Laws of Multiplication

Laws similar to those for addition also apply to multiplication. Special attention should be given to the multiplicative identity and inverse, M3 and M4.

M1. The product of any two real numbers a and b is again a real number, denoted a·b or ab.

M2. No matter how terms are grouped in carrying out multiplications, the product will always be the same: (ab)c = a(bc). This is called the associative law of multiplication.

M3. Given any real number a, there is a number one (1) called the multiplicative identity, such that a(1) = 1(a) = a.

M4. Given any nonzero real number a, there is a number (a^-1), or (1/a), called the multiplicative inverse, such that a(a^-1) = (a^-1)a = 1.

M5. No matter in what order multiplication is carried out, the product will always be the same: ab = ba. This is called the commutative law of multiplication.

Any set of elements obeying these five laws is said to be an Abelian, or commutative, group under multiplication. The set of all real numbers, excluding zero—because division by zero is inadmissible—forms such a commutative group under multiplication.

Laws of Addition

A1. The sum of any two real numbers a and b is again a real number, denoted a + b. The real numbers are closed under the operations of addition, subtraction, multiplication, division, and the extraction of roots; this means that applying any of these operations to real numbers yields a quantity that also is a real number.

A2. No matter how terms are grouped in carrying out additions, the sum will always be the same: (a + b) + c = a + (b + c). This is called the associative law of addition.

A3. Given any real number a, there is a real number zero (0) called the additive identity, such that a + 0 = 0 + a = a.

A4. Given any real number a, there is a number (-a), called the additive inverse of a, such that (a) + (-a) = 0.

A5. No matter in what order addition is carried out, the sum will always be the same: a + b = b + a. This is called the commutative law of addition.

Any set of numbers obeying laws A1 to A4 is said to form a group. If the set also obeys A5, it is said to be an Abelian, or commutative, group.

History of ALGEBRA

The history of algebra began in ancient Egypt and Babylonia, where people learned to solve linear (ax = b) and quadratic (ax² + bx = c) equations, as well as indeterminate equations such as x² + y² = z², whereby several unknowns are involved. The ancient Babylonians solved arbitrary quadratic equations by essentially the same procedures taught today. They could also solve some indeterminate equations.

The Alexandrian mathematicians Hero of Alexandria and Diophantus continued the traditions of Egypt and Babylon, but Diophantus' book Arithmetica is on a much higher level and gives many surprising solutions to difficult indeterminate equations. This ancient knowledge of solutions of equations in turn found a home early in the Islamic world, where it was known as the “science of restoration and balancing”. (The Arabic word for restoration, al-jabru, is the root of the word “algebra”.) In the 9th century, the Arab mathematician al-Khwarizmi wrote one of the first Arabic algebras, a systematic exposé of the basic theory of equations, with both examples and proofs. By the end of the 9th century, the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebra and solved such complicated problems as finding x, y, and z such that x + y + z = 10, x² + y² = z², and xz = y².

The Persian mathematician, astronomer, and poet Omar Khayyam showed how to express roots of cubic equations

Ancient civilizations wrote out algebraic expressions using only occasional abbreviations, but by medieval times Islamic mathematicians were able to talk about arbitrarily high powers of the unknown x, and, without yet using modern symbolism, work out the basic algebra of polynomials. This included the ability to multiply, divide, and find square roots of polynomials as well as a knowledge of the binomial theorem. The Persian mathematician, astronomer, and poet Omar Khayyam showed how to express roots of cubic equations by means of line segments obtained by intersecting conic sections, but he could not find a formula for the roots. A Latin translation of Al-Khwarizmi's Algebra appeared in the 12th century. In the early 13th century, the Italian mathematician Leonardo Fibonacci achieved a close approximation to the solution of the cubic equation x³ + 2x² + cx = d. Because Fibonacci had travelled in Islamic countries, he probably used an Arabic method of successive approximations.

Early in the 16th century, the Italian mathematicians Scipione del Ferro, Niccolò Tartaglia, and Gerolamo Cardano solved the general cubic equation in terms of the constants appearing in the equation. Tartaglia and Cardano's pupil, Ludovico Ferrari, soon found an exact solution to equations of the fourth degree, and as a result, mathematicians for the next several centuries tried to find a formula for the roots of equations of degree five, or higher. Early in the 19th century, however, the Norwegian mathematician Niels Abel and the French mathematician Évariste Galois proved that no such formula exists.

An important development in algebra in the 16th century was the introduction of symbols for the unknown and for algebraic powers and operations. As a result of this development, Book III of La Géometrie (1637), written by the French philosopher and mathematician René Descartes, looks much like a modern algebra text. Descartes's most significant contribution to mathematics, however, was his discovery of analytic geometry, which reduces the solution of geometric problems to the solution of algebraic ones. His geometry text also contained the essentials of a course on the theory of equations, including his so-called rule of signs for counting the number of what Descartes called the “true” (positive) and “false” (negative) roots of an equation. Work continued through the 18th century on the theory of equations, and in 1799 the German mathematician Carl Friedrich Gauss published the proof showing that every polynomial equation has at least one root in the complex plane (see Number: Complex Numbers).

By the time of Gauss, algebra had entered its modern phase. Attention shifted from solving polynomial equations to studying the structure of abstract mathematical systems whose axioms were based on the behaviour of mathematical objects, such as complex numbers, that mathematicians encountered when studying polynomial equations. Two examples of such systems are groups and quaternions, which share some of the properties of number systems but also depart from them in important ways. Groups began as systems of permutations and combinations of roots of polynomials, but went on to become one of the chief unifying concepts of 19th-century mathematics. Important contributions to their study were made by the French mathematicians Galois and Augustin Cauchy, the British mathematician Arthur Cayley, and the Norwegian mathematicians Niels Abel and Sophus Lie. Quaternions were discovered by British mathematician and astronomer William Rowan Hamilton, who extended the arithmetic of complex numbers to quaternions; while complex numbers are of the form a + bi, quaternions are of the form a + bi + cj + dk.

Immediately after Hamilton's discovery, the German mathematician Hermann Grassmann began investigating vectors. Despite its abstract character, American physicist J. W. Gibbs recognized in vector algebra a system of great utility for physicists, just as Hamilton had recognized the usefulness of quaternions. The widespread influence of this abstract approach led George Boole to write The Laws of Thought (1854), an algebraic treatment of basic logic. Since that time, modern algebra—also called abstract algebra—has continued to develop. Important new results have been discovered, and the subject has found applications in all branches of mathematics and in many of the sciences as well.

Introduction to ALGEBRA

Algebra, branch of mathematics in which symbols represent relationships. Classical algebra grew out of methods of solving equations; it represents numbers with symbols that combine according to the basic arithmetical operations of addition, subtraction, multiplication, division, and the extraction of roots. However, arithmetic cannot generalize mathematical relations such as Pythagoras' theorem, which states that in any right-angled triangle, the area of the square drawn on the hypotenuse is equal to the sum of the areas of the squares drawn on the other two sides. Arithmetic can produce only specific instances of these relations (for example, 3, 4, and 5, where 3² + 4² = 5²). Algebra, by contrast, can make a purely general statement that fulfils the conditions of the theorem: a² + b² = c². Any number multiplied by itself is termed squared, and is indicated by a superscript number 2. For example, 3 × 3 is notated 3²; similarly, a × a is equivalent to a².

Modern algebra has evolved from classical algebra by increasing its attention to the structures within mathematics. Mathematicians consider modern algebra to be a set of objects with rules for connecting or relating them. As such, in its most general form, algebra may fairly be described as the language of mathematics.