Algebra 2 Definitions

Deductive Reasoning

the process of applying rules - mathematics is this type of reasoning because there are mathematical rules yet to be discovered

Inductive Reasoning

the process of discovering rules - scientific endeavors are this type of reasoning; discovery of new mathematical relationships requires this type of reasoning

Numeral

a symbol or symbols used to express the idea of a number

Abstract

dealing with properties and ideas of things, just because they are things, regardless of feelings, emotions, and sensations we might connect with them; an idea that can describe an infinite number of concrete objects

Concrete

in mathematics, a word used to describe real objects - abstract ideas are always based on real things

Digit

any of the Hindu-Arabic numerals 1-9 and 0

Counting (Natural) Numbers

numbers used to count objects

Whole Numbers

counting numbers and the number 0

Integers

whole numbers and all negative numbers

Real Numbers

any number used to describe a positive or negative number; includes all integers, fractions, and decimals

Trichotomy Axiom

for any two real numbers a and b, exactly one of the following is true: a < b, a = b, a > b

Radical Sign

developed by Leonhard Euler in the 1700’s as a way to simplify the writing of factorial exponents

Square Root

if x is greater than 0, then the square root of x is the unique positive real number such that the square root of x squared is equal to ex

Imaginary Numbers

the result of taking the square root of a negative number

Complex Numbers

numbers that have a real part and an imaginary part, represented by a + bi, where i is Euler’s notation (the square root of -1)

X to the Power of 0 Equals 1

anything except 0 raised to the power of 0 equals 1

X to the -n Equals 1 Over X to the Power of n

anything except 0 raised to a negative power equals 1 over anything raised to the opposite of the power

Ratio

the size of one thing relative to another

Rational Number

a number that can be written as a fraction of integers

Irrational Number

a number that cannot be written as a fraction of integers

Fraction

part of a whole - a numerical quantity that indicates division of one number by another

Prime Number

a number that is only divisible by itself and 1

Numerator

the top value in a fraction

Denominator

the bottom value in a fraction

Logarithm

in any exponential relationship of the form N equals b to the power of L, the exponent, L can be referred to as this of N

Proportional

having a constant ratio

Analogy

resemblance in some particulars between things otherwise unlike; similar

Algebra

a generalization of arithmetic, where letters representing numbers are combined according to the rules of arithmetic, often to solve for an unknown value

Polynomial in One Unknown

one term, or a sum of individual terms in the form ax to the power of n, where a is a real number, x is an unknown quantity, and n is a whole number

Polynomial in More Than One Unknown

one term, or a sum of individual terms of the form ax to the power of n times y to the power of m times z to the power of p, where a is a real number, x, y, z, etc. are unknown quantities, and n, m, p, etc. are whole numbers

Quadratic Polynomial

a special type of trinomial of the for ax squared plus bx plus c

Degree of a Polynomial

for a polynomial of one variable, this refers to the largest exponent of that variable

Root of a Polynomial

for a polynomial in one variable, this is a number for which the polynomial equals zero

Point

that which has no part - its location is represented by a dot

Line

a widthless length - its location is represented on paper by using a pencil and straightedge

Line Segment

a line with a start point and end point

Plane

a flat surface having length and width only

Ray

a line with a starting point but no end point

Angle

two rays in the same plane that share a common starting point and do not overlap

Right Angle

formed when two lines or line segments intersect, forming four adjacent angles that are equal to each other

Obtuse Angle

an angle greater than a right angle

Acute Angle

an angle less than a right angle

Parallel

two lines that never intersect

Transversal

a line that intersects two or more parallel lines and is not perpendicular to those lines

Circle

a plane figure contained by one line called the circumference, such that all the straight line segments falling upon it are equal to each other

Diameter

any line segment drawn through the center of a circle and terminated in both directions by the circumference

Radius

half of the diameter of a circle

Polygon

simple, closed, coplanar geometric figures whose sides are straight lines

Regular Polygon

a polygon with sides of equal measure, and interior angles of equal measure

Vertex

where two polygon sides, or two angle rays, meet

Convex Polygon

a polygon that has no indentations

Concave Polygon

a polygon that has at least one indentation

Equilateral Triangle

all sides are congruent

Isosceles Triangle

two sides are congruent

Scalene Triangle

no sides are congruent

Right Triangle

has one right angle

Obtuse Triangle

has one angle greater than a right angle

Acute Triangle

has three acute angles

Parallelogram

has two pairs of parallel sides

Trapezoid

has one pair of parallel sides

Rectangle

a parallelogram with four right angles

Rhombus

an equilateral parallelogram

Square

a rhombus with four right angles

Polyhedron

a three-dimensional figure that is formed by four or more polygon that intersect only at their edges

Cone

a three-dimensional figure with a circular base and a lateral surface that comes to a point

Cylinder

a three-dimensional figure with two parallel circular bases connected by a lateral surface

Sphere

a three-dimensional figure which is the set of points a given distance from a given point called the center

Axiom

a self-evident statement about something obvious, normally common to all sciences

Postulate

a construction (drawing) of something, normally common to a particular science, that may not be obvious - assumed to be true without proof

Q.E.D.

acronym for the Latin phrase quod erat demonstrandum, “that which had to be demonstrated” - used to signal the end of a mathematical proof

Inscribed Angle

an angle whose vertex touches a circle, forming an intercepted arc on the opposite side

Minor Arc

the smaller arc formed outside of the inscribed angle

Major Arc

the remaining circumference of the circle not taken up by the minor arc

Chord

a line segment that connects any two points on a circle

Secant

a line that intersects a circle at two points (different than a chord because it is a line, rather than a line segment)

Tangent

a line in the same plane as the circle and touching the circle only once

Analytical Geometry

where algebra and geometry meet on the coordinate plane

Ordered Pair

a pair of numbers, written in a specific order; used to identify the location of a point on a coordinate plane

Cartesian Coordinate System

a plane containing a horizontal x-axis and a vertical y-axis; used to graph ordered pairs, functions, experimental data, etc.

Slope

a ratio expressing the change in the dependent variable (y) with respect to the independent variable (x)

X-Intercept

the location where a function crosses the x-axis

Y-Intercept

the location where a function crosses the y-axis

Function

a relationship where the output (y) depends on the input (x); each input (domain) maps to one and only one value in the output (range)

Relation

like a function, except the input (x) can map to more than one output (y)

Power Function

a function of the form f(x) = ax to the power of n, where n is a real number