abscissa
The first element in a coordinate pair. When graphed in the coordinate plane, it is the distance from the y-axis. Frequently called the x coordinate.
absolute value
The distance of a number from zero; the positive value of a number.
acid
Traditionally considered any chemical compound that, when dissolved in water, gives a solution with a hydrogen ion activity greater than in pure water (a pH less than ).
acute angle
A positive angle measuring less than degrees.
acute triangle
A triangle with all angles measuring less than degrees.
Calculating a sum by adding two or more numbers.
To solve a system of equations with the addition method, follow these steps:
1. Rewrite both equations in standard form if they aren’t in it already.
2. Multiply the terms of the equations by some constant that will cause the first variable to have the same coefficient in both equations, but with opposite signs, i.e., and (both have , but one of them has the opposite sign).
3. Add the equations together. The first variable will cancel out, leaving you with the second variable.
4. Solve the new equation for this variable.
5. Substitute the value you got in step 4 back into either one of the original equations, and solve for the remaining variable.
6. Plug both values back into the original equations and check the solutions.
If , , and are real numbers and , then: .
added to any number , is . Example: .
The additive inverse of any number is the number that gives zero when added to . Example: the additive inverse of is .
Two angles that share both a side and a vertex.
algebra
A branch of mathematics in which variables are substituted for unknown values to solve a particular problem.
algorithm
A step-by-step procedure for carrying out computation.
alternate angles
Two angles that are in opposite locations when lines are cut by a transversal.
altitude
Length from the uppermost point of a triangle to the line opposite.
angle
The union of two rays with a common endpoint, called the vertex.
anion
An ion with more electrons than protons, giving it a net negative charge.
annulus
The portion of a plane bounded by two concentric circles in the plane.
antiderivative
An antiderivative of a function is a function whose derivative is equal to . Example: .
approximate
Nearly correct or exact.
arc
A portion of the circumference of a circle.
area
The number of square units covering a shape or figure.
area of a circle
The area of a circle can be found with the formula , where is the radius.
area of a parallelogram
The area of a parallelogram can be found with formula , where is the base, and is the height.
area of a rectangle
The area of a rectangle can be found with formula , where is the length, and is the width.
area of a square
The area of a square can be found with formula , where is the side.
area of a trapezoid
The area of a trapezoid can be found with the formula , where is the maximum height, and and represent the bases of the trapezoid.
area of a triangle
The area of a triangle can be found with the formula , where is the base and is the height.
argument
The independent variable or expression of a function. Example: , .
arithmetic
Method of computing using addition, subtraction, multiplication, or division.
arithmetic sequence
A sequence with the difference between two consecutive terms constant. The difference is called the common difference.
To see how to calculate the sum of an arithmetic sequence, enter 'arithmetic sequence sum' into Mathway.
arithmetic sequence sum
To find the sum of the first terms of an arithmetic sequence, use the formula:
array
A set of numbers that will follow a specific pattern. An orderly arrangement often in rows, columns or a matrix.
associative property
When performing an operation on three or more numbers, the result is unchanged by the way the numbers are grouped.
associative property of multiplication
asymptote
On a graph, a line which is approached by a curve but never reached.
atom
A basic unit of matter consisting of a dense, central nucleus surrounded by a cloud of negatively charged electrons.
atomic mass
The mass of an atom, most often expressed in unified atomic mass units.
atomic number
The number of protons found in the nucleus of an atom and therefore identical to the charge number of the nucleus.
attribute
A characteristic to describe an object usually within a pattern. The attribute usually refers to the shape, size, or color.
average
A number that represents the characteristics of a data set, calculated by adding a group of numbers then dividing by the number of elements in that group.
axiom
A basic assumption about a mathematical system from which theorems can be deduced. For example, the system could be the points and lines in the plane. Then an axiom would be that given any two distinct points in the plane, there is a unique line through them.
axis
The horizontal and vertical lines that form the quadrants of the coordinate plane. The horizontal axis is usually called the x-axis, the vertical axis is usually called the y-axis.
axis of symmetry
A line that passes through a figure in such a way that the part of the figure on one side of the line is a mirror reflection of the part on the other side of the line.
bar graph
A visual representation of horizontal and vertical bars or lines to represent data.
base
1. The bottom of a plane figure or three-dimensional figure.
2. The number that is raised to various powers to generate the principal counting units of a number system.
3. An aqueous substance that can accept hydrogen ions.
base 10
The numbering system in common use, in which each place to the left or right of the decimal represents a power of .
bell curve
The shape of the graph that indicates the normal distribution.
benchmark
Point of reference used in estimation.
binomial
A polynomial with two terms. Example: .
binomial distribution
In probability, a binomial distribution gives the probabilities of outcomes (or outcomes ) in independent trials for a two-outcome experiment in which the possible outcomes are denoted and .
binomial theorem
In mathematics, a theorem that specifies the complete expansion of a binomial raised to any positive integer power.
, where is a positive integer.
bisect
To divide into two congruent parts.
box-and-whiskers plot
A type of graph used in data management particularly useful in showing the spread of the distribution of the data.
broken-line graph
A type of graph used in data management where the data points are joined by line segments.
calculate
To compute or simplify.
calculator
A machine used for computation.
calculus
The branch of mathematics involving derivatives and integrals. The study of motion in which changing values are studied.
capacity
The amount a container holds.
cardinal number
A whole number, used to answer the question how many?
Cartesian coordinates
A system in which points on a plane are identified by an ordered pair of numbers, representing the distances to two or three perpendicular axes.
cation
An ion with more protons than electrons.
Celsius
A temperature scale in which water freezes at and boils at .
census
Information gathered from all people in a population.
center of rotation
The point around which an object is rotated.
centi-
In the metric system, a prefix meaning hundredth.
central angle
An angle that has its vertex at the center of a circle.
chain rule
Used to differentiate composite functions. The chain rule is .
change of base formula
If , , and are positive, , and , then: .
chemical element
A pure chemical substance consisting of one type of atom distinguished by its atomic number, which is the number of protons in its nucleus.
chemical reaction
A process that leads to the transformation of one set of chemical substances to another.
chord
A line segment that connects two points on a curve.
circle
A figure whose points are all equidistant from a fixed point.
circle graph
A pictorial way of displaying how an entire thing, represented by the circle's interior, is distributed.
circumference
The distance around a circle.
circumference of a circle
The circumference can be found with the formula , where is the radius of the circle.
closed curve
A string of connected points in which the beginning of the string joins the end of the string.
coefficient
A constant that multiplies a variable.
cofactor
Defined for an matrix at an index in terms of the minor at that index as where the minor is .
collinear
Points are collinear if they lie on the same line.
combination
A selection in which order has no importance.
combine
To join, or bring together.
combined gas law
A gas law which combines Charles's law, Boyle's law, and Gay-Lussac's law.
commission
Earnings based on the amount of total sales.
common denominator
A denominator shared by two or more fractions.
common factor
A factor of two or more numbers.
common multiple
A multiple of two or more numbers.
commutative property
The order of numbers in a calculation does not affect the result.
commutative property of multiplication
compass
An instrument used for drawing circles, describing circles, or measuring distances. Consists of two hinged, movable legs.
compatible numbers
Numbers that are easy to manipulate mentally. Example: , .
compensation
Adjusting an estimated answer up or down to more closely approximate the value.
complement
The difference between a right angle and the angle.
complement set
A set whose elements do not belong to a given set.
complementary angles
Two angles whose sum is .
complex numbers
Numbers that have the form where and are real numbers and satisfies the equation .
composite
A natural number that is not prime.
compound bar graph
A bar graph that compares two or more quantities simultaneously.
compound event
The outcome of a probability experiment that involves more than one object. Example: when you roll two dice and the result is a on one and a on the other, this is a compound event.
compound inequality
Two or more inequalities that may have a common solution.
compound interest
The compound interest can be found with the formula: .
concave polygon
A polygon with at least one interior angle with measure greater than .
concentration
The measure of how much of a given substance there is mixed with another substance.
concentric
With reference to circles, having the same center.
cone
A three-dimensional figure with a circular base and one vertex.
congruent
Angles or figures that have the same size and shape.
conic section
The section formed by the intersection of a plane and a cone.
conjecture
An educated guess.
consecutive
Following, in succession, without interruption.
consistent system
A system of equations that has at least one solution.
constant
A fixed value that does not change.
convex polygon
A polygon with each interior angle measuring less than .
coordinate
A number in an ordered pair that names the location of a point on the coordinate plane. The first number in the ordered pair is called the abscissa and the second number is the ordinate.
coordinate plane
The plane determined by a horizontal number line, called the x-axis, and a vertical number line, called the y-axis, intersecting at a point called the origin. Each point in the coordinate plane can be specified by an ordered pair of numbers.
coplanar
Points that lie within the same plane.
correlation
A type of relationship between two variables. Two variables may be related as a positive correlation, a negative correlation, or illustrate no correlation.
corresponding angles
Angles that have the same relative positions.
cosecant
In a right triangle, the ratio of the length of the hypotenuse to the length of the opposite side; the reciprocal of the sine.
cosine
In a right triangle, the ratio of the length of the adjacent side to the length of the hypotenuse.
cotangent
In a right triangle, the ratio of the length of the adjacent side to the length of the opposite side; the reciprocal of the tangent.
counting numbers
The natural numbers, or the numbers used to count.
counting principle
If a first event has outcomes and a second event has outcomes, then the first event followed by the second event has times outcomes.
Cramer's Rule
A theorem in linear algebra, which gives the solution of a system of linear equations in terms of determinants.
critical points
A critical point of a function of a real variable is any value in the domain where either the function is not differentiable or its derivative is .
cross multiply
In a proportion, to rewrite the equation so that the product of the means equals the product of the extremes.
cross product
A product found by multiplying the numerator of one fraction by the denominator of another fraction and the denominator of the first fraction by the numerator of the second.
cube
A solid figure with six square faces.
cube root
A number that when cubed, raised to the power of , gives the original number.
cubic
Having the shape of a cube. When referring to volume, describing in terms of the volume of a cube with the indicated length edge.
curve
The graphic representation of an algebraic equation; a connected set of points.
cylinder
A three-dimensional figure having two parallel bases that are congruent circles.
Dalton's law
States that the total pressure exerted by a gaseous mixture is equal to the sum of the partial pressures of each individual component in a gas mixture.
data
Gathered information.
decimal number
The numbers in the base number system, having one or more places to the right of a decimal point.
decimal point
A symbol, looking exactly like a period, used to separate the whole number part, on the left, from the fractional part, on the right.
degree
A unit of measure for angles equal to of a full circle.
denominator
The bottom part of a fraction.
density
Mass per unit volume of a substance.
dependent events
Two events in which the outcome of the second is influenced by the outcome of the first.
dependent system
The equations of a system are dependent if all the solutions of one equation are also solutions of the other equation.
depreciation
A decrease in value.
derivative
A measurement of how a function changes when the values of its inputs change.
diagonal
A line segment connecting two nonadjacent vertices in a polygon.
diagram
A figure, usually a line drawing, that illustrates a geometrical theorem.
diameter
A line segment joining two points on a circle and passing through the center of the circle.
difference
The result of subtracting two numbers.
difference of cubes
difference of squares
digit
The ten symbols , , , , , , , , , and . The number has three digits: , , and .
dilatation
The enlargement or reduction of a plane figure.
dimension
A facet, aspect, or side of an object.
directrix
A fixed line associated with a parabola.
discriminant
In algebra, the discriminant of a polynomial with real or complex coefficients is a certain expression in the coefficients of the polynomial which is equal to zero if and only if the polynomial has a multiple root, a root with multiplicity greater than one, in the complex numbers.
distance
Length, as between two points.
distributive property
divide
To perform the operation of division.
dividend
A number to be divided by another number. In , the dividend is .
divisible
Capable of being evenly divided by a number, without a remainder.
division
The process of dividing two numbers.
division properties
If is any real number, then the following is true:
and , where .
division property of equality
If , , and are real numbers and , , then: .
If and are real numbers, then: , where .
divisor
A number in which another number is divided by. In , the divisor is .
domain
The set of all -values that a function passes through. Also the left coordinate in a coordinate pair.
double
To multiply by ; determine twice as much.
eccentricity
Deviation of a curve from circularity.
edge
The line segment where two faces of a polyhedron meet.
electron
A subatomic particle that carries a negative electric charge.
element
eliminate
To remove, to get rid of.
ellipse
The set of all points in a plane such that the sum of the distances to two fixed points is a constant.
empirical formula
A simple expression of the relative numbers of each type of atom in it, or the simplest whole number ratio of atoms of each element present in a compound.
empty set
A set that contains no elements.
endpoint
On a ray, segment, arc, or vector, a point at which the curve begins or ends; a point which touches only one other point on the curve.
equals
To be the same in value (symbol: ).
equation
A mathematical statement that says two expressions have the same value; any number sentence with an .
equation of a circle
With center at : .
With center at : .
equation of a hyperbola
For a hyperbola centered at there are two different standard equations:
To make the hyperbola open left and right: .
To make the hyperbola open up and down: .
equation of an ellipse
For an ellipse centered at there are two different standard equations:
Major axis parallel to the -axis: , .
Major axis parallel to the -axis: , .
equidistant
The same distance.
equilateral
A figure containing all equal sides.
equilateral triangle
A triangle that has three equal sides.
equivalent
Two or more expressions that have the same value.
equivalent equations
Two equations with the same solutions.
equivalent fractions
Fractions that reduce to the same number.
error of measurement
The difference between an approximate measurement and the actual measure taken.
estimate
An approximate calculation of a value.
evaluate
To substitute number values into an expression.
even number
A natural number that is evenly divisible by .
event
In probability, a set of outcomes.
expanded notation
Method of writing numbers as the sum of powers of ten or as the sum of its units, tens, hundreds, ...
exponent
A number that indicates the operation of repeated multiplication.
exponential function
A function in which the base , the base of the natural logs, is raised to some power.
exponential growth
Exponential growth can be calculated with the formula: .
expression
A mathematical symbol, or combination of symbols, representing a value, or relation. Example: .
exterior angle of a polygon
The angle outside a polygon formed by extending one of its sides.
face
A flat surface of a three-dimensional figure.
factor
One of two or more expressions that are multiplied together to get a product.
factor tree
A diagram representing a systematic way of determining all the prime factors of a number.
factoring
To break an expression into its factors. For factoring strategies, enter 'factoring strategy' into Mathway.
factoring a trinomial
1. Write the trinomial in descending order of powers of the variable being used.
2. Factor out the GCF. If the first term is negative, factor out a as well.
3. Check to see if the trinomial is now factorable.
4. If it is not yet factorable, manipulate the trinomial to take the form of a factorable trinomial, then factor.
5. Check that the factorization is valid by multiplying the binomials to result in the original trinomial.
factoring strategy
1. Factor out the common factors.
2. If there’s an expression with two terms, try to identify it as one of these problem types, if possible:
a. Difference of two squares: .
b. Sum of two cubes: .
c. Difference of two cubes: .
3. If an expression has four or more terms, try to factor it by grouping.
4. Factor the expression until each factor is prime, and check your final results by multiplying.
Fahrenheit
Temperature scale in which water boils at and freezes at .
Fibonacci Sequence
A sequence whereby each number is the sum of the two numbers preceding it.
figure
Two-dimensional shapes are often referred to as figures.
finding the inverse
To find the inverse of a one-to-one function, do the following:
1. If the function is written in function notation, replace with .
2. Swap any in the function with , and vice versa, i.e., becomes .
3. Solve for , then replace with .
finite
Not infinite. Finite has an end.
flip
A reflection of a two-dimensional shape, a mirror image of a shape.
focus
Imaginary point used in parabolas, hyperbolas, and ellipses.
FOIL
A technique for distributing two binomials. The letters FOIL stand for First, Outer, Inner, Last. First means multiply the terms which occur first in each binomial, Outer means multiply the outermost terms in the product, Inner means multiply the innermost terms, and Last means multiply the terms which occur last in each binomial.
formula
An equation that states a rule or a fact.
fraction
A number used to name a part of a group or a whole. The number below the division line is the denominator, and the number above the division line is the numerator.
frequency
The number of times a particular item occurs in a data set.
frequency table
A data listing which also lists the frequencies of the data.
function
A set of ordered pairs where each first element is paired with one and only one second element and no element in either pair is without a partner.
Fundamental Theorem of Algebra
Every polynomial equation having complex coefficients and degree greater than or equal to has at least one complex root.
fundamental theorem of calculus
A theorem in calculus with two results.
Part 1:
Part 2:
gas
A state of matter, without a definite shape or volume, consisting of a collection of particles (molecules, atoms, ions, electrons) that are in more or less random motion.
GCF
Greatest Common Factor; the largest number that divides two or more numbers evenly.
geometric sequence
A sequence with the ratio between two consecutive terms constant. This ratio is called the common ratio.
To see how to calculate the sum of an geometric sequence, enter 'infinite geometric sum' into Mathway.
geometric sequence sum
To find the sum of the first terms of a geometric sequence, use the formula: , where .
For the formula that finds the sum of an infinite geometric sequence, enter 'infinite geometric sum' into Mathway.
geometry
The study of lines, angles, shapes and their properties. Geometry is concerned with physical shapes and the dimensions of the objects.
golden rectangle
A rectangle where the ratio of its length to its width is , about .
A unit of measure for angles equal to of a full circle.
gram
A unit of mass, .
graph
A visual representation of data.
graphing method
To solve a system of two equations by graphing, follow these steps:
First, graph each equation on the same coordinate plane.
Next, find the coordinates of all the points at which the two lines intersect on the graph, if any.
If the lines only intersect at one point, then that point is the only solution.
If the lines are not overlapping and are parallel to one another, then no solutions exist for this system of equations.
If the lines coincide, or are completely on top of each other, then there is an infinite amount of solutions for this system of two equations.
greatest common factor
The largest number that divides two or more numbers evenly.
greatest integer function
The function which produces the greatest integer less than or equal to the number operated upon.
grouping symbols
Parentheses, brackets, braces, or fraction bars used to group together terms of an expression.
half
Either of the two quantities or pieces created when something is divided into two equal pieces.
height
The dimension used to describe the length from lowest point to highest point; how tall something is.
hexagon
A polygon that has six angles and six sides.
histogram
A type of statistical graph that uses bars, where each bar represents a range of values and the data is continuous.
horizontal
A line with zero slope.
hyperbola
A curved line where the difference of the distances from imaginary points, foci, to each point on the curve is constant.
hypotenuse
The side opposite the right angle in a right triangle.
identity
A number that when operating with it on any other number leaves the number unchanged.
identity matrix
A square matrix with 's along the diagonal from upper left to lower right and 's in all other positions.
The sum of any number and is that number.
identity property of multiplication
The product of and any number is that number.
image
The result of a transformation on an object.
imaginary number
An even root of a negative number; the square root of is symbolized by .
implicit differentiation
An application of the chain rule allowing one to calculate the derivative of a function given implicitly.
improper fraction
A fraction with a numerator that is greater than the denominator.
inclusive
All numbers including the ends. Example: List the odd numbers from to , inclusive: , , , .
inconsistent system
A system of equations is inconsistent if it does not have a solution.
increase
independent events
Two events in which the outcome of the second is not affected by the outcome of the first.
independent system
The equations of a system are independent if they share only one solution - the point of intersection.
indeterminate form
In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression whose limit cannot be evaluated by substituting the limits of the subexpressions.
index
The superscript at the beginning of a radical sign indicating the root to be taken, or extracted.
inequality
A mathematical expression which shows that two quantities are not equal.
infinite geometric sum
if .
infinity
A limitless quantity.
inflection point
A point on a curve at which the concavity changes sign from plus to minus or from minus to plus. The curve changes from being concave upwards to concave downwards, or vice versa.
inscribed angle
An angle placed inside a circle with its vertex on the circle and whose sides contain chords of the circle.
inscribed polygon
A polygon placed inside a circle so that each vertex of the polygon touches the circle.
integer
A whole number in the set of numbers containing zero, the natural numbers, and all of the negatives of the natural numbers.
integral
Given a function of a real variable and an interval of the real line, the integral is equal to the area of a region in the xy-plane bounded by the graph of , the x-axis, and the vertical lines and , with areas below the x-axis being subtracted.
intercept
The x-intercept of a line or curve is the point where it crosses the x-axis, and the y-intercept of a line or curve is the point where it crosses the y-axis.
intercepted arc
The arc of a circle within an inscribed angle.
interest
Amount paid or received for the loaning of money or the borrowing of money.
interior angles of a polygon
Angles within a polygon formed by the intersection of two sides.
interpolation
A method for estimating values that lie between two known values.
intersect
With lines or curves, to cross or have a point in common.
intersecting lines
Lines that have only one point in common.
intersection
With sets, the operation that creates a new set containing only those elements common to the original sets.
interval
A set of values between two endpoints.
inverse
Opposite in effect. is the additive inverse of , because their sum is zero. is the multiplicative inverse of , because their product is . For rules on find the inverse of a function, enter 'finding the inverse' into Mathway.
inverse operations
Two operations that have the opposite effect, such as addition and subtraction.
ion
An atom or molecule where the total number of electrons is not equal to the total number of protons, giving it a net positive or negative electrical charge.
irrational number
A number that cannot be expressed as the ratio of two integers.
isosceles
A polygon with two sides equal in length.
isosceles triangle
A triangle with at least two equal sides.
kilometer
A unit of measure that equals meters.
knot
A curve formed by an interlacing piece of spring by joining the ends.
l'Hospital's Rule
Rule that uses derivatives to help compute limits with indeterminate forms.
LCD
Least Common Denominator; the smallest multiple of the denominators of two or more fractions.
LCM
Least Common Multiple; the smallest non-zero number that is a multiple of two or more numbers.
least common denominator
The smallest multiple of the denominators of two or more fractions.
least common multiple
The smallest non-zero number that is a multiple of two or more numbers.
length
Measure of distance; a dimension of a solid or rectangle.
like fractions
Fractions that have the same denominator.
like terms
Terms that have the same variables raised to the same exponent. Example: and .
limit
A number that a function approaches as the independent variable of the function approaches a given value.
line
A straight set of points that extends into infinity in both directions.
line of symmetry
Line that divides a geometric figure into two congruent portions.
line segment
Two points on a line, and all the points between those two points.
linear equation
An equation whose graph is a line, that is, an equation that has a degree of one. Example: .
To see how to solve linear equations, enter 'solving linear equations' into Mathway.
locus
A path of points.
logarithmic function
Rule that returns for each argument the exponent to which the base must be raised in order to get the argument; the inverse of the exponential function.
logic
The study of sound reasoning.
lowest terms
Simplest form; when the GCF of the numerator and the denominator of a fraction is .
major arc
The larger of two arcs created when a circle is intersected at two points.
mantissa
Nonintegral, decimal part of a logarithm.
mass
The amount of matter in a particle or object.
matrix
A rectangular array of numbers, algebraic symbols, or mathematical functions.
A square matrix obtained from a given square matrix and having the property that its product with the given matrix is equal to the determinant of the given matrix times the identity matrix.
matter
maximum
Largest.
mean
In a data set, the sum of all the data points, divided by the number of data points; average.
measure
Dimension, capacity.
median
The middle number in a data set when the data are put in order.
midpoint formula
minimum
Smallest.
minor
Defined for an matrix at an index as the determinant of the matrix obtained by deleting row and column from .
minor arc
The smaller of two arcs made by the two point intersection of a circle.
minuend
In subtraction, the number which is decreased.
minus
Subtract; decrease by; lessen by.
minute
A unit of measure for angles equal to of a degree.
mixed number
A number written as a whole number and a fraction.
mixture
When two or more different substances are mixed together but not combined chemically.
mode
The number, or numbers, that occurs most frequently in a set of data.
molar mass
The mass of one mole of a substance, chemical element or chemical compound.
mole
The amount of substance of a system that contains as many elemental entities (atoms, molecules, ions, electrons) as there are atoms in of carbon-12. A mole has atoms or molecules of the pure substance being measured.
molecule
A sufficiently stable, electrically neutral group of at least two atoms in a definite arrangement held together by very strong, covalent, chemical bonds.
monatomic ion
An ion consisting of one or more atoms of a single element.
monomial
A number, a variable or a product of numbers and variables.
multiple
A multiple of a number is the product of that number and any other whole number. Zero is a multiple of every number.
multiplication
The process of repeating additions of the same number.
multiplication of 0
The product of and any number , is . Example: .
multiplication property of equality
If , , and are real numbers and , then: .
If and are real numbers, then: .
multiplicative identity
added to any number , is . Example: .
multiplicative inverse
The reciprocal of a number. The product of a number and its reciprocal is . Example: .
multiply
To compute a product; to perform a multiplication.
mutually exclusive events
Two or more events that cannot occur at the same time.
natural logarithm
A logarithm that has as a base.
natural numbers
The counting numbers.
negative number
A real number that is less than zero.
net
A plane figure obtained by opening and flattening a 3-D object.
neutron
A subatomic particle with no net electric charge and a mass slightly larger than that of a proton.
norm
The mean or the average, an established pattern or form.
normal
Perpendicular.
nth root
The nth root of a number is the number needed to multiply by itself times in order to get that number.
nucleus
The collection of protons and neutrons in the center of an atom, also composed of subatomic matter.
number line
A line on which every point represents a real number, usually increasing in value from left to right.
numeral
A written symbol referring to a number.
numerator
The top part of a fraction.
numeric
Referring to a number or numbers.
oblique angle
An angle that is neither a right, acute or obtuse angle.
obtuse angle
An angle whose measure is greater than .
obtuse triangle
A triangle with an obtuse angle.
octagon
A polygon with sides.
odd number
A whole number that is not evenly divisible by .
odds
The ratio of the probability that an event will occur compared with the probability of it not occurring.
operation
Addition, subtraction, multiplication, and division are the basic arithmetic operations.
operator
The symbol that expresses the operation to be performed.
opposites
Two numbers that are located the same distance from on the number line but in opposite directions. The sum of opposite numbers is .
To find the orbital radius, in meters, of a satellite circling the Earth, use the formula:
, where is the universal gravitational constant, is the Earth’s mass, and is the satellite’s period.
order of operations
The order of operations is a set of rules that can be remembered through PEMDAS.
First, simplify anything that's in parentheses. Then perform the following operations in the order shown below from left to right:
1. Evaluate exponents and radicals in the function.
2. Do all the multiplication and division operations within the function. If there are fractions present, simplify the numerators and denominators separately, then attempt to simplify the fraction itself.
3. Do all addition and subtraction operations.
4. When all groupings have been simplified and removed, repeat steps 1-3 until the entire function is simplified completely.
ordered pair
Set of two numbers in which the order has an agreed-upon meaning, such as the Cartesian coordinates , where the first coordinate represents the horizontal position, and the second coordinate represents the vertical position.
ordinal number
A number used to indicate place or position within a set or group.
ordinate
The second element in a coordinate pair. When graphed in the coordinate plane, it is the distance from the -axis. Frequently called the coordinate.
origin
The point on a coordinate plane, where the x-axis and the y-axis intersect.
outcome
In probability, a possible result of an experiment.
oxidation state
An indicator of the degree of oxidation of an atom in a chemical compound.
parabola
Set of points equally distant from a focus and a directrix.
parallel
Two lines are parallel if they are in the same plane and never intersect.
parallelogram
A quadrilateral with opposite sides parallel.
partial fractions
The process of decomposing a single fraction into multiple fractional terms.
pentagon
A five-sided polygon.
perfect square
A whole number that is the square of an integer. Example: is a perfect square because .
perimeter
The sum of the lengths of the sides of a polygon.
perimeter of a rectangle
The perimeter of a rectangle can be found with the formula , where is the length and is the width.
perimeter of a square
The perimeter of a square can be found with the formula , where is the length of one of its sides.
perimeter of a triangle
The perimeter of a triangle can be found with the formula , where , , and represent the lengths of the 3 sides of the triangle.
periodic table
A tabular display of the chemical elements.
permutation
A way to arrange things in which order is important.
perpendicular
Two lines are perpendicular if the angle between them is .
pH
A measure of the acidity or basicity of a solution.
Pi
The ratio of the circumference of a circle to its diameter (symbol: ), equaling ...
plane
A flat surface that stretches into infinity.
plane of symmetry
A plane that divides a 3-D object into two parts, each a mirror image of the other.
plot
To draw or graph a point on a number line or on a coordinate plane.
plus
pOH
Sometimes used as a measure of the concentration of hydroxide ions, OH-, or alkalinity.
point
A location in a plane or in space, having no dimensions.
point-slope equation of a line
An equation of the form , where is the slope and is a point on the line.
point-slope form
An equation of the form , where is the slope and is a point on the line.
polar
Expressed in terms of distance (from a point called the pole) and angle (with a ray as the initial side of the angle).
polyatomic ion
A charged species, ion, composed of two or more atoms covalently bonded or of a metal complex that can be considered as acting as a single unit in the context of acid and base chemistry or in the formation of salts.
polygon
A closed plane figure made up of several line segments that are joined together.
polyhedron
A three-dimensional solid that is bounded by plane polygons.
polynomial
An algebraic expression consisting of one or more summed terms, each term consisting of a constant multiplier and one or more variables raised to integral powers.
polynomial equation
An equation of the form , where is a polynomial.
population
In statistics, population refers to the entire group about which data are being collected.
positive number
A real number greater than zero.
power
A number that indicates the operation of repeated multiplication.
power rule
Used to differentiate expressions like . The power rule is .
prime
A natural number which has exactly two distinct natural number divisors: and itself.
prime factorization
Calculation of all prime factors in a number.
principal
In business, the amount lent or borrowed.
prism
A geometric solid with two bases that are congruent, parallel polygons and all other faces are parallelograms.
probability
For an experiment, the total number of successful events divided by the total number of possible events.
product
The result of two numbers being multiplied together.
Product Rule
In calculus, the product rule, also called Leibniz's law, governs the differentiation of products of differentiable functions. It may be stated as: .
proper fraction
A fraction whose numerator is less than its denominator.
properties of exponential functions
The domain of is the interval .
The range is .
The x-axis is an asymptote of this graph.
The graph has a y-intercept of .
The graph passes through the point .
properties of fractions
If , , , , and are real numbers, and if there is nothing being divided by , then the following properties of fractions apply:
if and only if
and
properties of inequalities
1. Any real number can be added to (or subtracted from) both sides of an inequality, producing a new inequality with the same direction as the original one.
2. Both sides of an inequality can be multiplied (or divided) by a positive number to produce a new inequality with the same direction as the original one.
3. Both sides of an inequality can be multiplied (or divided) by a negative number to produce a new inequality that has the opposite direction of the original one.
properties of logarithms
If , , , and are positive numbers and , then the following properties apply:
, where
If , then
proportion
An equation of fractions in the form:
proportional
A statement of equality in which each member is a fraction.
proton
A subatomic particle with an electric charge of elementary charge.
protractor
A device for measuring angles.
pyramid
A three-dimensional figure that has a polygon for a base and all of the faces are triangles having a common vertex.
Pythagorean Theorem
The theorem that relates the three sides of a right triangle: , where is the longest side.
One of the quarters of the plane of the Cartesian coordinate system.
A polynomial equation of the second degree. The general form is , where . For a method of solving quadratic equations, enter 'solving quadratic equations' into Mathway.
A polygon with four sides.
To multiply or to be multiplied by .
qualitative
A general description of properties that cannot be written in numbers.
quantity
An amount; a number or expression having value.
quartic
A polynomial having a degree of .
quartile
Any one of the values in a frequency distribution that divides the distribution into four parts of equal frequency.
quintic
A polynomial having a degree of .
quotient
The answer to a division problem.
Quotient Rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist.
In angle measure, of a revolution.
The number under the inclusion bar of the radical sign.
The distance from the center to a point on a circle; the line segment from the center to a point on a circle.
random
A number chosen without definite aim, reason, or pattern.
range
The set of all y-values that a function passes through. Also the right coordinate in a coordinate pair.
In statistics, the difference between the largest and the smallest numbers in a data set.
rate
A ratio that compares different kinds of units.
ratio
A pair of numbers that compares different types of units.
rational exponents
An exponent containing a rational number. Examples: , .
rational expression
The quotient of two polynomials. For help with solving rational equations, enter 'solving rational equations' into Mathway.
rational number
A number that can be expressed as the ratio of two integers.
ray
Part of a line, containing one endpoint and extending to infinity in one direction.
reaction
See chemical reaction.
reciprocal
The number which, when multiplied times a particular fraction, gives a result of .
rectangle
reference angle
In trigonometry, an acute angle which may be used as a reference or to compute the trigonometric functions of non-acute angles.
reflection
A transformation resulting from a flip.
reflex angle
An angle whose measure is between and .
regular polygon
A polygon in which all the angles are equal and all of the sides are equal.
remainder
The portion of the dividend that is not evenly divisible by the divisor.
repeating decimal
A decimal in which the digits endlessly repeat a pattern.
rhombus
A parallelogram with four equal sides.
right angle
An angle whose measure is .
right triangle
A triangle that contains a right angle.
rise
The vertical change between two points used to determine the slope of a line.
root
The root of an equation is the same as the solution to the equation.
rotation
A transformation in which a figure is rotated through a given angle, about a point.
run
The horizontal change between two points used to determine the slope of a line.
sample
Refers to a representative portion of the population from which information is gathered.
sample space
For an experiment, the sample space includes all the possible outcomes.
scale drawing
A drawing that is a reduction or enlargement of the original.
scale factor
The ratio of a distance measured on a scale drawing to the corresponding distance measured on the actual object.
scalene triangle
A triangle with three unequal sides.
scattergram
A graph with points plotted on a coordinate plane.
scientific notation
A method for writing extremely large or small numbers compactly in which the number is shown as the product of two factors.
secant
Ratio of the hypotenuse to the adjacent side of a right angled triangle.
secant of circle
A line that intersects a circle in two points.
second
A unit of measure for angles equal to of a minute.
second derivative
Measures how the rate of change of a quantity is itself changing. For example, the slope represents the rate of change, and the rate of change of the slope is the second derivative.
secondary data
Data obtained indirectly from sources such as a book or computer database.
sector
An area between an arc and two radiuses of a circle. Sometimes referred to as a wedge.
segment
A piece of a line with two endpoints.
sequence
A set of numbers, called terms, arranged in some particular order.
set
A well-defined group of objects.
significant figures
The digits that affect the accuracy of a measurement.
similar
Two polygons are similar if their corresponding sides are proportional.
simplest form (lowest terms)
A fraction is in simplest form if both its numerator and denominator are whole numbers and their only common factor is .
simplified fraction
A fraction in simplest form.
simplifying
Reducing to lowest terms.
sine
In a right triangle, the ratio of the length of the side opposite the angle to the length of the hypotenuse.
skew lines
Lines that are not in the same plane and that do not intersect.
slope
The steepness of a line expressed as a ratio, using any two points on the line.
slope-intercept
An equation of the form , where is the slope and is the y-intercept.
solute
The substance that is dissolved in a solution.
solution
1. The value of a variable that makes an equation true.
2. In chemistry, a solution is a homogeneous mixture composed of only one phase.
solving a system of three equations
1. Choose two of the three equations and eliminate a variable from them with any method of elimination.
2. Choose a different pair of two equations and eliminate the same variable from step 1.
3. Take the two new equations made in steps 4 and 5 and eliminate another variable, leaving you with one remaining variable, and solve for that variable.
4. Plug that value into either one of the two equations made in steps 4 and 5 that contain only two variables, and solve for the second variable.
5. To find the value of the third variable, plug your other two values into any of the three original equations, and solve for the third variable.
6. Check the solution in all three of the original equations.
solving linear equations
1. Start by removing all sets of parentheses with the distributive property, then combine any like terms.
2. Isolate the variables to one side of the equation, with the use of addition and subtraction. Combine like terms, if any.
3. Make the coefficient of the variable equal to one with the use of multiplication and division.
4. At this point, the variable should be alone on one side of the equation, meaning that any numbers on the other side are solutions of the equation for .
5. Test your answer by substituting it back into the original equation and solve. If it’s correct, both sides will equal the same value.
One method of solving quadratic equations is by the factoring method, shown below:
1. Rewrite the equation in quadratic form if it is not already.
2. Factor the polynomial to make it easier to solve.
3. Set each factor equal to zero with the zero-factor property.
4. Solve each equation and check the solutions in the original equation.
solving rational equations
1. Factor all the denominators in the equation.
2. Multiply both sides of the equation by the LCD of all rational expressions in the equation.
3. Remove all parentheses with the distributive property, and reduce any common factors. You should be left with a linear or quadratic equation.
4. Solve the equation and check the solution in the original equation.
solving three equations with three variables
1. Choose two of the three equations and eliminate a variable from them with any method of elimination.
2. Choose a different pair of two equations and eliminate the same variable from step 1.
3. Take the two new equations made in steps 4 and 5 and eliminate another variable, leaving you with one remaining variable, and solve for that variable.
4. Plug that value into either one of the two equations made in steps 4 and 5 that contain only two variables, and solve for the second variable.
5. To find the value of the third variable, plug your other two values into any of the three original equations, and solve for the third variable.
6. Check the solution in all three of the original equations.
specific heat
The measure of the heat energy required to increase the temperature of a unit quantity of a substance by a certain temperature interval.
sphere
A three-dimensional figure with all points in space a fixed distance from a given point, called the center.
A computer generated arrangement of data in rows and columns.
square
A quadrilateral with four equal sides and four angles.
square root
The square root of is the number that, when multiplied by itself, gives the number .
standard deviation
A statistic that measures the dispersion of a sample.
standard notation
Decimal notation.
statistics
The science of collecting, organizing, and analyzing data.
stem-and-leaf plot
In statistics, a way of recording, organizing and displaying numerical data so that the original data remains intact.
stoichiometry
The calculation of quantitative, measurable, relationships of the reactants and products in a balanced chemical reaction.
straight angle
An angle that measures .
subset
A set that forms one part of a larger set.
substitution method
To solve a system of two equations (containing two variables per equation) with the substitution method, we follow these steps:
1. If neither equation is already solved for a variable, choose one of the equations and solve it for one of the variables of your choice.
We will call the equation you chose ‘equation1’, and the other one will be 'equation2'.
2. Take equation1 and substitute it into equation2 in all places where equation2 has the variable that you solved for in equation1, and solve for the remaining variable.
3. Take the value you got in step 2 and plug it into equation1, which should be solved for a variable, and solve.
4. You should now have a solution set for the system of equations. Plug both of these values back into your two original equations and confirm that they work.
subtraction
The process of finding the difference between two numbers.
subtraction property of equality
If , , and are real numbers and , then: .
sum
sum of cubes
superset
A set that consists of a collection of smaller subsets.
supplementary angles
Two angles are supplementary if their sum is .
surface area
For a three-dimensional figure, the sum of the areas of all the faces.
symmetry
A correspondence of parts.
system of equations
A collection of two or more equations with a same set of unknowns.
tangent
In a right triangle, the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.
term
Any expression written as a product or quotient. Example: , , or .
terminating decimal
A fraction whose decimal representation contains a finite number of digits.
tessellate
The repeated use of geometric figures to completely cover a plane without overlapping.
theoretical probability
Probability that is determined on the basis of reasoning, not through experimentation.
transformation
A change in the position, shape, or size of a geometric figure.
translation
A transformation, or change in position, resulting from a slide with no turn.
transversal
A line that intersects two other lines.
trapezoid
A quadrilateral that has exactly two sides parallel.
tree diagram
A diagram that shows outcomes of an experiment.
trend
The general drift or tendency in a set of data.
triangle
A three-sided polygon.
trigonometry
Study of triangles, the measurements of their parts and of angle functions and relations.
trinomial
A polynomial consisting of three terms.
uniform
All the same. Having the same size, texture, color, design, ...
union
A set containing each of the elements of the two sets which were united.
unit
A standard quantity used in measurement. Example: an inch is a unit of length, a centimeter is a unit of length, and a pound is a unit of weight.
unit circle
A circle with a radius of one.
unit price
Price per unit of measure.
unlike terms
Terms with different variables or the same variables raised to different exponents. Example: and
variable
A letter used to represent a number value in an expression or an equation.
vector
Quantity that has magnitude (length) and direction. It may be represented as a directed line segment.
Venn Diagram
A Venn diagram is often two circles that overlap. The overlapping part usually contains information that is pertinent to the labels on both sides of the Venn diagram.
vertex
The point on an angle where the two sides intersect.
vertical
Perpendicular to horizontal; up and down as opposed to left and right.
vertical angles
A pair of opposite angles that is formed by intersecting lines.
vertical line test
A way of testing a graphed relation to determine if it is a function.
vertically opposite angles
Two angles formed by the intersection of two lines. They share a common vertex but no sides or interior points.
voltage
Commonly used as a short name for electrical potential difference. Its corresponding SI unit is the volt, .
volume
A measurement of space, or capacity.
volume of a cone
The volume of a cone can be found with the formula , where is the radius of the base, and is the height of the cone.
volume of a cube
The volume of a cube can be found with the formula , where is the length of one side of the cube.
volume of a cylinder
The volume of a cylinder can be found with the formula , where is the radius of the base, and is its height.
volume of a pyramid
The volume of a pyramid can be found with the formula , where is the length of the base, is the width of the base, and is its height.
volume of a rectangular prism
The volume of a rectangular prism can be found with the formula , where is its length, is its width, and is its height.
volume of a sphere
The volume of a sphere can be found with the formula , where is the radius of the sphere.
weight
A measure of how heavy something is.
whole number
The set of positive integers and zero.
width
Measure of a, usually horizontal, distance.
x-axis
The horizontal axis in a Cartesian coordinate plane.
x-coordinate
The abscissa.
x-intercept
The value of at the point where a line or curve crosses the x-axis.
y-axis
The vertical axis in a Cartesian coordinate system.
y-coordinate
The ordinate.
y-intercept
The value of at the point where a curve crosses the y-axis.
z-score
The number of standard deviations a data point is from the mean.
zero
The additive identity; the number that, when added to another number , gives .
zero factorial
A factorial can be used to represent the possible number of permutations for an expression. In the case of zero factorial, there is only one possible permutation: Zero. .
zero property of multiplication
The product of zero and any number is zero.
zero-factor property
If and are real numbers, and , then , , or .
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