All About Numbers
Square Roots – Undoing a Square
A square root undoes squaring: √n asks “what number, multiplied by itself, gives n?” Since 6 × 6 = 36, √36 = 6. Square roots are the reverse operation of the square numbers covered earlier in this section – every square number has a clean, whole-number square root, which is exactly why recognising them (the topic of the Perfect Squares page later in this section) is such a useful shortcut.
The radical symbol √ itself first appeared in print in 1525, in the German mathematician Christoff Rudolff's algebra textbook Coss. Historians believe the symbol likely evolved from a stylised lowercase letter “r”, standing for the Latin word radix, meaning “root” – the same root that gives us the modern English word “radical.” The little horizontal bar (called the vinculum) that extends over the number underneath was added later, to make clear exactly which part of an expression the root applied to.
Evaluating Square Roots
√n is the non-negative number that, multiplied by itself, equals n.
Geometrically, √25 answers the question: “what side length gives a square with an area of 25?” Fill a square with 25 unit squares, and you'll find it takes exactly 5 of them along each side.
So √25 = 5, because a square with side length 5 has area 5² = 25. Square roots and square numbers are exact opposites: squaring goes from a side length to an area, while a square root goes from an area back to the side length that produced it.
9 × 9 = 81, so √81 = 9.
Solving Equations with Square Roots
Take the square root of both sides: x = √144 = 12.
Real-Life Application
- The quadratic formula: solving quadratic equations uses square roots directly.
- The Pythagorean theorem: finding a triangle's side length requires a square root.
- Statistics: standard deviation is calculated as a square root.
Key Takeaways
- √n asks what number, squared, gives n.
- The √ symbol first appeared in print in 1525, likely derived from the letter “r” for radix.
- Cube roots, covered next, extend this same idea to undoing a cube instead of a square.