What is the square root of 5?

The square root of 5 (√5) is an irrational number, meaning it cannot be expressed as a simple fraction and its decimal representation goes on forever without repeating.

Here's what we know about it:

• Approximate Value: √5 ≈ 2.236067977... This is only an approximation; the decimal representation continues infinitely.

• Continued Fraction: √5 has a simple continued fraction representation: [2; 4, 4, 4, 4, ...] This means √5 = 2 + 1/(4 + 1/(4 + 1/(4 + ...))) . This representation is useful for finding rational approximations.

• Geometric Significance: The length of the diagonal of a rectangle with sides of length 1 and 2 is √5 (by the Pythagorean theorem). This relates it to the golden ratio (φ), as √5 appears in the formula for φ: φ = (1 + √5)/2

• Algebraic Properties: √5 is a root of the quadratic equation x² - 5 = 0. It's also part of various algebraic expressions and identities.

• Irrationality: It's relatively easy to prove that √5 is irrational using proof by contradiction (similar to the proof for the irrationality of √2).

In short, √5 is a fundamental mathematical constant with connections to geometry, algebra, and number theory, despite its seemingly simple definition.