## Idiot's Guides: Algebra I (2015)

### APPENDIX A. Glossary

**absolute value** The distance of a number from zero, without regard to direction. The size or magnitude of a number without regard to its sign.

**addend** Each of the numbers that are added in an addition problem.

**algorithm** A list of steps necessary to perform a process.

**associative property** A property of addition or multiplication that says that when adding or multiplying more than two numbers, you may group them in different ways without changing the result.

**asymptote** A line that a graph approaches very closely. Graphs never cross vertical asymptotes but may sometimes cross horizontal asymptotes.

**axis** A vertical or horizontal line that divides the coordinate plane into sections. The horizontal line is called the *x*-axis and the vertical line called the *y*-axis.

**axis of symmetry** An imaginary line passing through the vertex of a parabola, which divides the parabola into two sections that are reflections of one another.

**base** When an exponent is used to show repeated multiplication, the number to be multiplied is the base of the power.

**binary operation** A process that works on two numbers at a time.

**binary system** A place value system based on the number 2.

**canceling** The process of simplifying a multiplication of fractions by dividing a numerator and a denominator by a common factor.

**Cartesian plane** A system that identifies every point in the plane by an ordered pair of numbers. Also called the coordinate plane.

**closure** A property that says that adding or multiplying two numbers of a set results in another number in the set.

**coefficient** The numerical part of a term. Technically, a term is made up of a numerical coefficient and a variable coefficient, but usually the variable coefficient is just called the variable and the numerical coefficient is called the coefficient.

**common denominator** A multiple of the denominators of two or more fractions.

**common fraction** Fractions written as a quotient of two integers.

**commutative property** A property of addition or multiplication that says that reversing the order of the two numbers will not change the result.

**compound inequality** Two inequalities connected with a conjunction, either the word *and* or the word *or*.

**conjugate** The conjugate of an expression that is the sum of two terms is an expression that is the difference of those two terms. *a* − *b* is the conjugate of *a* + *b*, and *a* + *b* is the conjugate of *a* − *b*.

**conjunction** A compound inequality in which the word *and* is used.

**constants** Terms containing only numbers and no variables.

**coordinate system** A system that locates every point in the plane by an ordered pair of numbers, (*x*, *y*). Also called the Cartesian plane.

**counting numbers** The set of numbers {1, 2, 3, 4, …} They are the numbers used to count. The counting numbers are also called the natural numbers.

**cross-multiply** To find the product of the means and the product of the extremes of a proportion, and to say that those products are equal.

**decimal fraction** Fractions written in the base ten system with digits to the right of the decimal point.

**decimal system** A place value system in which each position in which a digit can be placed is worth ten times as much as the place to its right.

**degree** The degree of a monomial with one variable is the power of the variable.

**denominator** The number below the division bar in a fraction that tells how many parts the whole was broken into, or what kind of fraction you have.

**difference** The result of a subtraction problem.

**digit** A single symbol that tells how many.

**discontinuity** The value at which the rational function is undefined is called a discontinuity, because there is a break in the graph at that value.

**discriminant** The discriminant is the portion of the quadratic formula that appears under the radical, *b*^{2} − 4*ac*. It can be used to tell how many solutions (and what type of solutions) a quadratic equation will produce.

**disjunction** A compound inequality in which the two inequalities are joined by *or*.

**distributive property** A property that says that for any three numbers *a*, *b*, and *c*, *c*(*a* + *b*) = *ca* + *cb*. The answer you get by first adding *a* and *b* and then multiplying the sum by *c* will be the same as the answer you get by multiplying *a* by *c* and *b* by *c* and then adding the results.

**dividend** In a division problem, the number that is divided.

**divisor** In a division problem, the number by which you divide.

**domain** The set of all values that can be substituted in a function for *x*, that is, all possible inputs.

**domain, rational expression** The set of all real numbers that can be substituted in a rational expression for the variable without making the denominator equal to zero.

**double root** A solution that occurs twice for the same equation, because the equation was a perfect square trinomial. Also called a double solution.

**equation** A mathematical sentence, which often contains a variable.

**exponent** The small raised number that tells how many times to use the base number as a factor.

**exponentiation** The operation of raising to a power.

**expression** Any mathematical calculation.

**extended ratio** Several related ratios condensed into one statement. The ratios *a:b*, *b:c*, and *a:c* make the extended ratio *a:b:c*.

**extraneous solution** A solution produced by correct algebraic procedures that does not satisfy the equation.

**extremes** The first and last numbers of a proportion.

**factor** Each of the numbers or variables being multiplied.

**factor tree** A method of finding the prime factorization of a number by starting with one factor pair and then factoring each of those factors, continuing until no possible factoring remains.

**factoring** The process of rewriting an integer as the product of two integers, or a polynomial as the product of two polynomials of lesser degree.

**FOIL** An acronym for First, Outer, Inner, Last. It summarizes the four multiplications necessary when multiplying two binomials.

**fraction** A symbol that represents part of a whole.

**function** A relation in which each number in the domain has only one partner from the range.

**function, linear** A function of the form *y* = *mx* + *b*. It defines the output variable as a multiple of the input variable, possibly plus or minus a constant.

**greatest common factor** The greatest common factor of two numbers is the largest number that is a factor of both.

**identity** A property that says that adding 0 or multiplying by 1 leaves a number unchanged.

**imaginary numbers** Numbers that do not fit in the real number system, but that mathematicians use to solve certain types of problems.

**improper fraction** A fraction whose value is more than 1. The numerator is larger than the denominator.

**index** A small number appearing in the crook of the radical sign that tells what power is being undone. If no index appears, the radical indicates a square root.

**inequality** A sentence that compares two expressions that are not equal and shows which one is larger.

**inequaltity, compound** Two inequalities connected with a conjunction, either the word *and* or the word *or*.

**inequality, linear** A statement that defines a relation in which the multiple outputs are less than, or greater than, or equal to, an expression involving the input value.

**input** The numbers from the domain that are put into a function.

**integers** The set of numbers that includes all the positive whole numbers, and their opposites, the negative whole numbers, and zero.

**interest** Money you pay for the use of money you borrow, or money you receive because you’ve put your money into a bank account or other investment.

**interest rate** The percent of the principal that will be paid in interest each year.

**inverse** A property that says that every operation can be reversed. Every number has an opposite and adding the opposite to a number brings you back to 0. Every number except 0 has a reciprocal and multiplying by the reciprocal gets back to 1. Zero is its own opposite but has no reciprocal.

**inverse operation** An operation that reverses the work of another.

**irrational numbers** Numbers that cannot be written as the quotient of two integers.

**least common denominator** The least common multiple of two or more denominators.

**least common multiple** The smallest number that has each of two or more numbers as a factor.

**like terms** Terms that have the same variable, raised to the same power.

**line** A set of points that has length but no width or height.

**linear function** A function of the form *y* = *mx* + *b*. It defines the value of the output variable as a multiple of the input variable, possibly plus or minus a constant.

**linear inequality** A statement that defines a relation (not a function) in which the output is less than, or greater than, or less than or equal to, or greater than or equal to, some expression involving the input value.

**linear system** A set of two linear equations, each involving the same two variables. The solution of the system is the one pair of values that satisfy both equations.

**mean** The arithmetic average of a group of numbers, found by adding all the numbers and dividing by the number of numbers in the group.

**means of a proportion** The two middle numbers in a proportion.

**minuend** The first number in a subtraction problem.

**mixed number** A whole number and a fraction, written side by side, representing the whole number plus the fraction.

**monomial** A monomial is a constant, a variable, or a product of constants and variables.

**natural numbers** The set of numbers {1, 2, 3, 4, …} They are the numbers you use to count. Also called the counting numbers.

**negative exponent** A negative exponent on a non-zero base represents a fraction with a numerator of 1 and a denominator of the base raised to the corresponding positive power.

**number line** A line divided into segments of equal length, labeled with numbers, usually the integers. Positive numbers increase to the right of zero, and negative numbers go down to the left.

**numerator** The number above the bar in a fraction, which tells you how many of that denomination are present.

**order of operations** An agreement among mathematicians regarding the order in which operations are performed when solving an equation. Operations enclosed in parentheses or other grouping symbols are performed first, followed by evaluating exponents. After that, do multiplication and division as you meet them moving left to right, and finally do addition and subtraction as you meet them, moving left to right.

**ordered pair** Two numbers, usually designated as *x* and *y*, that locate a point in a coordinate system.

**origin** The point at which the *x*-axis and *y*-axis intersect in a coordinate plane.

**output** The numbers in the range produced by a function.

**parabola** A cup-shaped graph characteristic of quadratic functions. Technically, it is defined as a set of points, each of which is equidistant from a predetermined point and line.

**parallel lines** Lines on the same plane that never intersect.

**partial product** In a multiplication of polynomials, or in the multiplication of multi-digit numbers, the polynomial produced by multiplying by just one digit or term is a partial product.

**PEMDAS** A mnemonic, or memory device, to help you remember that the order of operations is parentheses, exponents, multiplication and division, addition and subtraction.

**percent** A ratio that compares numbers to 100. 42 percent means 42 out of 100, or 42:100.

**perpendicular lines** Lines that meet to form a right angle.

**place value system** A number system in which the value of a symbol depends on where it is placed in a string of symbols.

**plane** A flat surface that has length and width but no thickness.

**point** A position in space that has no length, width, or height.

**polynomial** A polynomial is a sum of monomials.

**power** A way to tell how many times a number should be used in repeated multiplication. The number to be multiplied is the base of the power, and the small raised number that tells how many times to use it is called the exponent.

**power of ten** A number formed by multiplying several 10s. The first power of ten is 10. The second power of ten is 100, and the third power of 10 is 1,000.

**prime factorization** The prime factorization of a number is a multiplication that uses only prime numbers and produces the original number as its product.

**prime number** A whole number whose only factors are itself and 1.

**principal** The amount of money borrowed or invested. The rate is the percent of the principal that will be paid in interest each year.

**product** The result of multiplying.

**proper fraction** A fraction whose value is less than one.

**proportion** Two equal ratios. The means of a proportion are the two middle numbers. The extremes are the first and last number.

**quadrants** The four sections into which the coordinate plane is divided by the axes.

**quadratic** An equation that includes a term in which the variable is squared. There may also be a variable term in which the variable appears but is not squared, and a constant term.

**quadratic formula** The formula , which can be used to find solutions for any quadratic equation of the form *ax*^{2} + *bx* + *c* = 0 by substituting the values of *a*, *b*, and *c* and simplifying.

**quotient** The result of division.

**radical** The symbol for the square root. From the Latin word *radix*, meaning root.

**radicand** The number or expression under a radical.

**range** The set of all outputs, all values of *y*.

**rate** A comparison of two quantities in different units, for example, miles per hour or dollars per day.

**ratio** A comparison of two numbers by division.

**rational equation** An equation that contains one or more rational expressions.

**rational expression** A quotient of two polynomials, provided that the polynomial in the denominator is not zero.

**rational numbers** The set of all numbers that can be written as the quotient of two integers.

**rationalizing the denominator** The process of changing the appearance, but not the value, of a quotient so that no radicals remain in the denominator.

**real numbers** The name given to the set of all rational numbers and all irrational numbers.

**reciprocal** Two numbers are reciprocals if their product is 1. Each number is the reciprocal of the other.

**relation** A pairing of numbers from one set, called the domain, with numbers from another set, called the range.

**relatively prime** Two numbers are relatively prime if the only factor they have in common is 1.

**remainder** A remainder is the number left over at the end of a division problem. It’s the difference between the dividend and the product of the divisor and quotient.

**root** The opposite operation of a power, a way of undoing a power.

**slope** The slope of a line is a number that compares the rise or fall of a line to its horizontal movement.

**solving an equation** An equation is a mathematical sentence, which often contains a variable. Solving an equation is a process of isolating the variable to find the value that can replace the variable to make a true statement.

**square numbers** Numbers created by raising a number to the second power.

**standard form** The standard form of a polynomial writes the terms in order from highest degree to the lowest.

**subtrahend** In a subtraction problem, the number that is taken away.

**sum** The result of addition.

**system** A set of two linear equations, each involving the same two variables. The solution of the system is the one pair of values that satisfy both equations.

**term** An algebraic expression made up of numbers, variables, or both, that are connected only by multiplication.

**unlike terms** Terms with different variables, such as *x* and *y*.

**variable** A letter or symbol that takes the place of a number.

**vertex** The turning point of a graph.

**whole numbers** The set of numbers {0, 1, 2, 3, 4, …} formed by adding a zero to the counting numbers.

** x-coordinate** The first number in an ordered pair, which indicates horizontal movement.

** y-coordinate** The second number in an ordered pair, which indicates vertical movement.

**zero** **exponent** Any non-zero number to the 0 power is 1.