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Square Root of 2 — Definition, Formula & Examples

The square root of 2 is the positive number that, when multiplied by itself, gives exactly 2. Its decimal expansion begins 1.41421356… and continues forever without repeating, making it an irrational number.

The principal square root of 2, denoted 2\sqrt{2}, is the unique positive real number xx satisfying x2=2x^2 = 2. It is algebraic of degree 2 over the rationals and was historically the first number proven to be irrational.

Key Formula

2≈1.41421356…\sqrt{2} \approx 1.41421356\ldots
Where:
  • 2\sqrt{2} = The positive number whose square equals 2

How It Works

You encounter 2\sqrt{2} whenever a square has side length 1 — its diagonal measures exactly 2\sqrt{2} units, by the Pythagorean theorem. Because 12=11^2 = 1 (too small) and 22=42^2 = 4 (too large), 2\sqrt{2} must lie somewhere between 1 and 2. No fraction ab\frac{a}{b} with integers aa and bb can equal 2\sqrt{2} exactly, which is what it means to be irrational. The classic proof of this fact uses contradiction: assume 2=ab\sqrt{2} = \frac{a}{b} in lowest terms, then show both aa and bb must be even, contradicting the lowest-terms assumption.

Worked Example

Problem: Find the length of the diagonal of a square with side length 5.
Apply the Pythagorean theorem: The diagonal of a square with side ss forms a right triangle with two sides of length ss.
d=s2+s2=s2d = \sqrt{s^2 + s^2} = s\sqrt{2}
Substitute s = 5: Plug in the side length and compute.
d=52≈5×1.4142=7.071d = 5\sqrt{2} \approx 5 \times 1.4142 = 7.071
Answer: The diagonal is 52≈7.0715\sqrt{2} \approx 7.071 units.

Why It Matters

The discovery that 2\sqrt{2} is irrational shocked ancient Greek mathematicians and reshaped the foundations of number theory. In geometry and trigonometry courses, 2\sqrt{2} appears constantly — in 45-45-90 triangles, rotation matrices, and the unit circle. Engineers and architects use it whenever computing diagonals of square cross-sections or designing structures with right-angle symmetry.

Common Mistakes

Mistake: Believing that 2=1.414\sqrt{2} = 1.414 exactly.
Correction: The value 1.414 is only an approximation. Since 2\sqrt{2} is irrational, its decimal expansion never terminates or repeats. In exact work, leave your answer as 2\sqrt{2} rather than rounding.