What is "square number"

Square \Square\ (skw[^a]r), a.

1. (Geom.) Having four equal sides and four right angles; as, a square figure.

2. Forming a right angle; as, a square corner.

3. Having a shape broad for the height, with rectilineal and angular rather than curving outlines; as, a man of a square frame.

4. Exactly suitable or correspondent; true; just.

   She's a most triumphant lady, if report be square to her.
   --Shak.

5. Rendering equal justice; exact; fair; honest; as, square dealing.

6. Even; leaving no balance; as, to make or leave the accounts square.

7. Leaving nothing; hearty; vigorous.

   By Heaven, square eaters. More meat, I say.
   --Beau. & Fl.

8. (Naut.) At right angles with the mast or the keel, and parallel to the horizon; -- said of the yards of a square-rigged vessel when they are so braced.

   Note: Square is often used in self-explaining compounds or combinations, as in square-built, square-cornered, square-cut, square-nosed, etc.

   Square foot, an area equal to that of a square the sides of which are twelve inches; 144 square inches.

   Square knot, a knot in which the terminal and standing parts are parallel to each other; a reef knot. See Illust. under Knot.

   Square measure, the measure of a superficies or surface which depends on the length and breadth taken conjointly. The units of square measure are squares whose sides are the linear measures; as, square inches, square feet, square meters, etc.

   Square number. See Square, n., 6.

   Square root of a number or Square root of a quantity (Math.), that number or quantity which, multiplied by itself, produces the given number or quantity.

   Square sail (Naut.), a four-sided sail extended upon a yard suspended by the middle; sometimes, the foresail of a schooner set upon a yard; also, a cutter's or sloop's sail boomed out. See Illust. of Sail.

   Square stern (Naut.), a stern having a transom and joining the counter timbers at an angle, as distinguished from a round stern, which has no transom.

   Three-square, Five-square, etc., having three, five, etc., equal sides; as, a three-square file.

   To get square with, to get even with; to pay off. [Colloq.]

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be written as .

The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name square number comes from the name of the shape; see below.

Square numbers are non-negative. Another way of saying that a (non-negative) integer is a square number, is that its square root is again an integer. For example, = 3, so 9 is a square number.

A positive integer that has no perfect square divisors except 1 is called square-free.

For a non-negative integer , the th square number is , with being the zeroth one. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square, e.g., $\textstyle \frac{4}{9} = \left(\frac{2}{3}\right)^2$.

Starting with 1, there are square numbers up to and including , where the expression represents the floor of the number .