The Handy Math Answer Book, Second Edition (2012)

ALGEBRA

THE BASICS OF ALGEBRA

What is the origin of the word “algebra”?

The word “algebra” comes from the title of the book Al jabr w’al muqābalah, by Persian mathematician Muhammad ibn Musa al-Khuwarizmi (783-c. 850; also seen as al-Khowarizmi or al-Khwarizmi). The book is roughly translated as Transposition and Reduction, in which he explained the basics of algebraic methods. (For more information about the history of algebra, see “History of Mathematics.”)

What early mathematicians are thought to be responsible for originating the use of algebraic methods and ideas?

To some scholars, Greek (Hellenic) mathematician Diophantus (c. 210-c. 290) is considered the “father of algebra,” as he developed his own algebraic notation. His words were noted and preserved by the Arabs; the translation of his words into Latin in the 16th century led to many algebraic advances. In more “modern” times, French mathematician François Viète (1540-1603; also known by his Latin name, Franciscus Vieta) is often credited as the “founder of modern algebra.” (For more information about Diophantus and Viète, see below.)

What is algebra?

Depending on whether one is a student or a professional mathematician, the word can mean either of the following (both of which are further described elsewhere in this chapter). School algebra is what mathematicians refer to as the algebra we learn in middle and high school and call “arithmetic.” But for most of people, algebra means solving polynomial equations with one or more variables; the solutions to such equations are often obtained by the operations of addition, subtraction, multiplication, division, raising to a power, and extracting a root. (For more information about all these operations, see “Math Basics.”). This also includes determining the properties of functions and graphs.

But mathematicians use the word “algebra” most often in reference to the abstract study of number systems and operations within them, such as groups, rings, and invariant theory. This is called abstract algebra.

How else is the word algebra used?

Algebra may be defined as the subjects of arithmetic and abstract algebra, but there are other meanings. These algebras involve vectors and matrices, the algebra of real numbers, complex numbers, and quaternions (an operator or factor that changes one vector into another). There are also those exotic algebras “invented” by mathematicians—usually named after the inventor—with the majority not truly understood except, perhaps, by their creators.

ALGEBRA EXPLAINED

What is an expression in mathematics?

An expression in mathematics is a statement that uses either numbers, variables, or both. For example, the following are all mathematical expressions:

y

4

6 - 4

5 × x - 7

4 + 5 (3 - 2)

x + 4 (7 - x)

In order to write an expression from a written mathematical problem, one has to interpret the text. For example, one person weighs 100 pounds and another weighs y pounds; the expression for their combined weights would be 100 + y.

What are equations?

In its simplest form, an equation is represented by expressions written with an equal sign in between; the two entities on either side are equal to each other. They are among the simplest mathematical problems most people deal with; in fact, most people have solved equations in their daily lives without realizing it. For example, when students first learned addition in school, they typically work on equations such as: ____ + 5 = 7, in which the blank needs to be filled. This problem could also be expressed as x + 5 = 7, a simple equation. In this case, the equation is solved when x equals 2. The following are also equations:

6 = 6

x = 8

y + 8 = 14

x - 4 - 15 - x

5xy - 8xy² + 4

There are also some fundamental properties of equations that are good to know. They include symmetric properties (if a = b, then b = a); substitution (if a = b, then a may be replaced by b); addition (if a = b, and c is a number, then a + c = b + c); and multiplication (if a = b, and c is a number, then a × c = b × c).

What are algebraic equations?

An algebraic equation, as with the equations defined above, is a statement in which two numbers, letters, or expressions are equal. But algebraic equations take the idea even further: Most of the time, you are trying to simplify the numbers and one (or more) variables in the equation. They can further be defined as any combination of variables or constants linked together by any operation—addition, subtraction, multiplication, division (except division by zero). This type of algebraic equation is often referred to as a polynomial (see elsewhere in this chapter for more about polynomials).

Who was the first to write and solve general algebraic equations?

French mathematician François Viète (1540-1603; also known as Franciscus Vieta, his Latin name) is often referred to as the “founder of modern algebra.” He was not a professional mathematician, but contributed a great deal to the understanding and spread of modern symbolic algebra. Although some of his work paid tribute to ancient mathematical traditions, Viète created a kind of “new math.” It was not one based on the traditional geometric visualizations, but rather expressed as abstract formulas and general rules. But Viète still divided algebra into distinct branches partially derived from Greek mathematics: zetetics (translating a problem into an equation), poristics (proving theorems through equations), and exegetics (solving equations). He also was the first to combine algebra with geometry and trigonometry. (For more about Viète, see “History of Mathematics” and “Geometry and Trigonometry.”)

What are variables in algebraic equations?

Variables are the symbols (usually a letter such as x or y) used in algebraic equations that represent an unknown number—and on whose value a function, polynomial, and so on, depends. Variables remain unknown until the equation is solved; thus, they are sometimes referred to as unknowns in an algebraic equation.

You can use algebra to solve word problems by converting known and unknown data into figures and variables. For instance, a word problem might involve two trains, one going 120 mph and one traveling at 112 mph. If the first train is 250 miles from a destination and the second train is 220 miles from the same destination, which train will arrive first?

It is not always easy to work with variables, as there are so many letters used throughout various equations. But in many mathematical and scientific texts, there are some variables that are customary to use. They are listed as follows:

n indicates natural numbers or integers

x represents real numbers

z stands for complex numbers.

What are some other terms used in dealing with algebraic equations?

There are many terms in algebra, including those dealing with equations. The following lists some of the more common ones:

Equality and inequality—An equality is a mathematical statement that shows the equivalence of two quantities. For example, if a is equal to b, it is written as the equality a = b. An inequality is just the opposite: a does not equal b, or a ≠ b.

Formula—A formula is a rule, fact, or principle expressed in terms of mathematical symbols, including equations, equalities, identities, or inequalities. (Note: the plural of formula in Latin is “formulae,” but it is more readily accepted today as “formulas.”)

Identity—An identity is a mathematical relationship equating one quantity to another that initially may appear to differ; it also means an equation that is always true, such as the Pythagorean theorem (for more about identities, see below).

How are word problems expressed as equations?

There seem to be a gazillion word problems out there—just ask anyone taking such tests in grade or high school! The one thing most have in common is that they can be expressed as an equation—many as algebraic equations—in which there are known and unknown quantities. In almost all instances, there are key words that lead the reader to determine not only the numbers and variables, but what operations to use in the equation to determine the answer. The following lists some simple key words and the corresponding operation:

Common Key Words Used in Word Problems

What are independent and dependent variables in algebra?

Variables can be broken down into independent and dependent variables. An independent variable is a quantity that increases or decreases (is variable), or that has an infinite number of values in the same expression. For example, in the expression x² + y² = r², x and y are variables. A dependent variable is a quantity that varies, but is produced by changes in the independent variable. In other words, the dependent variable’s value is dependent on the independent variable. For example, in the expression f(x) = y, x is the independent variable and y is a dependent variable (because y is dependent on the value of x).

Are there differences between independent and dependent variables in mathematics and statistics?

Yes, there are subtle differences between these two types of variables in mathematics and statistics. In mathematics, independent variables are those whose value determines the value of other variables; in statistics, they are a manipulated variable in an experiment or study whose presence or degree determines the change in the dependent variable.

François Viète was the first to write and solve general algebraic equations by introducing the systematic use of letters as algebraic symbols.

Dependent variables in mathematics are those variables whose value is determined by an independent variable; in statistics, they are the observed variables in an experiment or study whose changes are determined by the presence or degree of one or more independent variables. (For more information about variables in statistics, see “Applied Mathematics.”)

What is a solution?

When an equation has a variable, the number that replaces the unknown and makes the equation true is called a solution. For example, for the equation (5 × y) + 2 = 12, y would be 2, or “2” is the solution that makes the equation true. Working it out would be: (5 × 2) + 2 = 12; 10 + 2 = 12; and finally, 12 = 12. Remember, when finding a solution for an equation, one must follow any parentheses, exponents, multiplications, divisions, additions, and subtractions in the correct order of operation.

How do you simplify an algebraic equation?

The best way to simplify an equation is to combine like terms, which makes the equation simpler to solve. Numbers may be combined, as well as any terms with the same variable.

Terms can be combined either by adding or subtracting variables of the same kind. One can also use multiplication and division to simplify an equation by multiplying or dividing each side by the same number (except 0). The following are some examples:

· Add the like terms: 4x + 3x = 14, simplifies to 7x = 14.

· The equation 4 + 8x + 10 - 4x - 2 = 20 can be simplified by combining the like terms, which gives the simplified result 12 + 4x = 20.

· If necessary, do a combination of addition, subtraction, multiplication, or division. For example, the equation 2x - 2 = 4x + 3, simplifies to 2x = 4x + 5 (by adding 2 to both sides); then subtract 2x from both sides, simplifying to 0 = 2x + 5. (Note: Since subtracting any number is the same as adding its negative, it is often more helpful to replace subtractions with additions of a negative number.) Finally, subtract 5 from both sides (or 2x = -5) divide both sides by 2, resulting in the solution of x = -5/2 (or -2.5).

· To simplify expressions raised to a power, certain rules should be followed. For example, for (x + 3)² - 4x, square x + 3, or (x + 3)(x + 3), first squaring the first term (x squared equals x²), then the second (3 squared equals 9), then multiply and add the inner and outer terms together (3x + 3x = 6x). By combining like terms, the entire equation results in the simplified expression x² + 6x + 9 - 4x, which finally equals x² + 2x + 9.

How did symbols for unknowns and knowns in algebraic equations develop?

In 1591, François Viète was the first to write and solve general algebraic equations by introducing the systematic use of letters as algebraic symbols. He used vowels (a, e, i, o, u) for the unknowns and consonants (the rest of the alphabet) for the coefficients (or knowns).

But it was René Descartes who introduced a new way of using letters in the alphabet in his work, La Gèometrie. He used the letters at the end of the alphabet (x, y, …) for unknowns and beginning of the alphabet (a, b, …) for knowns (in many instances, these letters are italicized). This standard is still in algebraic use today.

What is the order of operations when you simplify an algebraic equation?

There is a certain order to working out an algebraic equation. The following lists the operations in their correct order:

· First: do what is inside the parentheses

· Second: do the exponents

· Third: do all the multiplications and divisions from left to right

· Finally: do all the additions and subtractions from left to right

There are also rules when it comes to grouping in algebraic equations—especially when working with parentheses. When an expression has parentheses within parentheses, work from the inside out, removing the innermost parentheses first. And remember—if there is a positive sign in front of the parentheses, it does not change any sign inside the parentheses; if there is a negative sign in front of the parentheses, it will change all the signs within the parentheses.

What are some examples of algebraic equation solutions?

The following lists some simple solutions to selected algebraic equations:

· Solve for the equation 4x - 4 = 12

add 4 to each side: 4x = 16

divide both sides by 4: x = 16/4

solve for x: x = 4

· Solve for the equation (x + 3)² - 4x = (x—1)² + 3

expand each side by doing the operations in the parentheses: x² + 2x + 9 =

x² - 2x + 4

then add -x² to each side (another way of subtracting):

2x + 9 = -2x + 4

then add 2x to each side: 4x + 9 = 4

then add -9 to each side: 4x = -5

then multiply each side by 1/4 (divide by 4): x = -5/4

What are functions?

Functions are mathematical expressions describing the relationship between variables and involve only algebraic operations. If there is one independent variable, the dependent variable, for example y, can be determined through the function. This is often seen written as y = f(x), spoken as, “y equals fof x” (functions also use notation with x; for example, f(x) = 2x + 1 is a function). Functions within equations are also common; for example, in the equation -x² + y = 3, the y represents a function of x. To see this, note that this equation can also be written as y = 3 + x².

Functions do not always have to be in terms of f(x). They can just as easily be termed g(x), depending on the equation. But note: The equation x² + y² = 9 is not a function, as x and yare both independent variables.

Are there different types of functions?

Yes, there are different types of functions—so many, in fact, that the topic of “functions” is a book in itself. In particular, algebraic equations include polynomial and rational expression functions. For example, polynomial equations include linear (first degree) functions, such as f(x) = 2x; a quadratic (second degree) function example is f(x) = x² (for more about polynomials and degrees, see below).

But “algebraic and polynomial functions” are not the only use of the term “function”—so don’t get confused. There are also non-algebraic functions called exponential functions, and the inverses of exponential functions, called logarithmic functions. Set theory emphasizes the use of functions (for more about functions and sets, see “Foundations of Mathematics”); and there are trigonomic functions that include the relationships of sine, cosine, and tangent functions (for more information about trigonometry, see “Geometry and Trigonometry”). There also are continuous or discontinuous functions, transcendental functions, and even real and complex functions (all this may or may not be connected to algebra). The list goes on, and it is easy to see that mathematicians love the word “function.”

What is a coefficient?

In an algebraic equation, a coefficient is simply a multiplicative factor. In the majority of cases, the coefficient is the numerical part (most often a constant) of the equation. Thus, it is called a numerical coefficient. For example, in 3x= 6, the coefficient is 3; in -3x = 6, the coefficient is -3, as the coefficient takes on the sign of the operation. Terms such as xy may not appear to have a numerical coefficient, but it is 1—a number that is not written, but assumed.

Coefficients do not have to be just numbers: In the equation 5x³y, the coefficient of x³y is 5. But in addition, the coefficient of x is 5x²y, and the coefficient of y is 5x³. Coefficients are also seen in functions; for example, in the function f(x) = 2x, the 2 is the coefficient.

A 1621 Latin edition of Greek mathematician Diophan-tus’s Arithmetica.

What is a diophantine equation?

The first mention of diophantine equations was by Greek (Hellenic) mathematician Diophantus (c. 210-c. 290 C.E.). In his treatise Arithmetica, he solved equations with several variables for integral solutions—or what we call diophantine equations today. (For more about Dio-phantus in history, see “History of Mathematics.”) These are represented by one equation with at least two variables, such as x and y, and whose solutions have to be whole numbers (or integers). These equations either have no solutions, or an infinite or finite number of solutions. Diophantine analysis is the mathematical term for how to determine integer solutions for such algebraic equations.

What is a linear equation?

As the term suggests, linear equations have to do with lines; and in algebra, a linear equation means certain equations (or functions) whose graph is a line (for an extensive explanation of graphs, see “Geometry and Trigonometry”). More specifically, in algebra, a linear equation is one that contains simply the variable, which makes them one of the simplest types of equations. For example, a linear equation in one variable has one unknown (the variable) represented by a letter; this letter, usually x, is always to the power of 1, meaning there is no x² or x³ in the equation.

For instance, x + 3 = 9 is a simple linear equation. To solve such an equation, one must either add, subtract, multiply, and/or divide both sides of the equation by numbers and variables—and do this in the correct order—to end up with a solution: a single variable and single number on opposite sides of the equals sign. In this case, the solution to the linear equation is x = 6.

Finally, linear equations can be further broken down. For example, in the linear equation ax + by + cz + dw = h, in which a, b, c, and d are known numbers and x, y, z, and w are unknown numbers, if h = 0, the linear equation is said to be homogeneous.

What is the absolute value of a number?

The absolute value of a real number is the number stripped of any negative value. Therefore, the absolute value of a number will always be greater than or equal to zero. (Formally, the absolute value is considered the distance of a number from zero on a number line.) The symbol for “absolute value” is the number inside two parallel vertical lines (| |). For example, the absolute value of x is given by |x|. If the number is negative within the absolute value sign, it will automatically become positive. In numerical form, |3| equals 3 and |-3| equals 3.

When discussing complex numbers, the absolute value often means squaring the numbers, then taking the square root of those numbers. For example, the common way of writing complex equations is z = a + bi; the absolute value of z becomes . For instance, if z = 3 - 4i, then

What is a system of equations?

A system of equations is any set of simultaneous equations—two or more—that are intertwined and have to be determined together. They are usually a finite set of equations with the same unknowns, all of which have common solutions. A set of linear equations is said to be a linear system, while a set of homogeneous linear equations is called a homogeneous linear system. The number of equations is finite, meaning the solutions don’t go on forever like some with billions of answers. But a word of caution: Some problems use systems with hundreds of equations and just about the same number of variables.

ALGEBRAIC OPERATIONS

What are the common types of mathematical operations?

Operations are ways to obtain a single entity from one or more entities by manipulating numbers (and letters) in certain ways. They include the elementary operations most of us are familiar with: addition, subtraction, multiplication, division, cubing, squaring, and integer root extraction. There are numerous types of other operations, including binary operations, in which two quantities or expressions x and y interact in set theory (for more about set theory, see “Math Basics”).

What is the algebraic concept of inverse?

“Inverses” in algebra are operations (or numbers) that “undo” each other. For example, if one multiplies 4 times its inverse, or 1/4, the solution is 1; thus, the rule for multiplicative inverse for x (with x ≠ 0) is 1/x, as in x(1/x) = 1. In addition, if you add -4 to 4, you get zero (0); thus, the additive inverse of x is -x, as in x + (-x) = 0.

What are identity and conditional equations?

Identity and conditional equations are ways in which numbers associate with each other. When an equation is true for every value of the variable, then the equation is called an identity equation. It is often denoted as I or E (the E is from the German Einheit, or “unity”). For example, 3x = 3x is an identity equation, because x will always be the same number. Zero is the identity element for addition, because any number added to 0 does not change the value of any of the other numbers in the operation (or x + 0 = x). The number 1 is the identity element of multiplication, as any number in an operation multiplied by 1 does not change the value of that number. Multiple identity is often written as x × 1 = x.

When an equation is false for at least one value, it is called a conditional equation. For example, 6x = 12 is conditional because it is false when x = 3 (and any number other than 2). In other words, if at least one value can be found in which the equation is false (or the right side is not equal to the left side) then the equation is called a conditional equation.

How do numbers associate with each other?

Generally in mathematics, there are certain properties of operations that determine how numbers associate with each other. Closure is a property of an operation that reveals how numbers associate with each other; in particular, when two whole numbers are added, their sum will be a whole number. Closure as a property of multiplication occurs when two whole numbers are multiplied and their resulting product is a whole number.

What does iteration mean?

As in most English texts, iteration means repeating, and this hold true for mathematics. In the case of numbers, iteration means a procedure in which the result is fed back and the procedure repeated; in some instances, it is repeated over and over. For example, to find the square root of 39, you can use iteration—or repeat a procedure to find the solution. Knowing that the solution must be close to 6 (or the square root of 36, a number close to 39), one can divide 39 by 6 (39/6) and get 6.5. Next, average 6 and 6.5 to get 6.25; then iterate again, dividing 39 by 6.25 (39/6.25) = 6.24 (the actual square root of 39 is 6.244997 …).

One of the most obvious places in which iterations take place is in your calculator or computer. For example, in order to get the square root of 39, as in the example above, a calculator (or computer) automatically uses iteration to calculate the answer to a certain decimal place. The more numbers in a procedure, the more iterations are needed, which is why supercomputing has become such a great asset not only to mathematics, but to many other sciences as well.

An associative property means that for a given operation that combines three quantities (two at a time), the initial pairing of the quantities is arbitrary. For example, when doing an addition operation, the numbers can be combined in two ways: (a + b) + c = a + (b + c). Thus, when adding the numbers 3, 4, and 5, this means that they may be combined as (3 + 4) + 5 = 12 or 3 + (4 + 5) = 12. Following the same logic for multiplication, the associative law states that (a × b) × c = a × (b × c). In fact, in an associative operation, the parentheses that indicate what quantities are to be first combined can be omitted; an example of the associative law for addition is 3 + 4 + 5 = 12, and for multiplication, 2 × 3 × 4 = 24. But not all operations are associative. One good example is division: You can’t divide in the same way as you added or multiplied above. For example, the result of dividing three numbers differs. The operation (96÷12) ÷ 4 = 2 is not the same as 96 ÷ 4 (12 ÷ 4) = 32.

Like the associative property, the commutative property is another way of looking at how numbers associate with each other in operations. In particular, this law holds that for a given operation that combines two quantities, the order of the quantities is arbitrary. For example, in addition, adding 4 + 5 can be written either as 4 + 5 = 9 or 5 + 4 = 9, or expressed as a + b = b + a. When working on a multiplication operation, the same rule applies, as in a × b = b × a. Again, not all operations are commutative. For example, subtraction is not, as 6 - 3 = 3 is not the same as 3 - 6 = -3. Division also is not commutative, as 6 ÷ 3 = 2 is not the same as 3 ÷ 6 = ½.

The final property of an operation is the distributive property. In this rule, for any two operations, the first is distributive over the second. For example, multiplication is distributive over addition; for any numbers a, b, and c, a × (b + c) = (a × b) + (a × c). For the numbers 2, 3, and 4, you would have 2 × (3 + 4) = 14 or (2 × 3 ) + (2 × 4) = 14. Formally, there is a right and left distribution—left is listed above; right is (a + b) × c = (a × c) + (b × c). In most cases, both are commonly referred to as distributivity. Again, not all operations are distributive. For example, addition is not distributive over multiplication, as in a + (b × c) (a + b) × (a + c).

What is the factorial of a number?

A factorial is the product of consecutive natural numbers for all integers greater than or equal to 0. Factorials usually start with 1; the symbol for factorial is an exclamation point (!). For example, 4 factorial (4!) is 1 × 2 × 3 × 4, or 24. In the numerical system, consecutive factorials are 1 (1! = 1), 2 (2! = 1 × 2), 6 (3! = 1 × 2 × 3), 24 (4! = 1 × 2 × 3 × 4), 120 (5! = 1 × 2 × 3 × 4 × 5), 720 (6! = 1 × 2 × 3 × 4 × 5 × 6), 5040 (7! = 1 × 2 × 3 × 4 × 5 × 6 × 7), 40,320 (8! = 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8), 362,880 (9! = 1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9), and so on.

There are two additional rules: The number 0 factorial (0!) = 1; and the factorial values for negative integers are not defined. Factorials are most often used in reference to counting numbers, statistics (especially in probability calculations), calculus, and physics.

EXPONENTS AND LOGARITHMS

What is an exponent in terms of algebra?

An exponent is actually raising a number to a certain power; this is written as a superscript to the right of a real number, such as 3⁴, expressed as “three raised to the fourth power,” or “three with an exponent of four.” (For more information on exponents, see “Math Basics.”) The exponent represents the number of times a number is being multiplied. The above example actually means “3 × 3 × 3 × 3,” which is equal to 81. The powers can be an integer (negative or positive numbers), real number, or even a complex number. This can also be thought of as taking the quantity b, the base number, to the power of another quantity often called e, the exponent. (In many computer-oriented texts, this is written as b ∧ e.)

Exponents are important to algebra as they are often included in most algebraic equations. The process of performing the operation of raising to a power is known as exponentiation. Exponents are also often associated with functions. For example, in the function f(x) = x², the 2 is the exponent.

What is a base in algebra?

The base is used in algebra in connection with powers. In fact, it is called the base of a power—or the number that is used as a factor a given number of times. In the example above, 3⁴, 3 is the base. The base can either be the number used with an exponent to create a power, such as the 3 in 3⁴; or a number written as a subscript, such as with a logarithm, for example, log_ax, in which a is the base number. (See below for more information about logarithms; for more information about bases, see “Math Basics”).

What are some simple rules of exponents?

There are several simple rules when it comes to exponents. These include the following:

· The equation x¹ = x (or a number raised to the 1 power is the number itself; this is also called the “rules of 1”).

· The equation x⁰ = 1 (unless x = 0, then this is referred to as undefined; this is also called the “zero rule”).

· A number without an exponent has an exponent of 1, as in 20 = 20¹.

· A negative exponent means to divide by that number of factors instead of multiplying. For example, 3^-3 is equal to 1/(3³). But there is a restriction to this rule: x^-n = 1/xⁿ only when x is not zero; if x is 0, then xⁿ is undefined.

What are some rules for combining exponents?

There are also rules for combining exponents (also called the laws of indices):

· To multiply exponents with identical bases, add the exponents, such as 3² × 3³ = 3⁵ (3 is the base).

· To multiply like exponents, combine terms, such as 10² × 2² = (10 × 2)² = 400.

· To divide identical bases, subtract the exponents, such as 103/10 = 10^3-1 = 10² = 100 (the denominator 10 has an assumed exponent of 1).

What are the connections between logarithms and algebra?

Logarithms are the numbers of the power to which a base must be raised in order to get a given positive number. For example, the logarithm of 100 to the base 10 is 2, or log₁₀100 = 2; this is because 10² = 100. Common logarithmsare positive numbers that use the number 10 as the base; they are written as log x. Those using the number symbolized by e as the base are called natural logarithms (also phrased as logarithms with a base e); the natural logarithm of a number x is written as ln x.

What’s the connection? Because logarithms are really exponents, they satisfy all the usual rules of exponents. Consequently, tedious and long algebraic calculations such as those involving multiplication and division can be replaced by the simpler processes of adding or subtracting the corresponding logarithms. In general, logarithmic tables are usually used for this purpose—although calculators, computers, and Internet mathematics and computational sites usually replace the need for such tables.

Even the simple act of cranking up your sound system involves math. The decibel scale used by your loudspeakers and amplifiers employs the concept of logarithms.

What was the progression of logarithm development?

The invention of logarithms was a long process, starting with Scottish mathematician John Napier (1550-1617; also known as Laird of Merchiston), who first came up with the idea of logarithms in 1594. But the actual invention and announcement of logarithms would take another 20 years: In 1614, Napier would publish Mirifici logarithmorum canonis descripto (Description of the Wonderful Canon of Logarithms), offering tables and rules for their use.

Not long afterward, in 1617, English mathematician Henry Briggs (1561–1630) published Logarithmorum chilias prima (Logarithms of Numbers from 1 to 1,000), introducing the concept of common logarithms—or logarithms based on the powers of ten. And finally, independently from Briggs and Napier came Swiss mathematician Joost Bürgi (1552–1632), who in 1620, presented Arithmetische und geometrische progress-tabulen, a German work presenting the discovery of logarithms.

The discoveries differed in several ways: Napier’s approach was algebraic; Bürgi’s was geometric. There were differences from the common and natural logarithms used today. And neither Napier nor Bürgi mentioned the concept of a logarithmic base— something that Briggs presented.

By 1624, Briggs would write Arithmetica logarithmica (The Arithmetic of Logarithms), extending his common log tables from 1 to 20,000 and from 90,000 to 100,000. But the work on logarithms did not end with Napier, Briggs, or Bürgi. Natural logarithms eventually evolved out of Napier’s original work. And defining logarithms as exponents was finally recognized by English mathematician John Wallis (1616–1703), who presented them in his 1685 publication De algebra tractatus (Treatise of Algebra).

For what other invention was John Napier known?

Scottish mathematician John Napier may have been known for his contributions toward logarithms, but was also the inventor of a tool called Napier’s Bones (also called Napier’s Rods). These were multiplication tables inscribed on strips of animal bone or wood. Wilhelm Schickardt would eventually build the first calculating machine based on Napier’s Bones, a device that could add, subtract, and—with help— multiply or divide.

Napier was also the “instigator” in another discovery: In 1621, English mathematician and clergyman William Oughtred (1575–1660) used Napier’s logarithms as the basis for the slide rule (a ruler-like instrument used long before hand-held calculators came in vogue). Oughtred not only invented the standard rectilinear slide rule, but also the circular slide rule, which was an extremely useful tool that remained in common usage for more than three hundred years. (For more about Oughtred, see “Math Basics.”)

What are the properties of logarithms?

Logarithms have certain properties depending on interpretations of an equation. The following lists some of the most common properties (these rules are the same for all positive bases):

· log_a 1 = 0, because a⁰ = 1. For example, in the equation 14⁰ = 1, the base is 14 and the exponent is 0. Because a logarithm is an exponent, this would mean the equation can be written as a logarithmic equation, or log₁₄ 1 = 0 (zero is the exponent).

· log_a a = 1, because a¹ = a. For example, in the equation 3¹ = 3, the base is 3 and the exponent is 1; the result is 3, with the corresponding logarithmic equation being log₃ 3 = 1.

· log_a a^x = x, because a^x = a^x. For example, 3⁴ = 3⁴, with the base as 3. The logarithmic equation becomes log₃ 3⁴ = 4.

What are some examples of logarithm use?

Logarithms are used in many areas of science and engineering, especially in those areas in which quantities vary over a large range. For example, the decibel scale for the loudness of sound and the astronomical scale of stellar brightness are both logarithmic scales.

In this example of an exponential function, when 2 is raised to the power of x, the line representing the function always lies above the x axis.

What is an exponential function?

Along with exponents come exponential functions, or the relationship between values of a variable and the numbers formed by raising some positive number to the power of those values. In functional notation, an exponential function is written f(x) = a^x, in which a is a positive number; for example, the function f(x) = 2^x is an exponential function.

In logarithmic terms, an exponential function is most commonly written as exp (x) or e^x, in which e is called the base of the natural logarithm. These types of functions are usually shown on a graph. (For more information about graphs, see “Geometry and Trigonometry.”) As a function of the real variable x, the resulting graph of e^x is always positive, or above the x axis and increasing from left to right. Although the line of such a function never touches the x axis, it gets very close to it.

What is e in terms of logarithms?

No, e is not the code name in a James Bond movie. When talking about logarithms (or logs), in the majority of mathematical circles, it means the base of the natural logarithm. It is yet another irrational, transcendental number (such as pi, or π) that has a plethora of names: It has been called everything from the logarithmic constant and Napier’s number to Euler’s constant and the natural logarithmic base. One of the best ways to define e is to use the expression (1 + x)^(1/x); e is the number that this expression approaches as x gets smaller and smaller. Substituting in values for x gives you the idea: if x = 1, the result is 2; if x = 0.5, the result is 2.25; when x = 0.25, the result is 2.4414…; if x = 0.125, the result is 2.56578 …; if x = 0.0625, the result is 2.63792…; and so on. This is why approximations are often used in solving equations using e.

What are the rules for combining logarithms?

There are certain rules for combining logarithms. In the following cases, let a be a positive number that does not equal 0; n be a real number; and u and v be positive real numbers:

Logarithmic Rule 1: log_a(uv) = log_a(u) + log_a(v)

Logarithmic Rule 2: log_a(u/v) = log_a(u) - log_a(v)

Logarithmic Rule 3: log_a(u)ⁿ = nlog_a(u)

This can be expressed as follows: In rule one, multiplication inside the log is turned into addition outside the log (and vice versa); in rule two, division inside the log is turned into subtraction outside the log (and vice versa); and in rule three, an exponent on anything inside the log can be moved to the front of the log as a multiplier (and vice versa). But remember, these rules only apply if the bases are the same. For example, because the bases are not the same in log_a(u) + log_b(v), this expression can’t be simplified.

How do you expand logarithms?

Like an algebraic expression, it is possible to expand logarithms, which is a way of “picking apart” an expression. The following lists two examples of expanding a logarithmic expression:

· To expand log₂(3x) = log₂(3) + log₂(x)

· To expand log₂(12/x) = log₂(12) - log₂(x)

Is it possible to simplify logarithms?

Yes, as in algebraic equations, it is possible to simplify logarithms, but in different ways. The following lists some examples:

· To simplify log₃(x) + log₃(y) = log₃(xy)

· To simplify log₃(6) - log₃(4) = log₃(6/4)

· To simplify 2log₃(x) = log₃(x²)

How can the base of logarithms be changed?

The base of logarithms can be changed from one that is not 10 or e to an equivalent logarithm with base 10 or e. The following gives the formula for such a transition, in which a, b, and x are real positive numbers (but neither a nor b are equal to 1, and x is greater than 0):

Convert log_a x to the base b by using the formula (log_bx)/(log_ba).

How are equations with exponents and logarithms solved?

The way to solve an exponential equation is relatively easy: Take the log of both sides of the equation, then solve for the variable. For example, to solve for x in the equation e^x = 60.

1. First, take the natural log (ln) of both sides:

ln(e^x) = ln(60)

2. Simplify using the logarithmic rule #3 (see above) for the left side:

x ln(e) = ln(60)

3. Then simplify again, since ln(e) = 1 to:

x= ln(60) = 4.094344562

4. And finally, check your answer (using log tables or your calculator) in the original equation e^x = 60:

e ^4.094344562 = 60 is definitely true.

The way to solve a logarithmic equation is equally easy: Just rewrite the equation in exponential form and solve for the variable. For example, to solve for x in the equation ln(x) = 11:

1. 1.First, change both sides so they are exponents of the base e:

e^ln(x) = e¹¹

2. When the bases of the exponent and logarithm are the same, the left part of the equation becomes x, thus, it can be written:

x = e¹¹

3. To obtain x, determine the solution for e¹¹, or

x is approximately 59,874.14172.

4. And finally, check your answer (using tables or your calculator) in the original equation ln(x) = 11:

ln(59,874.14172) = 11 is definitely true.

POLYNOMIAL EQUATIONS

What is a polynomial?

In its simplest form, a polynomial is a mathematical equation that involves a sum of powers in one or more variable, all multiplied by coefficients. In such equations, variables and numbers on both sides of the equal sign are considered polynomials. For example, by expanding out the expression (x - 2)³ (or multiplying out the equation) we discover that (x - 2)³ = x³ - 6x² + 12x - 8, which is a polynomial equation.

Are there other ways of describing polynomials?

Yes, there are other ways to describe polynomials. In particular, a polynomial with only one variable is called a univariate polynomial. A multivariate polynomial is one with more than one variable. There are other terms to further define polynomials, including the following:

Monomial—A monomial is a one-term polynomial (mono means “one”). For example, 3x is a monomial.

Binomial—A binomial is a two-term polynomial (bi meaning “two”). For example, 3x² - 10 is a binomial.

Trinomial—A trinomial is a three-term polynomial (tri means “three”). For example, 4x³ + 3x + 6 is a trinomial.

What is the degree of a polynomial equation?

The highest order power (or exponent) used in a univariate polynomial is called its polynomial degree or order; it is also defined as the largest (or maximum) sum of exponents that appear on the variables in the equation’s terms. Commonly, the word “degree” is used more frequently because “order” has another meaning when discussing polynomials.

You can also talk about degrees in reference to monomials, binomials, and trinomials. The degree of a monomial is the sum of the degrees of the variable in the equation; the degrees of the other polynomial equations are the greatest of the degrees in terms after the equation has been simplified. For example, a third degree trinomial equation is x³ - 3x - 2 = 0.

What are quartic equations?

Quartic equations are polynomial equations whose highest power of the unknown variable is four. Or put another way, quartic equations are algebraic equations whose highest exponent (or degree or order) is four. But take note: Quartic equations are not the same as quadratic equations—or second degree equations in one variable—so don’t mix them up.

What are the names of polynomial equations with different degrees?

The polynomial equations with different degrees (or orders)—especially the lowest degrees—are as follows:

· first degree (1) polynomial = linear

· second degree (2) polynomial = quadratic

· third degree (3) polynomial = cubic

· fourth degree (4) polynomial = quartic

· fifth degree (5) polynomial = quintic

· sixth degree (6) polynomial = sextic

How do you multiply polynomial equations?

To multiply two monomials, multiply the coefficients and then multiply the variables (and when multiplying the variables, keep the variables and add the exponents). The following are several examples of multiplying various polynomial equations:

Multiplying monomials:

(5x)(6x²)

= (5 × 6)(x²⁺¹) (multiply the numbers and add the exponents)

= 30x³

Multiplying a monomial and polynomial (binomial):

4y(2y-8)
= 4y(2y) - 4y(8) (multiply 4y times both terms)

= 8y² - 32y (depending on what is on the other side of the equation, you can further simplify by dividing by 8, or y² - 4y)

Multiplying polynomials:

6y³(8y⁶ + 5y⁴ - 3y³)

= 6y³(8y⁶) + 6y³(5y⁴) - 6y³(3y³)

= 48y⁹ + 30y⁷ - 18y⁶

How do you divide polynomials?

To divide a polynomial by a monomial, divide each term in the polynomial by the monomial (to divide monomials, we divide the coefficients and then subtract the exponents) or:

(A + B + C)/M = A/M + B/M + C/M

For example, to solve the equation

10x⁵/2x³ = (10/2) (x^5-3) = 5x²

What does factoring polynomials mean?

When a polynomial is written as the product of two or more polynomial equations, the polynomial has been factored. This allows a complicated polynomial to be broken up into easier, lower degree pieces; and it makes the equation easier to solve. One way to look at it is that factoring a polynomial is the opposite process from multiplying polynomials.

One of the most basic ways to factor a polynomial is similar to factoring a number. When a number is factored, the result will be the prime factors that multiply together to give the number (for example, 6 = 2 × 3 , or 12 = 2 × 2 × 3; see “Math Basics” to learn more about prime factors). With polynomials, this is often called “taking out a common factor”: If every term in a polynomial expression has several factors, and if every term has at least one factor that is the same, then that factor is called a common factor. If this is the case, then the common factor can be removed from every term and multiplied by the whole remaining expression.

For example, for the equation 2x² + 8x, the first term has factors of 2 and x, while the second term has factors of 2, 4, and x. The common factors are 2 and x, making 2x the overall common factor. This makes the expression equal to 2x(x + 4). Thus, it is easy to see that when a polynomial is factored, it results in simpler polynomials that can be multiplied together to give the initial polynomial.

What is the difference and sum of cubes?

When factoring polynomials, there is the difference and sum of cubes. The difference of cubes takes the form: a³ - b³, and can be factored into (a - b)( a² + ab + b²). Thus, if an expression resembles a³ - b³, then (a - b) is a factor; use long division to find the remaining factor(s).

The sum of cubes takes the form a³ + b³, and can be factored into (a + b)(a² + ab + b²). Thus, if an expression resembles a³ + b³, then (a + b) is a factor. Again, use long division to find the remaining factor(s).

How do you find the roots of a polynomial?

Finding the root, also called a zero, of a polynomial is one way to solve for the equation. In other words, the root of an equation is simply a number (or numbers) that solves the equation.

For example, for second degree polynomials, we can find the roots by completing the square. Picking apart an equation is the best way to see this:

1. 3x²- 4x + 1 = 0

2. (1/3)( 3x² - 4x + 1) = (1/3)0 (making the coefficient of the x² term into a 1)

3. x² - (4/3)x + 1/3 = 0

4. (x² - (4/3)x) + 1/3 = 0 (group the x and x² terms together)

5. (x² - (4/3)x + (-2/3)²) - (-2/3)² + 1/3 = 0 (determine the coefficient of the x term, divide it by 2 and then square; add and subtract that term)

6. (x - 2/3) ²- 4/9 + 1/3 = 0

7. (x - 2/3) ² - 1/9 = 0 (add together the 4/9 + 1/3 by converting the denominator to 9, in which 1/3 becomes 3/9)

8. (x - 2/3) ² = 1/9 (move the 1/9 to the other side of the equation by subtracting it from both sides)

9. x - 2/3 = 1/3 or × - 2/3 = -1/3

What is a perfect square?

There are many equations that can be factored into a perfect square. Any expression written in the form x² + 2ax + a² is a perfect square—an expression written as [something]². To determine if an expression is a perfect square, first see if the constant term is a square number—in other words, can you take the square root of it and get an integer for an answer. If so, determine if the square root of the constant, multiplied by 2 gives the coefficient of the linear term (or the x term). If it does, the original expression may be factored into a perfect square. (Note: The above procedure only works when the coefficient of x² is 1.)

For example, in the equation x² + 8x + 16, the constant term (16) is already a perfect square (the square root of 16 is 4). Since 2(4) = 8, the original expression can be written as a perfect square. Because we know x² + 2ax + a² is a perfect square, and equals (x + a)², by substituting the common factor 4 into the equation, we find that x² + 8x + 16 = (x + 4)².

That means that x = 1 or x = 1/3 are the two roots that make the equation true (just substitute either number into the initial equation to see that they are both true).

What are examples of polynomials with one root and no roots?

The following is an example of a polynomial with only one root:

x² + 6x + 9 = 0

(x² + 6x + (6/2)²) - (6/2)² + 9 = 0

(x + 3)² - 9 + 9 = 0

(x + 3) ² = 0

x + 3 = 0

x = -3, or the polynomial has only one root x = -3

What is the Fundamental Theorem of Algebra?

The Fundamental Theorem of Algebra (FTA) is nothing new; it was first proved by mathematician Carl Friedrich Gauss (1777–1855) in 1799. The equation was as follows:

a_nxⁿ + a_n-1x^n-1 + … + a₁x¹ + a₀ = 0 (as long as n is greater than or equal to 1 and a_n is not zero, and has at least one root in the complex numbers).

The proof of this theorem goes on for pages—far beyond the scope of this book. What all those proofs, numbers, and letters boil down to is that a polynomial equation must have at least one number in its solution. It also tells us when we have factored a polynomial completely. Simple enough, but like much of mathematics, someone had to prove it.

But that is not all: The FTA is not constructive, and therefore it does not tell us how to completely factor a polynomial. In other words, in reality, no one really knows how to factor a polynomial, we only know how to apply techniques to certain kinds of polynomials. In fact, French mathematician Evariste Galois (1811–1832), who died tragically in a duel, proved that there will never be a general formula that will solve fifth degree or higher polynomials.

But not all polynomials have roots. The following is an example of a polynomial with no root:

2x² - 6x + 8 = 0

(½)(2x²- 6x + 8) = (½)0

x² - 3x + 4 = 0

(x² - 3x + (-3/2)²) - (-3/2)² + 4 = 0

(x - 3/2)²- 9/4 + 4 = 0

(x - 3/2)² + 7/4 = 0

(x - 3/2)² = -7/4

Because a real number squared is greater than or equal to 0, that means (x - 3/2)² will always be greater than or equal to 0. Therefore, the answer can’t be -7/4, a negative number, and there are no real roots for this polynomial.

What is a quadratic equation?

A quadratic equation is a second degree (order) polynomial equation, thus it is guaranteed to have two solutions, both of which may be real or complex. This is seen in the standard form ax² + bx + c = 0.

The roots of x can be found in the equation by factoring and completing the square (a method of transforming a quadratic equation so that it is in the form of a perfect square). For example, to solve the equation x² - 3x = 4 by factoring, you would do the following:

First, write the equation in standard quadratic equation form:

x²- 3x - 4 = 0

Then factor the form:

(x - 4)(x + 1) = 0

In order for these numbers to equal zero, we determine that for (x - 4), x would be 4; and for (x + 1), x would equal -1. Thus, the solutions are 4 and -1.

Can all quadratic equations be solved by factoring?

Don’t be fooled: Not all quadratic equations can be solved by factoring. For example, x² - 3x = 3 is not solvable with this method. One way to solve quadratic equations is by completing the square; still another method is to graph the solution (a quadratic graph forms a parabola—a U-shaped line seen on the graph). But one of the most well-known ways is by using the quadratic formula (see below).

For example, if we want to find the roots of the polynomial x² + 2x - 7, we can replace the “corresponding” numbers from the initial equation into the quadratic equation; thus, a = 1, b = 2, and c = -7. Substituting these numbers into the quadratic formula, we solve for:

That equals

What is the discriminant of a quadratic equation?

In the quadratic equation ax² + bx + c = 0, the value of b² - 4ac is the discriminant—the same numbers and letters that are under the square root sign of the quadratic formula. This is actually the products of the squares of the polynomial root differences. In other words, this quantity characterizes certain properties of the quantity’s roots. The discriminant is often used for such mathematical concepts as metrics, modules, quadratic fields, and polynomials.

MORE ALGEBRA

What is an array?

A mathematical array is a list of lists, or a rectangular arrangement of objects; it is a useful way of keeping a collection of things. Arrays are orderly arrangements of objects in columns and rows, with the objects most often being numbers. For example, the most common arrays are two dimensional with a certain amount of rows and columns:

In this case, this is a 2-by-3 array (rows are always mentioned before columns). The order of numbers and/or elements in an array may sometimes, but not always, be significant (as it is in a matrix; see below).

What is a matrix?

A matrix is a concise way of representing and working with linear transformations; also, it is a rectangular array or grid of numbers or variables that allows the user to perform certain mathematical operations. They are usually symbolized as large parentheses or two large pair of parallel double lines surrounding the array of numbers or variables. These numbers can be manipulated to solve systems of equations or problems with many different variables or numbers, by addition, subtraction, multiplication or other methods. Each row and column of a given matrix must have the same number of elements.

Any time one has a list of numbers, or a table of numbers in a specific order, concerning anything at all (prices, grades, populations, coordinates of points, production tables …), it can be considered to be a matrix. When the idea of the matrix was first conceived, its development dealt with transformation of geometric objects and solution of systems of linear equations. Historically, the early emphasis was on the determinant (see below), not the matrix; today, especially in linear algebra, matrices are considered first.

Who invented matrices?

Although a simple form of matrices may have been used by the Mayans (and maybe other cultures; see below), the true mathematical use of a matrix was first formulated around 1850, by English mathematician, poet, and musician James Sylvester (1814–1897). In his 1850 paper, Sylvester wrote, “For this purpose we must commence, not with a square, but with an oblong arrangement of terms consisting, suppose, of m lines and n columns. This will not in itself represent a determinant, but is, as it were, a Matrix out of which we may form various systems of determinants by fixing upon a number p, and selecting at will p lines and p columns, the squares corresponding of pth order.” In this case, Sylvester used the term matrix to describe its conventional use, or “the place from which something else originates.”

The ancient Mayans may have been among the first cultures to use a simple form of matrices.

But the matrix story was not all about Sylvester. In 1845, Sylvester’s collaborator, English mathematician Arthur Cayley (1821–1895), used a form of matrices in his work, On the Theory of Linear Transformations; by 1855 and 1858, Cayley began to use the term “matrix” in its modern mathematical sense. Although he was an avid mountaineer and a lawyer for close to a decade and a half (which is how he met Sylvester), during his free time Cayley published more than 200 mathematical papers. He also contributed a great deal to the field of algebra, initiated analytic geometry of n-dimensional spaces, and developed the theory of invariants, among other mathematical feats.

Sylvester also remained brilliant throughout his life. He founded the American Journal of Mathematics in 1878; and at the ripe age of 71, he invented the theory of reciprocants (differential invariants).

What are some examples of a matrix?

The dimensions of a matrix are the number of rows (horizontal numbers) and columns (vertical numbers); it is written as the rows first, columns second. The following are some simple examples of matrices—all differing in dimensions:

A 3 by 2 (3 × 2) matrix:

A 2 by 3 (2 × 3) matrix:

A 4 by 4 (4 × 4) matrix (when the row and column dimensions are the same, it is called a “square matrix”):

How do you add matrices?

It is pretty easy to add matrices. For example, for (1 2) + (-1 -2), you add the matrices, (or in this case, row numbers), simply by adding the corresponding terms in each matrix. Thus, the matrix of (1 2) + (-1 -2) = (1-1 2-2) = (0 0).

What is the identity matrix?

The identity matrix is the n-by-n matrix that has all ones down the main diagonal and zeroes everywhere else; it must also be a square. The following is the identity matrix:

Or in more formal and explicit terms:

What happens when you multiply a matrix by the identity matrix?

When you multiply any n-by-n matrix by the identity matrix, you get that same matrix back again. Therefore, let the letter I represent the n-by-n identity matrix, and A represent any other n-by-n matrix; then we have A × I = A and I× A = A. This is much like the situation in the real numbers: x × 1 = x and 1 × x = x.

Did the Mayans know about matrices?

Some scientists believe that the Mayan culture had a good working idea about matrices long before they became a part of mathematics. It is thought that the Mayans discovered how to place a set of “numbers” in columns and rows, then performed diverse operations on them, such as a method of adding and subtracting along diagonals to solve “equations” with unknown quantities. They may have even used the matrices for multiplication, division, and calculations of square and cubic roots on a matrix array by using a series of dots in a matrix to perform the operations. And there is evidence of these mathematical squares on their monuments, paintings, and even garments, including those of the Mayan priests and on crests of higher officials. But not everyone agrees with this notion; some scientists believe that the Mayan grids are merely mimicking objects in nature, such as the back of a turtle’s shell.

How are matrices used?

Matrices are used in a multitude of fields, from mathematics and science to certain humanities fields. For example, they are used in physics to determine the equilibrium of rigid bodies; in graph theory, fractals, and solutions of systems of linear equations in mathematics; and in forest management, computer graphics, cryptology—even electrical networks.

ABSTRACT ALGEBRA

What is abstract algebra?

Abstract algebra is a collection of mathematical topics that deal with algebraic structures rather than the usual number systems. These structures include groups, rings, and fields; branches of these topics include communtative and homological algebras. In addition, linear algebra and even elementary number theory (see “Math Basics”) are often included under abstract algebra.

What is an algebraic structure?

An algebraic structure is made up of a set (collection of objects called elements; for more information about sets, see “Foundations of Mathematics”) together with one or more operations on the set that satisfy certain axioms. The algebraic structures get their names depending on the operations and axioms. For example, algebraic structures include fields, groups, and rings—and many other structures with strange names such as loops, monoids, groupoids, semigroups, and quasigroups.

What is a field?

A field is an algebraic structure that shares the common rules for operations (addition, subtraction, multiplication, and division, except division by zero) of the rational, real, and complex numbers (but not integers, see below under “ring”). A field must have two operations, must have at least two elements, and must be commutative, distributive, and associative (see above for definitions). Formerly called “rational domain,” a field in both French (corps) and German (Körper) appropriately means “body.” A field with a finite number of members is called a Galois or finite field. Fields are useful to define such concepts as vectors and matrices.

What is a group in abstract algebra?

A group, usually referred to as G, is a finite or infinite set of elements together with a binary operation (often called the group operation) that together satisfy the four fundamental properties—closure, associativity, and the identity and inverse properties (for more information about these properties, see elsewhere in this chapter). A great many of the objects investigated in mathematics turn out to be groups, including familiar number systems—such as the integers, rational, real, and complex numbers under addition; non-zero rational, real, and complex numbers under multiplication; non-singular matrices under multiplication; and so on. The branch of mathematics that studies groups is called group theory, an important area of mathematics that has many applications to mathematical physics (such as particle theory).

What is a ring?

A ring is an algebraic structure (some definitions say a set) in which two binary operators (addition and multiplication) in various combinations must satisfy either the additive associative, commutative, identity, and inverse properties, the multiplicative associative property, or the left and right distributivity properties. For example, the elements of one operation, such as addition, must form a group that is commutative, also known as an abelian group. The multiplicative operation must produce unique answers that have the associative property. These two operations are further connected by requiring the multiplication to have a distributive property with respect to the addition. This can be written as follows, with a, b, and c elements of the ring:

Rings are usually named after one or more of their investigators. But such a practice usually makes understanding the properties of the various associated rings difficult for anyone other than the mathematician working on the ring.

What is linear algebra?

Linear algebra is the study of linear sets of equations. It encompasses their transformation properties, and includes the analysis of rotations in space, least squares fitting, and numerous other problems in mathematics, physics, and engineering.

What is Boolean algebra?

Boolean algebra is an abstract mathematical system used to express the relationship between sets (groups of objects or concepts; see “Math Basics”). It is important in the study of information theory, the theory of probability, and the geometry of sets. The use of Boolean notation in electrical networks aided the development of switching theory and the eventual design of computers.

It was English mathematician George Boole (1815–1864) who first developed this type of logic by demonstrating the algebraic manipulation of logical statements, showing whether or not a statement is true, and showing how a statement can be made into a simpler, more convenient form without changing its overall meaning. Today, this way of looking at logic is called Boolean algebra. (For more information about Boole, see “History of Mathematics.”)

Though Charles Dodgson is more commonly known as the author of Alice’s Adventures in Wonderland under the pen name of Lewis Carroll, he was also a brilliant mathematician. Among other accomplishments, he devised ways to refine Boolean algebra notations.

Boolean algebra did not end there: In 1881, the English logician and mathematician John Venn (1834–1923) interpreted Boole’s work and introduced a new way of diagramming Boole’s notation in his treatise, Symbolic Logic.This was later refined by the English mathematician Charles Dodgson (1832–1898), better known as the writer of Alice’s Adventures in Wonderland (under the pseudonym Lewis Carroll). Today when studying sets, we call this method not the Boole, Carroll, or Dodgson diagram, but the Venn diagram. Thus, Boolean notation demonstrates the relationship between groups, indicating what is in each set alone, what is jointly contained in both, and what is present in neither.

How is algebra used to solve problems in various fields— in mathematics and otherwise?

Algebra has been used extensively to solve a myriad of mathematical and scientific problems in recent years. For example, a problem that has plagued two mathematical areas—topology and algebra—was recently solved: the relationship between different mathematical structures. First formulated by German mathematician Friedrich Hirzebruch in the mid-20th century, the problem was to determine which numbers were topological invariants of complex-algebraic varieties (or the zero-sets of polynomials). The researchers found that no such numbers were topologically invariant, and thus, the numbers depend on the algebraic structure of the variety, not the topological properties. Still another study used algebra in the field of DNA hybridization—a process central to most biotechnological devices that monitor changes in cells’ gene expressions. Based on their calculations and applications of algebra within their model, the researchers were able to monitor cellular changes.

Why are mathematicians so interested in transcendental numbers?

Transcendental numbers are those that are not the root of any integer polynomial, or that are not an algebraic number of any degree. Thus all transcendental numbers are irrational (rational numbers are algebraic numbers of degree one). The importance of such numbers translates through more than two millennia of history: For example, they provided the first proof that circle squaring was insoluble, which is one of the geometric problems that has baffled mathematicians throughout antiquity.