Squares and square roots
Squares and square roots are fundamental mathematical operations that form the basis of numerous mathematical concepts and applications. The square of a number \( x \), represented as \( x^2 \), is calculated as \( x \times x \), while the square root of a number \( y \) is the number \( x \) such that \( y = x^2 \) and is denoted as \( x = \sqrt{y} \). Historically, the study of squares and square roots has contributed significantly to the development of various numerical systems and problem-solving methods. These operations are not just theoretical; they appear in practical applications, such as calculating standard deviations in statistics and are essential in fields like engineering and carpentry.
Geometrically, a square represents the area of a square with side length \( x \), which ties back to ancient mathematical practices, such as those seen in the Pythagorean theorem. The exploration of square roots led to the discovery of irrational numbers, notably \( \sqrt{2} \), which cannot be expressed as a ratio of integers. In modern mathematics, the concept of imaginary numbers, introduced by the mathematician Rafael Bombelli, expands the understanding of square roots to include negative numbers, leading to the development of complex numbers.
The significance of squares and square roots also extends into number theory, where perfect squares and their properties have intrigued mathematicians for centuries, influencing theories about representations of integers and the classification of quadratic forms. Overall, squares and square roots are not only foundational concepts in mathematics but also tools that facilitate a deeper understanding of numbers and their relationships.
Squares and square roots
Summary: Squares and square roots have long challenged mathematicians and have led to various expansions of the number system and developments in number theory.
The square of a number x, denoted x2, is the number x×x. The inverse operation is called the square root: the number x is a square root of y if y=x2, the notation used being x=√y. Historically, these operations have been a major source of new problems, ideas, and systems of numbers in the early and modern development of mathematics. Square roots have also appeared in many applications, such as computing the standard deviation of a data set, and have often presented a challenge to scientists and mathematicians in the days before readily available calculating technology. Middle-grade students in the twenty-first century continue to use squares and square roots to simplify computations and solve problems, as do carpenters and engineers.
Definition
Geometrically, the “square” of a number x measures the area of a square whose side has length x. This idea explains the name and is likely the way that ancient civilizations were first confronted with the operation. The Pythagorean theorem is an equality between sums of areas of squares constructed on the sides of a right triangles, namely a2+b2=c2 if a and b are the two legs and c is the hypotenuse. Applied to the triangle obtained by halving a square of side length one along one of its diagonals, it shows that such a diagonal has length equal to √2.
A member of the Pythagorean School sometimes identified as Hippasus of Metapontum (c. fifth century b.c.e.) discovered that this number cannot be expressed as the ratio of two integers—it is irrational. The discovery was a sensation amid the Pythagorean School where it was preached that all numbers were rational and called for an extension of the number system.
In the centuries that followed, extensions of the number system would include all numbers expressible with an infinite number of decimal digits, so that each positive number has a square root (for example, √2=1.4142136…) and negative numbers, which can be multiplied according to the usual associative rules, and the following additional ones governing signs: -1×x=-x; (-1)×(-1)=1, which implies that (-1)2=1 so that -1should also be counted as a square root of 1.
More generally, both extensions can be combined to yield the system of real numbers, which are the numbers with sign and infinite decimal expansions. In this system, each square of a number is a positive number (or zero), and each positive number has exactly two square roots, which differ by a sign. For example, 2 has as square roots the numbers 1.4142136…and -1.4142136…, a fact denoted by the expression √2=±1.4142136… .
Computation
Square roots can be computed by hand, by calculator, or by computer (up to the desired numerical approximation) by several methods, including those using sequences, exponentials, logarithms, or continued fractions. Mathematicians in ancient Egypt and Babylonia are some of the first who are thought to have extracted square roots. Early Chinese, Indian, and Greek mathematicians also contributed to this area. According to some historians, the first method to be introduced in Europe was that of Aryabhata the Elder, a Hindu mathematician and astronomer. One of the oldest ones, still at the basis of many currently used algorithms, is the so called Babylonian method (which is also an instance of the modern Newton–Raphson method for solving general equations in one variable). Given a positive number S and choosing an initial “guess” x0, the method produces a sequence of numbers xn converging to the square root of S by the rule
For example, the first approximations to √2 starting from x0=1 are x1=15, x2=1.416…, x3=1.414215…, x4=1.4142135623746…, the last one already having 11 correct decimal digits.
Solving the Quadratic Equation
Square roots are used to solve the general quadratic equationax2+bx+c=0, where a, b, and c are parameters, and a is not zero. The formula, at least partially known to the ancient Greek, Babylonian, Chinese, and Indian mathematicians, is
provided that the so-called discriminant of the equation, the number b2-4ac, is not negative.
Imaginary Numbers
The Italian mathematician Rafael Bombelli, in his book L’Algebra written in 1569, proposed the introduction of a new number i, which should denote the square root of -1. Multiplying the number i by real numbers would yield square roots of negative real numbers. The new numbers so obtained are called “imaginary numbers,” a name introduced by René Descartes (who meant it to bear a derogatory connotation). A new number system is obtained with the numbers formed by adding a real and an imaginary number; such numbers are called “complex numbers.” Complex numbers can be added, multiplied, and divided, and the preceding quadratic formula shows that any quadratic equation has two solutions that are complex numbers; this remains true even if the parameters a, b, c are allowed to be complex number themselves. Actually, a stronger result holds true: Carl Friedrich Gauss (1777–1855) discovered that any equation of the form a0xd+a1xd-1+•••+ad-1x+ad has d solutions in complex numbers, an important theorem known as the fundamental theorem of algebra. Partly thanks to this property, complex numbers are of fundamental importance in modern mathematics and in many fields of science and engineering, such as telecommunications.
Implications in Number Theory
Questions regarding squares, square roots, and quadratic forms have played a particularly important role in number theory, often giving rise to the simplest instances of rich theories. Numbers that are squares of integers are called “perfect squares,” the first examples being 1, 4, 9, 16, 25,… . Galileo Galilei examined perfect squares in the attempt to understand infinity. Leonardo Fibonacci wrote a number theory book called Liber Qudratorum, the book of squares.
The problem of representing integers as sums of perfect squares has also received much attention. Pierre de Fermat (c. 1607–1665) proved that the odd prime numbers that are sums of two perfect squares are exactly those that have remainder 1 when divided by 4, an example being 13=22+32 (whereas, for example, the prime number 7 has no such representation). Joseph Louis Lagrange (1736–1813) proved that every positive integer can be written as the sum of at most four perfect squares (for example, 15=9+4+1+1);three squares suffice only for those numbers which are not of the form 4k(8m+7), as was later proved by Adrien-Marie Legendre.
In his 1801 masterpiece Disquisitiones Arithmeticae, written at the age of 21, Gauss investigated two problems whose generalizations are still major topics of current research. The first one is related to the question of representing integers as the sum of squares and asks for a classification of binary quadratic forms, which are functions of two variables x and y of the shape f(x,y)=ax2+2bxy+cy2, where a, b, and c are integer parameters, in terms of the set of integers they represent—the set of possible values of f(x,y) as x and y range among the integers. The second problem considered by Gauss is the following: given two odd prime numbers p and q, is it possible to write p as the difference of a perfect square and a multiple of q (in symbols p=n2-mq)? Conversely, is it possible to write q as the difference of a perfect square and a multiple of p? Gauss proved that if at least one of p, q leaves remainder 1 when divided by four, then the two questions have the same answer; and that if p and q both leave remainder 3 when divided by 4, then the answer to the second question is “no” whenever the answer to the first question is “yes” and vice versa.
As a consequence of this result (known as the “quadratic reciprocity law”) he was able to give an efficient method for answering the question. In fact, Gauss found not one but eight different proofs of this fact, which is so central in modern number theory that about 200 more proofs were later found.
Bibliography
Conway, John H., and Francis Y. C. Fung. The Sensual (Quadratic) Form. Washington, DC: Mathematical Association of America, 1991.
Mazur, Barry. Imagining Numbers: (Particularly the Square Root of Minus Fifteen). New York: Farrar, Straus and Giroux, 2003.
Nahin, Paul J. An Imaginary Tale: The Story of i. Princeton NJ: Princeton University Press, 2010.