Math 2.0

definition

Square root

Definition

Let xx be a real number with x≥0x \ge 0. The square root of xx, written x\sqrt{x}, is the unique real number yy with y≥0y \ge 0 and y2=xy^2 = x.

Remarks

The existence of square roots shows that exactly one such yy exists. The function x↦xx \mapsto \sqrt{x} on [0,∞)[0, \infty) is increasing, and ab=ab\sqrt{a b} = \sqrt{a} \sqrt{b} for a,b≥0a, b \ge 0. For each real xx, x2=∣x∣\sqrt{x^2} = |x|, the absolute value of xx.

Explanations

L1 human

The square root of xx is the side length of a square with area xx. The equation y2=25y^2 = 25 has two solutions, 55 and −5-5. The symbol 25\sqrt{25} means the positive one, 55.

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Real numbers, object, no evidence levelReal numbersobjectSet bounded above, definition, no evidence levelSet bounded abovedefinitionSupremum, definition, no evidence levelSupremumdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinition
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  1. Real numbers object
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  5. Existence of square roots of nonnegative reals statement
  6. Square root definition

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Sources

  1. Web https://en.wikipedia.org/wiki/Square_root, lead section
  2. Web https://en.wikipedia.org/wiki/Square_root, Properties and uses

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