GLOSSARY

associative law of addition: Typically written a + (b + c) = (a + b) + c, this rule means that it doesn’t matter how you group added numbers, or which two you add first.

associative law of multiplication: Written (a × b) × c = a × (b × c), it means that it doesn’t matter how you group multiplied numbers, or which two you multiply first.

commutative law of addition: Written a + b = b + a, it means switching the order of two numbers when adding them doesn’t change the result.

commutative law of multiplication: Written a × b = b × a, it means that switching the order of two numbers when multiplying them doesn’t change the result.

complex plane: A two-dimensional space containing two number lines, or axes, that intersect at right angles—an x axis for real numbers, and an i axis for imaginary numbers.

complex number: Two-part, hybrid numbers that are usually written in the form a + bi, where a is a real number and bi, or b times i, is an imaginary number. Each complex number is associated with a point on the complex plane.

constant: A particular number, such as e, i, π, 1, or 0, which are the five constants in Euler’s formula.

cosine function: Written cos θ, it’s a function that effectively takes as input the size of an angle of a right triangle and that outputs the ratio between the length of the side adjacent to the angle and the length of the triangle’s hypotenuse. Like the sine function, it can also be defined in terms of the coordinates of points on the unit circle.

cubic equation: An equation that includes a variable such as x raised to the third power (that is, having an exponent of three) but no x’s raised to higher powers. Example: x3 + 2x2 5x + 8 = 0.

distributive law: An arithmetic rule often written a × (b + c) = (a × b) + (a × c). It means that when you multiply a sum by a number, you get the same result that you would get if you separately multiplied the number times each of the summed numbers and then added the products together.

e: A constant, defined as the number that the expression (1 + 1/n)n approaches as ever larger integers are plugged in for n. Called Euler’s number, it often crops up in math, sometimes quite unexpectedly. It is an irrational number as well as a transcendental one.

exponent: In basic math, an exponent is defined as a positive integer written as a superscript next to a constant or variable that designates how many times to multiply the constant or variable times itself. For instance, 102, which is spoken “ten squared” or “ten raised to the second power,” means 10 times 10, or 100, and 103 means 10 times 10 times 10, or 1,000. Mathematicians have repeatedly expanded this definition to allow zero, negative integers, non-integer real numbers (both rational and irrational), imaginary numbers, and complex numbers as exponents.

factorial: Symbolized by ! written adjacent to an integer, the factorial operator means “multiply together all the positive integers up to and including the specified integer.” Thus, 3!, which is spoken “three factorial,” is shorthand for 1 × 2 × 3, or 6. Both 0! and 1! are defined as 1.

function: As used in this book, the term function means an expression with variables, such as x + 5. Functions resemble computer programs that convert input numbers into output numbers in specified ways. They’re designated with equations such as f(x) = x + 5, where f(x) means “a function whose variable is x.”

hypotenuse: The side opposite the 90-degree angle of a right triangle.

i: Defined as the square root of 1, i is the one-unit imaginary number on which all the imaginary numbers are based.

imaginary number: A number of the form a × i, where a is a real number, and i stands for the square root of 1. Each imaginary number is the counterpart of a real number. For instance, i, the unit imaginary number, is the imaginary counterpart of 1, and i is the imaginary counterpart of 1. The imaginary number π times i is e’s exponent in Euler’s formula.

irrational number: A number that can’t be expressed as a fraction. An irrational number’s decimal representation includes an infinite number of patternless digits to the right of the decimal point. Both π and e are irrational numbers.

nth root: The nth root of a number is another number that when multiplied times itself n times is equal to the original number. For instance, 2 is a 4th root of 16, since 2 × 2 × 2 × 2 = 16.

oscillation: Movement back and forth at a constant speed. Cyclic phenomena such as sound and radio waves, and alternating current, involve oscillation.

origin: The point at which the x and y axes meet in the xy plane, and at which the x and i axes meet in the complex plane. Its coordinate pair is (0,0) in the xy plane. The complex plane’s origin is associated with the complex number 0 + 0i.

parallelogram: A four-sided polygon whose opposite sides are parallel.

pi: Designated by the Greek letter π, pi is a constant that’s equal to the circumference of any circle divided by its diameter. An irrational number, it’s approximately 3.14159.

polygon: A many-sided figure. Regular polygons have equal-length sides and equal-sized internal angles, such as stop-sign octagons.

protractor: A tool for measuring angles, typically consisting of a half circle made of transparent plastic, with angles marked off on the outer edge from 0 to 180 degrees.

Pythagorean theorem: It states that the sum of the squared lengths of the two shorter sides of any right triangle is equal to the square of the hypotenuse’s length. It is typically expressed with an equation like x2 + y2 = z2, where x, y, and z represent side lengths of a right triangle.

radian: The angle swept out from the center of a circle by an arc along the circle that’s equal in length to its radius. This implies that 2π radians = 360o, π radians = 180o, and π/2 radians = 90o.

radius: Half of a circle’s diameter, pictured as a line segment between the circle and its center.

ratio: Typically written in the format 2 to 3, or 2:3, ratios express the same numerical relationships that fractions do. For instance, if a recipe specified a sugar-to-flour ratio of 1 to 3, you could say that it calls for a mixture consisting of 1/4 sugar and 3/4 flour.

real number: A number that lies along the familiar number line. The reals include positive and negative integers, zero, rational numbers (fractions), irrational numbers, and transcendental numbers. The rationals include all the integers; the irrationals include the transcendentals.

right triangle: A triangle with a 90-degree angle and two smaller angles.

sine function: Written sin θ, it’s a function that effectively takes as input the size of an angle of a right triangle and that outputs the ratio between the length of the side opposite to the angle and the length of the triangle’s hypotenuse. Like the cosine function, it can also be defined in terms of coordinate pairs of points on the unit circle.

square root: The square root of a number x is a number that when multiplied times itself equals x. For example, both 2 and 2 are square roots of 4.

transcendental number: A number that cannot be the solution of any polynomial equation with integer constants multiplied times the variables. An example of a polynomial equation is x2 2x 35 = 0. The fact that 7 solves this equation rules out that number as a transcendental. Both π and e were demonstrated in the nineteenth century to be transcendental.

unit circle: A one-unit-radius circle whose center is the origin of the xy plane, or the origin of the complex plane. Radius-like line segments can be pictured in the unit circle sweeping out angles.

variable: Usually designated by letters such as x, variables are proxies for numbers that haven’t been pinned down. When they appear in equations such as x 2 = 4, they represent “unknowns” that can be determined by finding numbers that, when substituted for the variables, make the equations true.

vector: An arrow-like visual representation of complex numbers on the two-dimensional complex plane. Vectors can also be used as visual proxies of coordinate pairs on the xy plane. Vectors are used in physics to represent things such as the speed and direction of moving objects.

x and y coordinates: Pairs of numbers written in parentheses that are associated with points on the xy plane. The first of the two coordinates is measured along the x axis, specifying a point’s distance from the y axis. The second coordinate is measured along the y axis and specifies the point’s distance from the x axis.

xy plane: A two-dimensional surface featuring a horizontal real-number line called the x axis, and a vertical real-number line called the y axis. The xy plane enables the mapping of arithmetic and algebraic concepts onto geometric counterparts, such as the representation of coordinate pairs of numbers as points on the plane.

Zeno’s paradox: Zeno, an ancient Greek philosopher, proposed several thought experiments that led to preposterous conclusions. The “race course paradox,” one of his most famous, suggests that a runner can never complete a race because he must first get half of the way to the finish line, then half of the remaining distance after that, and then half of the remainder after that, and so on—an infinite number of steps that was seemingly impossible to complete.