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Mathematics LibreTexts

Glossary

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    51369
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    Example and Directions
    Words (or words that have the same definition)The definition is case sensitive(Optional) Image to display with the definition [Not displayed in Glossary, only in pop-up on pages](Optional) Caption for Image(Optional) External or Internal Link(Optional) Source for Definition
    (Eg. "Genetic, Hereditary, DNA ...")(Eg. "Relating to genes or heredity")The infamous double helix https://bio.libretexts.org/CC-BY-SA; Delmar Larsen
    Glossary Entries
    Word(s)DefinitionImageCaptionLinkSource
    compound inequalityA compound inequality is made up of two inequalities connected by the word “and” or the word “or.”    
    conditional equationAn equation that is true for one or more values of the variable and false for all other values of the variable is a conditional equation.    
    contradictionAn equation that is false for all values of the variable is called a contradiction. A contradiction has no solution.    
    identityAn equation that is true for any value of the variable is called an Identity. The solution of an identity is all real numbers.    
    linear equationA linear equation is an equation in one variable that can be written, where \(a\) and \(b\) are real numbers and \(a≠0\), as \(ax+b=0\).    
    solution of an equationA solution of an equation is a value of a variable that makes a true statement when substituted into the equation.    
    boundary lineThe line with equation \(Ax+By=C\) is the boundary line that separates the region where \(Ax+By>C\) from the region where \(Ax+By<C\).    
    domain of a relationThe domain of a relation is all the \(x\)-values in the ordered pairs of the relation.    
    functionA function is a relation that assigns to each element in its domain exactly one element in the range.    
    horizontal lineA horizontal line is the graph of an equation of the form \(y=b\). The line passes through the y-axis at \((0,b)\).    
    intercepts of a lineThe points where a line crosses the \(x\)-axis and the \(y\)-axis are called the intercepts of the line.    
    linear equationAn equation of the form \(Ax+By=C\), where \(A\) and \(B\) are not both zero, is called a linear equation in two variables.    
    linear inequalityA linear inequality is an inequality that can be written in one of the following forms: \(Ax+By>C\), \(Ax+By≥C\), \(Ax+By<C\), or \(Ax+By≤C\), where \(A\) and \(B\) are not both zero.    
    mappingA mapping is sometimes used to show a relation. The arrows show the pairing of the elements of the domain with the elements of the range.    
    ordered pairAn ordered pair, \((x,y)\) gives the coordinates of a point in a rectangular coordinate system. The first number is the \(x\)-coordinate. The second number is the \(y\)-coordinate.    
    originThe point \((0,0)\) is called the origin. It is the point where the \(x\)-axis and \(y\)-axis intersect.    
    parallel linesParallel lines are lines in the same plane that do not intersect.    
    perpendicular linesPerpendicular lines are lines in the same plane that form a right angle.    
    point-slope formThe point-slope form of an equation of a line with slope \(m\) and containing the point \((x_1,y_1)\) is \(y−y_1=m(x−x_1)\).    
    range of a relationThe range of a relation is all the \(y\)-values in the ordered pairs of the relation.    
    relationA relation is any set of ordered pairs, \((x,y)\). All the \(x\)-values in the ordered pairs together make up the domain. All the \(y\)-values in the ordered pairs together make up the range.    
    solution of a linear equation in two variablesAn ordered pair \((x,y)\) is a solution of the linear equation \(Ax+By=C\), if the equation is a true statement when the \(x\)- and \(y\)-values of the ordered pair are substituted into the equation.    
    solution to a linear inequalityAn ordered pair \((x,y)\) is a solution to a linear inequality if the inequality is true when we substitute the values of \(x\) and \(y\).    
    standard form of a linear equationA linear equation is in standard form when it is written \(Ax+By=C\).    
    vertical lineA vertical line is the graph of an equation of the form \(x=a\). The line passes through the \(x\)-axis at \((𝑎,0)\).    
    break-even pointThe point at which the revenue equals the costs is the break-even point; \(C(x)=R(x)\).    
    coincident linesCoincident lines have the same slope and same \(y\)-intercept.    
    complementary anglesTwo angles are complementary if the sum of the measures of their angles is \(90\) degrees.    
    consistent and inconsistent systemsConsistent system of equations is a system of equations with at least one solution; inconsistent system of equations is a system of equations with no solution.    
    cost functionThe cost function is the cost to manufacture each unit times \(x\), the number of units manufactured, plus the fixed costs; \(C(x) = (\text{cost per unit})x+ \text{fixed costs}\).    
    determinantEach square matrix has a real number associated with it called its determinant.    
    matrixA matrix is a rectangular array of numbers arranged in rows and columns.    
    minor of an entry in a \(3×3\) determinantThe minor of an entry in a \(3×3\) determinant is the \(2×2\) determinant found by eliminating the row and column in the \(3×3\) determinant that contains the entry.    
    revenueThe revenue is the selling price of each unit times \(x\), the number of units sold; \(R(x) = (\text{selling price per unit})x\).    
    row-echelon formA matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a \(1\) and all entries below the diagonal are zeros.    
    solutions of a system of equationsSolutions of a system of equations are the values of the variables that make all the equations true; solution is represented by an ordered pair \((x,y)\).    
    solutions of a system of linear equations with three variablesThe solutions of a system of equations are the values of the variables that make all the equations true; a solution is represented by an ordered triple \((x,y,z)\).    
    square matrixA square matrix is a matrix with the same number of rows and columns.    
    supplementary anglesTwo angles are supplementary if the sum of the measures of their angles is \(180\) degrees.    
    system of linear equationsWhen two or more linear equations are grouped together, they form a system of linear equations.    
    system of linear inequalitiesTwo or more linear inequalities grouped together form a system of linear inequalities.    
    binomialA binomial is a polynomial with exactly two terms.    
    conjugate pairA conjugate pair is two binomials of the form \((a−b), (a+b)\). The pair of binomials each have the same first term and the same last term, but one binomial is a sum and the other is a difference.    
    degree of a constantThe degree of any constant is \(0\).    
    degree of a polynomialThe degree of a polynomial is the highest degree of all its terms.    
    degree of a termThe degree of a term is the sum of the exponents of its variables.    
    monomialA monomial is an algebraic expression with one term. A monomial in one variable is a term of the form \(ax^m\), where \(a\) is a constant and \(m\) is a whole number.    
    polynomialA monomial or two or more monomials combined by addition or subtraction is a polynomial.    
    polynomial functionA polynomial function is a function whose range values are defined by a polynomial.    
    Power PropertyAccording to the Power Property, \(a\) to the \(m\) to the \(n\) equals \(a\) to the \(m\) times \(n\).    
    Product PropertyAccording to the Product Property, \(a\) to the \(m\) times \(a\) to the \(n\) equals \(a\) to the \(m\) plus \(n\).    
    Product to a PowerAccording to the Product to a Power Property, \(a\) times \(b\) in parentheses to the \(m\) equals \(a\) to the \(m\) times \(b\) to the \(m\).    
    Properties of Negative ExponentsAccording to the Properties of Negative Exponents, \(a\) to the negative \(n\) equals \(1\) divided by \(a\) to the \(n\) and \(1\) divided by \(a\) to the negative \(n\) equals \(a\) to the \(n\).    
    Quotient PropertyAccording to the Quotient Property, \(a\) to the \(m\) divided by \(a\) to the \(n\) equals \(a\) to the \(m\) minus \(n\) as long as \(a\) is not zero.    
    Quotient to a Negative ExponentRaising a quotient to a negative exponent occurs when \(a\) divided by \(b\) in parentheses to the power of negative \(n\) equals \(b\) divided by \(a\) in parentheses to the power of \(n\).    
    Quotient to a Power PropertyAccording to the Quotient to a Power Property, \(a\) divided by \(b\) in parentheses to the power of \(m\) is equal to \(a\) to the \(m\) divided by \(b\) to the \(m\) as long as \(b\) is not zero.    
    standard form of a polynomialA polynomial is in standard form when the terms of a polynomial are written in descending order of degrees.    
    trinomialA trinomial is a polynomial with exactly three terms.    
    Zero Exponent PropertyAccording to the Zero Exponent Property, \(a\) to the zero is \(1\) as long as \(a\) is not zero.    
    degree of the polynomial equationThe degree of the polynomial equation is the degree of the polynomial.    
    factoringSplitting a product into factors is called factoring.    
    greatest common factorThe greatest common factor (GCF) of two or more expressions is the largest expression that is a factor of all the expressions.    
    polynomial equationA polynomial equation is an equation that contains a polynomial expression.    
    quadratic equationPolynomial equations of degree two are called quadratic equations.    
    zero of the functionA value of \(x\) where the function is \(0\), is called a zero of the function.    
    Zero Product PropertyThe Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero.    
    complex rational expressionA complex rational expression is a rational expression in which the numerator and/or denominator contains a rational expression.    
    critical point of a rational inequalityThe critical point of a rational inequality is a number which makes the rational expression zero or undefined.    
    extraneous solution to a rational equationAn extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined.    
    proportionWhen two rational expressions are equal, the equation relating them is called a proportion.    
    rational equationA rational equation is an equation that contains a rational expression.    
    rational expressionA rational expression is an expression of the form \(\frac{p}{q}\), where \(p\) and \(q\) are polynomials and \(q≠0\).    
    rational functionA rational function is a function of the form \(R(x)=\frac{p(x)}{q(x)}\) where \(p(x)\) and \(q(x)\) are polynomial functions and \(q(x)\) is not zero.    
    rational inequalityA rational inequality is an inequality that contains a rational expression.    
    similar figuresTwo figures are similar if the measures of their corresponding angles are equal and their corresponding sides have the same ratio.    
    simplified rational expressionA simplified rational expression has no common factors, other than \(1\), in its numerator and denominator.    
    complex conjugate pairA complex conjugate pair is of the form \(a+bi, a-bi\)    
    complex numberA complex number is of the form \(a+bi\), where \(a\) and \(b\) are real numbers. We call \(a\) the real part and \(b\) the imaginary part.    
    complex number systemThe complex number system is made up of both the real numbers and the imaginary numbers.    
    imaginary unitThe imaginary unit \(i\) is the number whose square is \(–1\). \(i^2 = -1\) or \(i=\sqrt{-1}\).    
    like radicalsLike radicals are radical expressions with the same index and the same radicand.    
    radical equationAn equation in which a variable is in the radicand of a radical expression is called a radical equation.    
    radical functionA radical function is a function that is defined by a radical expression.    
    rationalizing the denominatorRationalizing the denominator is the process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer.    
    square of a numberIf \(n^2=m\), then \(m\) is the square of \(n\).    
    square root of a numberIf \(n^2=m\), then \(n\) is a square root of \(m\).    
    standard formA complex number is in standard form when written as \(a+bi\), where \(a\), \(b\) are real numbers.    
    discriminantIn the Quadratic Formula, \(x=\frac{-b±\sqrt{b^2-4ac}}{2a}\), the quantity \(b^2-4ac\) is called the discriminant.    
    quadratic functionA quadratic function, where \(a\), \(b\), and \(c\) are real numbers and \(a≠0\), is a function of the form \(f(x)=ax^2+bx+c\).    
    quadratic inequalityA quadratic inequality is an inequality that contains a quadratic expression.    
    asymptoteA line which a graph of a function approaches closely but never touches.    
    common logarithmic functionThe function \(f(x)=\log{x}\) is the common logarithmic function with base10, where \(x>0\). \[y=\log{x} \text{ is equivalent to } x=10^y\]    
    exponential functionAn exponential function, where \(a>0\) and \(a≠1\), is a function of the form \(f(x)=a^x\).    
    logarithmic functionThe function \(f(x)=\log_a{x}\) is the logarithmic function with base \(a\), where \(a>0\), \(x>0\), and \(a≠1\). \[y=\log_a{x} \text{ is equivalent to } x=a^y\]    
    natural baseThe number \(e\) is defined as the value of \((1+\frac{1}{n})^n\), as \(n\) gets larger and larger. We say, as \(n\) increases without bound, \(e≈2.718281827...\)    
    natural exponential functionThe natural exponential function is an exponential function whose base is \(e\): \(f(x)=e^x\). The domain is \((−∞,∞)\) and the range is \((0,∞)\).    
    natural logarithmic functionThe function \(f(x)=\ln(x)\) is the natural logarithmic function with base \(e\), where \(x>0\). \[y=\ln{x} \text{ is equivalent to } x=e^y\]    
    one-to-one functionA function is one-to-one if each value in the range has exactly one element in the domain. For each ordered pair in the function, each \(y\)-value is matched with only one \(x\)-value.    
    circleA circle is all points in a plane that are a fixed distance from a fixed point in the plane.    
    ellipseAn ellipse is all points in a plane where the sum of the distances from two fixed points is constant.    
    hyperbolaA hyperbola is defined as all points in a plane where the difference of their distances from two fixed points is constant.    
    parabolaA parabola is all points in a plane that are the same distance from a fixed point and a fixed line.    
    system of nonlinear equationsA system of nonlinear equations is a system where at least one of the equations is not linear.    
    annuityAn annuity is an investment that is a sequence of equal periodic deposits.    
    arithmetic sequenceAn arithmetic sequence is a sequence where the difference between consecutive terms is constant.    
    common differenceThe difference between consecutive terms in an arithmetic sequence, \(a_n−a_{n−1}\), is \(d\), the common difference, for \(n\) greater than or equal to two.    
    common ratioThe ratio between consecutive terms in a geometric sequence, \(\frac{a_n}{a_{n−1}}\), is \(r\), the common ratio, where \(n\) is greater than or equal to two.    
    finite sequenceA sequence with a domain that is limited to a finite number of counting numbers.    
    general term of a sequenceThe general term of the sequence is the formula for writing the \(n\)th term of the sequence. The \(n\)th term of the sequence, \(a_n\), is the term in the \(n\)th position where \(n\) is a value in the domain.    
    geometric sequenceA geometric sequence is a sequence where the ratio between consecutive terms is always the same    
    infinite geometric seriesAn infinite geometric series is an infinite sum infinite geometric sequence.    
    infinite sequenceA sequence whose domain is all counting numbers and there is an infinite number of counting numbers.    
    partial sumWhen we add a finite number of terms of a sequence, we call the sum a partial sum.    
    sequenceA sequence is a function whose domain is the counting numbers.    
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