Square (algebra) in the context of "Hypotenuse"

In geometry, a square is a regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal sides. As with all rectangles, a square's angles are right angles (90 degrees, or π/2 radians), making adjacent sides perpendicular. The area of a square is the side length multiplied by itself, and so in algebra, multiplying a number by itself is called squaring.

Equal squares can tile the plane edge-to-edge in the square tiling. Square tilings are ubiquitous in tiled floors and walls, graph paper, image pixels, and game boards. Square shapes are also often seen in building floor plans, origami paper, food servings, in graphic design and heraldry, and in instant photos and fine art.

In physical science, an inverse-square law is any scientific law stating that the observed "intensity" of a specified physical quantity (being nothing more than the value of the physical quantity) is inversely proportional to the square of the distance from the source of that physical quantity. The fundamental cause for this can be understood as geometric dilution corresponding to point-source radiation into three-dimensional space.

Radar energy expands during both the signal transmission and the reflected return, so the inverse square for both paths means that the radar will receive energy according to the inverse fourth power of the range.

In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as$${\displaystyle ax^{2}+bx+c=0\,,}$$where the variable ⁠${\displaystyle x}$⁠ represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equation is linear, not quadratic.) The numbers a, b, and c are the coefficients of the equation and may be distinguished by respectively calling them, the quadratic coefficient, the linear coefficient and the constant coefficient or free term.

The values of ⁠${\displaystyle x}$⁠ that satisfy the equation are called solutions of the equation, and roots or zeros of the quadratic function on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a real double root, or two complex solutions that are complex conjugates of each other. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation $${\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0}$$where r and s are the solutions for ⁠${\displaystyle x}$⁠.

An imaginary number is the product of a real number and the imaginary unit i, which is defined by its property i = −1. The square of an imaginary number bi is −b. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary.

Originally coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler in the 18th century, and Augustin-Louis Cauchy and Carl Friedrich Gauss in the early 19th century.

In mathematics, a square root of a number x is a number y such that ${\displaystyle y^{2}=x}$; in other words, a number y whose square (the result of multiplying the number by itself, or ${\displaystyle y\cdot y}$) is x. For example, 4 and −4 are square roots of 16 because ${\displaystyle 4^{2}=(-4)^{2}=16}$.

Every nonnegative real number x has a unique nonnegative square root, called the principal square root or simply the square root (with a definite article, see below), which is denoted by ${\displaystyle {\sqrt {x}},}$ where the symbol "${\displaystyle {\sqrt {~^{~}}}}$" is called the radical sign or radix. For example, to express the fact that the principal square root of 9 is 3, we write ${\displaystyle {\sqrt {9}}=3}$. The term (or number) whose square root is being considered is known as the radicand. The radicand is the number or expression underneath the radical sign, in this case, 9. For non-negative x, the principal square root can also be written in exponent notation, as ${\displaystyle x^{1/2}}$.