What is the square root of 7?

The square root of 7 (√7) 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: √7 ≈ 2.645751311... This is just an approximation; the actual value has infinitely many decimal places.

• Irrationality: As mentioned, it's irrational. This means it cannot be written as a fraction a/b where 'a' and 'b' are integers and 'b' is not zero. This can be proven using proof by contradiction, similar to the proof for the irrationality of √2.

• Continued Fraction Representation: √7 can be represented as a continued fraction: [2; 1, 1, 1, 4, 1, 1, 1, 4, ...] This means 2 + 1/(1 + 1/(1 + 1/(1 + 1/(4 + ...)))) . This representation is infinite and periodic, reflecting the irrational nature of the number.

• Geometric Interpretation: √7 represents the length of the hypotenuse of a right-angled triangle with legs of length 2 and √3 (or other Pythagorean triples that result in 7).

• Computational Methods: Approximations of √7 can be found using various numerical methods like the Babylonian method (also known as Heron's method), Newton-Raphson method, or using calculators or computer programs.

In summary, while we can't express √7 exactly as a decimal or fraction, we can approximate it and understand its properties as an irrational number with various mathematical representations.