Square roots in the context of Square (algebra)

⭐ Core Definition: Square roots

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

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 ${\sqrt {x}},$ where the symbol " ${\sqrt {~^{~}}}$ " is called the radical sign or radix. For example, to express the fact that the principal square root of 9 is 3, we write ${\sqrt {9}}=3$ . 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 $x^{1/2}$ .

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Square roots in the context of Number line

A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely. The association between numbers and points on the line links arithmetical operations on numbers to geometric relations between points, and provides a conceptual framework for learning mathematics.

In elementary mathematics, the number line is initially used to teach addition and subtraction of integers, especially involving negative numbers. As students progress, more kinds of numbers can be placed on the line, including fractions, decimal fractions, square roots, and transcendental numbers such as the circle constant $π$ : Every point of the number line corresponds to a unique real number, and every real number to a unique point.

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Square roots in the context of Constructible number

In geometry and algebra, a real number $r$ is constructible if and only if, given a line segment of unit length, a line segment of length $|r|$ can be constructed with compass and straightedge in a finite number of steps. Equivalently, $r$ is constructible if and only if there is a closed-form expression for $r$ using only integers and the operations for addition, subtraction, multiplication, division, and square roots.

The geometric definition of constructible numbers motivates a corresponding definition of constructible points, which can again be described either geometrically or algebraically. A point is constructible if it can be produced as one of the points of a compass and straightedge construction (an endpoint of a line segment or crossing point of two lines or circles), starting from a given unit length segment. Alternatively and equivalently, taking the two endpoints of the given segment to be the points (0, 0) and (1, 0) of a Cartesian coordinate system, a point is constructible if and only if its Cartesian coordinates are both constructible numbers. Constructible numbers and points have also been called ruler and compass numbers and ruler and compass points, to distinguish them from numbers and points that may be constructed using other processes.

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Square roots in the context of Square (algebra)

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⭐ Core Definition: Square roots

Square roots in the context of Number line

Square roots in the context of Constructible number