 1, there are no natural numbers between 1 and 2. This is evident from the sequence of squares of natural numbers:

12  1  √1  1 (principal roots only)22  4  √4  2

Here, √2 lies between √1 and √4. Since the principal root of a natural number forms a bijection among natural numbers, meaning no natural number has two principal roots and no natural numbers square to be the same number, √2 cannot be a natural number. Instead, it falls in the realm of irrational numbers.

Real Numbers

“Really” is not a mathematical concept; the term “real number” is a formal mathematical construction. According to definitions such as the extension of the rationals by Dedekind cuts, √2 is indeed a real number. Real numbers include all rational and irrational numbers, making √2 a legitimate member of the real number set.

Approximation and Irrationality

The value of √2 can be approximated through the Heron’s method. For instance:

approximately dfrac{3^2}{2^2 cdot 2}  dfrac{17}{12}  1.41 overline{6} or dfrac{17^2 - 12^2 cdot 2}{2 cdot 17 cdot 12}  dfrac{577}{408}  1.4142156863. The latter, when squared, results in a very close 2.0000060074.

These approximations illustrate why √2 is an irrational number, a conclusion reached by showing it cannot be expressed as a ratio of two integers. The proof is as follows:

Proof of Irrationality

Definitions: Even numbers: 2 times any integer. Odd numbers: Any even number plus one. If you square an odd number, it is also odd. If you square an even number, it is even. A rational number is a number that can be expressed as a ratio of two co-prime integers.

Assume √2 is Rational

Assume √2 can be expressed as (frac{a}{b}) where (a) and (b) are co-prime. (2 frac{a^2}{b^2}) implies (a^2 2b^2). Thus, (a 2c) for some integer (c), leading to (frac{a}{b} frac{2c}{b}). (2 frac{4c^2}{b^2}) implies (2b^2 4c^2) or (b^2 2c^2). Therefore, (b 2d) for some integer (d), leading to (frac{a}{b} frac{2c}{2d} frac{c}{d}). Since (a) and (b) were co-prime, (c) and (d) must also be co-prime. This is a contradiction, as (c) and (d) are co-prime but share a factor of two.

This contradiction proves that √2 cannot be expressed as a ratio of two integers, thus proving its irrationality. By extension, √2 is not a natural number, not an integer, and not a rational number, but an irrational number and a real number.

Understanding the nature of √2 is crucial for grasping the broader landscape of numbers and their properties. Whether you are a student, a teacher, or a mathematician, the study of √2 is an intriguing journey into the world of irrational and real numbers.