1.1 integers and operations

Cluster

A sample in which groups which share common characteristics are selected

circumference

The circumference of a circle is its outside edge. The word circumference also means the distance around the outside edge a circle.

degree

The degree of a monomial involving a single variable is the value of the variable's exponent. Notice that the degree of a constant is 0, since x0 = 1.

Comparison symbols include:

> (greater than) < (less than) (greater than or equal to) (less than or equal to) = (equal to) (not equal to)

Fermi Question

A Fermi Question is one which is posed so that there is no easy way to determine its exact answer. Instead, estimation is used to approximate the solution.

binomials

A binomial is a polynomial having exactly two terms.

composite

A composite number is an integer that can be factored. This means that it can be divided evenly by a positive integer other than 1 and itself. A positive integer that is not composite is called a prime number.

conversions

A conversion is a rule that changes one unit into another.

Cartesian coordinate system

A coordinate system formed by two number lines, one horizontal and one vertical, that intersect at the zero point of each line. The number lines are called axes and are usually labeled as the x-axis and y-axis. For this reason, a Cartesian coordinate system is often referred to as the xy-plane. It provides a method for naming points in that plane.

dependent variable

A dependent variable in a relationship is a variable whose value depends on that of another variable. It is graphed on the vertical axis, or y-axis, of the xy-plane.

derived unit

A derived unit is built from a combination of base units. For example, the unit "miles per hour" is derived from the base units "miles" and "hours."

difference of squares

A difference of squares is an expression (polynomial or otherwise) that contains two perfect squares, with one subtracted from the other.This type of expression can always be factored as follows: A2 - B2 = (A + B)(A - B)

difference of two cubes

A difference of two cubes is an expression that contains two perfect cubes, with one subtracted from the other. This type of expression can always be factored as follows: A3 - B3 = (A - B)(A2 + AB + B2)

box-and-whisker plot

A display that shows the minimum, first quartile, median, third quartile, and maximum of a set of data.

bimodal

A distribution that has two values or classes that are more common than the others.

factorial

A factorial is a product of a certain integer and all positive integers smaller than that integer.

descriptive statistics

A field of statistics that focuses on describing data that have already been collected, using graphs and numerical analysis.

inferential statistics

A field of statistics that uses sample data to make inferences about a larger population.

function

A function is a relationship between two quantities where one quantity, the dependent variable, depends on the other quantity, the independent variable. A function takes in values of the independent variable, called inputs, and gives back values of the dependent variable, called outputs. A function has exactly one output for each input.

line graph

A graph that uses line segments or curves to show how a quantity changes over time.

bar graph

A graphical display that uses bars to show frequencies of categorical, or qualitative, data.

histogram

A graphical display that uses bars to show frequencies of numerical data.

bar graph

A graphical display used to show the relative sizes of quantities under different conditions. Each bar represents a quantity, and the height of the bar shows the quantity's size. Bar graphs can help you visualize the information contained in a function.

half-plane

A half-plane is the graph of a linear inequality. If it is a strict inequality, ("less than" or "greater than", then the line is dashed to indicate that it is not included in the graph of the inequality. If it is not a strict inequality, ("less than or equal to" or "greater than or equal to"), then the line is solid.

kilowatt-hour

A kilowatt-hour is the amount of energy needed to provide 1000 watts of power for a period of an hour.

line graph

A line graph shows how one quantity changes as conditions change. It makes it easier to see the general patterns in the data it represents.

linear equation

A linear equation is any equation whose graph is a straight line.

linear polynomials

A linear polynomial is a polynomial of degree 1. A linear polynomial has a variable term and possibly a constant term.

mathematical sentence

A mathematical sentence is a comparison of two mathematical expressions. The sentence is called an equation if the two expressions are equal.

matrix

A matrix is a rectangular array of numbers providing a short-hand notation for representing and manipulating data. One way that a matrix is used is to represent the coefficients of a system of linear equations. Solving a system of equations using a matrix involves manipulating the rows of the matrix to get it into reduced form.

maximum

A maximum is a value at which a function or polynomial is larger than at any other nearby value. On a graph, a maximum looks like the top of a hill. An extreme point of a graph is either a maximum or a minimum.

minimum

A minimum is a value at which a function or polynomial is smaller than at any other nearby value. On a graph, a minimum looks like the bottom of a valley. An extreme point of a graph is either a maximum or a minimum.

expression

A mix of numbers, variables, and operations. It does not contain an equals sign or inequality sign. An expression with variables is often called an algebraic expression.

monomial

A monomial is a number, a power of a variable, or a product of the two. A monomial term has the form axn, where a is any number, and n is a non-negative integer. The number a is called the coefficient of the term and n is called the degree of the term.

constant

A number that does not change (unlike a variable).

order of magnitude

A number that represents the power of ten associated with the scale of a large or small number.

ordered pair

A pair of numbers representing the coordinates of a point in a Cartesian coordinate system. These numbers are written in a particular order and are enclosed in parentheses. By convention, the horizontal coordinate is the first number in the ordered pair and the vertical coordinate is the second number.

descending order

A polynomial is written in descending order if its terms are arranged with the exponents decreasing from left to right. You will find that writing polynomials in descending order can help you work with them more easily. This is especially true for doing division: polynomials need to be in descending order before you can divide them.

Convenience

A sample in which the most easily accessible subjects are reached.

Census

A sample that includes all members of a population.

exponent

A small raised number that tells you how many times to multiply the base number by itself.

bar graph

A tool used to show the relative sizes of quantities under different conditions. Each bar represents a quantity, and the height of the bar shows the quantity's size. Bar graphs can help you visualize the information contained in a function.

accurate

Accuracy is the degree to which a measured value is close to the expected or accepted true value.

elements

An element is an individual item of a set. For example, the elements of the set of months of the year are "January," "February," "March," etc. Elements are often grouped together in curly braces "{ }" to show that they belong to a set. For example, the set of positive even integers is usually written {2, 4, 6, 8, 10,...}. The ellipses (...) show that these numbers go on forever.

equation

An equation is a mathematical statement that has an equal sign. In an equation, the amounts or expressions on either side of the equal sign have the same value.

exponential expression

An exponential expression is an expression that involves an exponent. The expression may contain variables, which may be the base or the exponent.

independent variable

An independent variable in a relationship is a variable whose value determines that of another variable. It is graphed on the horizontal axis, or x-axis, of the xy-plane.

inequality

An inequality is a mathematical sentence that compares two unequal values. The symbols for an inequality are <, >, or ≠.

input-output diagram

An input-output diagram relates the inputs of a function to outputs. Input values of a function are shown on the left with arrows pointing to their corresponding outputs. The values on the left are elements of the function's domain, and the values on the right are elements of the function's range.

input-output table

An input-output table is a two-column table whose first column contains input values of a function and whose second column contains output values. The values in the first column are elements of the function's domain, and the values in the second column are elements of the function's range.

common multiple

An integer is a common multiple of two other integers if it is a multiple of both.

basic units

Basic units are those units which cannot be broken down into smaller units. They are the units on which a system is built.

canceling units

Canceling units involves identifying common units in the numerator and denominator of a fraction and removing them from a problem in order to convert from one unit to another.

complete the square

Completing the square is a method of solving quadratic equations. It involves adding a number to both sides of an equation to make one side a perfect square trinomial. The equation can then be solved by taking the square root of both sides and simplifying.

decimal

Decimal means based on powers of ten. For example the number 346 has 3 hundreds, 4 tens, and 6 ones. Hundreds, tens, and ones are all powers of ten.

equivalent

Equivalent fractions are fractions that have the same value.

factor it out

Factoring out a common factor from an expression is writing the expression as the product of a factor (common to all of its terms) and some other expression. This is equivalent to using the distributive property.

function notation

Function notation is used for those equations that define functions. The notation indicates that one quantity, the dependent variable, depends on another quantity, the independent variable. The independent variable is written in parentheses.

combinations

Groupings of objects selected from a larger group of elements without regard to their ordering.

direct variation

In a relationship of direct variation, one quantity is a constant multiple of another. If data from such a relationship is plotted on a Cartesian coordinate system, it will lie on a line that passes through the origin (0,0). An equation expressing direct variation has the form: y = mx, where m is any non-zero number.

leg

In an isosceles triangle, the two congruent sides are known as legs. In a right triangle, the legs are the sides adjacent to the right angle.

base

In exponential form, the base is the number on the bottom -- the number being multiplied.

exponent

In exponential form, the exponent is the small, raised number used to show repeated multiplication -- the number of times you multiply the base.

isolate the variable

Isolating the variable means getting the variable alone on one side of an equation or inequality in order to solve it. This is done by performing the appropriate reverse operations to both sides of the equation.

irrational numbers

Numbers that cannot be written in the form , where a and b are integers. Irrational numbers cannot be written as terminating decimals or repeating decimals

GCF

Out of the common factors shared by two numbers, the greatest common factor is the factor with the largest value.

empirical probability

Probability that is based on experimentation or testing. It is a result of completing a number of trials, counting successes among those trials, and calculating a probability based on those outcomes.

comparison symbols

Symbols used to compare two numbers or expressions.

British System of Units

The British System of units was developed based upon common measurements — like the length of one's foot or forearm. It was implemented in the US and is now used officially in only three countries in the world.

Division Property of Equality

The Division Property of Equality states that if you divide both sides of an equation by the same nonzero number, the sides will remain equal.

FOIL

The FOIL method helps you to organize the multiplication of each term of the first binomial by each term of the second. The letters stand for First, Outer, Inner, and Last, and describe the order in which you multiply the terms of the two binomials together.

Identity Property of Multiplication

The Identity Property of Multiplication states that the product of any number and one is that number. This property can also prove very useful when converting units, because it allows for multiplication of the original number by conversion ratios that are equal to 1.

Metric System

The Metric System is a system of measurement based on powers of 10. In this manner, it can use prefixes attached to base units to cover scales from the very small to the very large. It was first developed and adopted in France in the 18th Century.

Multiplication by One

The Rule of Multiplication by One states that when converting units, multiplying any measurement by a ratio that equals 1 will change the form of the measurement but not its value.

absolute value

The absolute value of a number is the distance between that number and zero on the number line. An absolute value is always positive, because it represents a distance. The symbol for absolute value is a pair of vertical bars surrounding the number value.

area

The area of a figure is the amount of space inside a flat shape. It is measured in square units, such as square inches, square centimeters, or square feet.

mean

The arithmetic average of all values in a data set.

axes

The axes in a Cartesian coordinate system are the two number lines, one horizontal and one vertical, that form the coordinate system. The axes are often called the x-axis (horizontal) and y-axis (vertical), but may be labeled differently for specific applications. The axes divide the coordinate system into four quadrants. The point where the axes intersect is called the origin.

best-fit line

The best-fit line for a collection of points is a line that best approximates the points. That is, the best-fit line is the line that is closest to all of the points.

coefficient

The coefficient of an expression is the real number in any expression formed by the product of one or more variables and that real number.

complex

The complex numbers are the set of all numbers of the form a + bi, where a and b are any real numbers, and i is the imaginary part. Complex numbers can sometimes be found when using the quadratic formula to solve quadratic equations. They are useful in many fields, including physics, astronomy, economics, and engineering.

diameter

The diameter of a circle is a chord that passes through the center of the circle and has endpoints on the circle. The term diameter can also refer to the length of the chord.

interquartile range (IQR)

The difference between the upper quartile (Q3) and the lower quartile (Q1) in a data set.

discriminant

The discriminant is the number under the radical sign in the quadratic formula. distance formula The formula that allows you to find the distance between two points in the xy-plane. For example, the distance between points (x2, y2) and (x1, y1) is shown below. _/(x2 - x1) 2+(y2 - y1) 2

domain

The domain of a function is the set of all values that the function will take as inputs. Graphically, the domain is the set of all x-values of points that appear in the function's graph.

equation of the line

The equation of a line is a mathematical expression that shows the relationship between the coordinates of every point on that line.

extreme value

The extreme values of a function occur at the points on the graph which have the largest or smallest value of F(x). These points are called the maximum and minimum.

graph of the polynomial

The graph of a polynomial is the graph of an equation where the right side is the polynomial. The graph of a polynomial equation is continuous--it doesn't break anywhere. It is also smooth--it doesn't have any sharp corners. Higher degree polynomials usually have more complex graphs.

graph

The graph of an equation is a line or curve in the xy-plane such that the coordinates of every point on the line or curve can be substituted for the appropriate variables, and keep the equation true. Each point on the graph is an element of the solution set of the equation.

hypotenuse

The hypotenuse of a right triangle is the side opposite the right angle. It is the longest side of the right triangle.

imaginary

The imaginary numbers are the set of all numbers of the form bi, where b is any real number and i is the imaginary part.

integers

The integers are the whole numbers and their opposites. They are the numbers {..., -3, -2, -1, 0, 1, 2, 3,...}. The ellipses (...) indicate that the integers go on forever in both directions. The positive integers, negative integers, and whole numbers are all subsets of the set of integers.

negative integers

The integers that are less than zero. They are the opposites of the positive integers.

least common multiple

The least common multiple of two numbers, abbreviated LCM, is the smallest positive common multiple of those numbers

median

The middle value in a data set. Half of the values fall below the median, and half fall above.

numerator

The number above the fraction bar in a fraction. It shows the number of parts of the whole that are being considered.

denominator

The number below the bar in a fraction. It shows the number of equal parts into which the whole is divided.

mode

The value, or class, that appears most often in a data set.

evaluate

To substitute a numeric value into a function and solve it.

negative reciprocals

Two numbers with the product of -1. If the slopes of two lines are negative reciprocals of each other, then the lines are perpendicular.