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a number representing part of a whole; numerator/denominator (number of pieces, intervals, or parts) / (number of pieces, intervals or parts in the whole)

In a fraction, something written in the form

, the number or expression that makes up the top of the fraction, ,is called the numerator.In a fraction, something written in the form

, the number or expression that makes up the bottom of the fraction, , is called the denominator.the least common multiple of a set of denominators; the least common multiple is the smallest multiple common to all the numbers (in the example below the least common multiple is 20)

The multiples of 4 are: 4, 8, 12, 16, **20**, 24, 28, 32, 36, 40, 44,...

The multiples of 5 are: 5, 10, 15, **20**, 25, 30, 35, 40, 45, 50,...

for positive exponents, the exponent of a number tells us how many times to use the base in a multiplication; for negative exponents, the exponent tell us how many times to divide 1 by the base; an exponent is usually written as a superscript to the right of the base; exponents are also called powers or indices

**Scientific notation** is a special way of writing numbers that is particularly convenient for numbers that are either very large or very small. A number written in scientific notation will be in the form

,

where

and is an integer.when working with decimals, moving the decimal point one place to the right represents division by ten; moving the decimal point one place to the left represents multiplication by 10; in this sense, decimal notation refers to a base-10 positional notation; to convert a decimal to a percent, multiply the decimal number by 100

percent literally means "divided by 100"; to convert a percent to a decimal, divide the percent number by 100

The order of operations establishes the order in which different mathematical procedures should be performed. This is often expressed with the acronym PEMDAS (or perhaps BEDMAS, BIDMAS, or BODMAS). PEMDAS, which stands for Parentheses Exponents Multiplication Division Addition Subtraction, reminds us that when simplifying expressions, we should start by simplifying all expressions inside parentheses, then apply exponents, then multiply or divide, and lastly add or subtract.

- Level 1 Parentheses, Brackets ("Containers") and Exponents
- Level 2 Multiplication and Division
- Level 3 Addition and Subtraction
Do the order of operations starting with Level 1, then do Level 2, then do Level 3. It can be good practice to solve the problem from left to right.

**Common Misconception**: Multiplication is performed before division. These two operations have equal priority. Do any multiplication or division as you move from left to right in the problem. For example, if we had , we would simplify it as follows: .

**Common Misconception**: Addition is performed before subtraction. These two operations have equal priority. Do any addition or subtraction as you move from left to right in the problem. For example, if we had , we would simplify it as follows: .

The order of operations gives us a set of general guidelines that we must apply carefully.

A math statement combining numbers or variables using sums, differences, products, quotients (including fractions), exponents, roots, logarithms, trig functions, parentheses, brackets, functions, or other mathematical operations. Expressions may not contain the equal sign (=) or any type of inequality (< or >)

The absolute value

is the magnitude of , that is, the number with any negative sign removed. For example, and . Note that any calculation within the absolute value is done first and then the sign ignored eg . The absolute value of a number also represents the distance of the number from zero.The factors of a term or expression are numbers, variables, or combinations of numbers and variables that, when multiplied together, produce that term or expression. For example,

and are factors of , and , , , and are factors of .Each of the quantities connected by a **term**.

For addition:
The **commutative property of addition** states that changing the order in which numbers are added does not affect the final result. For example, . This is true no matter how many terms are being added together.
For multiplication:
The **commutative property of multiplication** states that changing the order in which numbers are multiplied does not affect the final result. For example, . This is true no matter how many terms are being multiplied together.

A variable is a symbol, generally a letter, that serves as a placeholder for a number. The numeric value of a variable may change, depending on the context in which it is used. For example, in the expression

In an equation like , however, represents a certain number whose value we just don't know yet. If we modify the equation to get by itself on one side, then we can find out that value: .

An algebraic expression combines constants and variables using addition, subtraction, multiplication, and division.

A variable term is a term whose value is not fixed, due to the presence of a variable in it. In other words, a variable term will have one or more variable factors. In the expression

, is a variable term.A constant term is a term whose value is fixed, since it does not have any variables in it (i.e. it does not have any variable factors). In the expression

, is a constant term.A coefficient is the numerical factor of a variable term. For instance, for the term

, the coefficient is -5, and for the term , the coefficient is 1.Evaluating an algebraic expression means substituting numeric values for each of the variables in the expression. If we have the expression

, and we let , then the value of the expression will be , or .A math statement that states the values of two mathematical expressions are equal. The statement includes the equals sign symbol (=).

terms that contain the same powers of variables; like terms are identical except for their numerical factor or coefficient, such as 2xy and -4xy

For an expression of the form

Examples:

Isolating a variable means rearranging and modifying an equation, according to the rules of algebra, so that a variable is all by itself on one side of the equation. If we started off with the equation

, we could add to both sides of the equation to end up with . The variable is now isolated.The real number line is a horizontal line used to represent the real numbers. Every point on the line represents a different real number, with values increasing from left to right. The arrows on either end indicate that the line extends forever in either direction, which allows it to include all values from

to .Whereas an equation relates two quantities that are of equal value, an inequality is a way to mathematically relate two quantities that do not have the same value. An inequality may use one of the following symbols:

- means that is not equal to .
- means that is greater than .
- means that is less than .
- means that is greater than or equal to .
*solution set*of that inequality. Solution sets are often represented by graphs on number lines.

To write with MathQuill, type \neq and then a space. To write , type \le and then a space, and to write , type \ge and then a space.
means that is less than or equal to .
The set of all numbers that satisfy an inequality (i.e. can be plugged in for the variables to make the inequality true) is called the

If we know that

and that , then we can use a compound inequality to combine these two inequalities into one statement: .An interval is a set of real numbers that includes all real numbers between a lower bound and an upper bound. If an interval represents a set of values that a variable, let’s say

Let’s take a look at an example. If we have , we could also write that lies in the interval . Here, can equal , since that has a square bracket next to it, but it can’t equal , since that is by a round parenthesis.

the length around a 2-Dimensional closed, planar figure measured in units; the distance around a two dimensional shape

a measure of 2-Dimensional space inside a closed, planar figure measured in square units; the measure of space or size of a surface

The length and width of a rectangular area. The length is considered to be the longer side length. The width is the shorter side length.

A way of comparing one quantity to another; for example, the ratio of

A proportion is an equation that equates two ratios

A method used to solve for a missing value in a proportion. The method comes from multiplying both sides of the equation by the product of the two denominators.

Two angles whose sum is 90º. One angle is said to be the **complement** of the other.

Two angles whose sum is 180º. One angle is said to be the **supplement** of the other.

Supplementary angles that are side by side or that share a vertex form a straight line

A set can be represented by a single letter or name, and the elements of the set can be listed inside braces separated by a commas. For example the set of even numbers between 1 and 11 is {2, 4, 6, 8, 10}.

The union of two sets is the set of all values that belong to either of those two sets, and we denote the union of two sets

and by writing . So if we have { } and { }, then { }. In this course, we will most often use unions to join together intervals that make up solution sets.The intersection of two sets is the set of all values that belong to both of the sets. The elements must appear in both sets in order to be part of the intersection, and we denote the intersection of two sets

and by writing So if we have { } and { }, then { }.A compound inequality connects two or more inequalities by using the words “and” or “or”. In this course, we’ll limit ourselves to compound inequalities with just two inequalities.

- If the word “and” is used, then a value must satisfy both of the inequalities in order to be in the solution set. Example: If we start out with and , then we can simply to have and , so the solution is , written as a chained inequality. In interval notation, this is . Some values that would satisfy this compound inequality, then, are , , and . Some values that would not satisfy it are , , and .
- If the word “or” is used, then a value may satisfy either or both of the inequalities in order to be in the solution set. Example: Let’s say we begin with or . We could also write this as , where , typed as \or in MathQuill, just means “or”. We can then simplify to have . In interval notation, we would write this as . Some values that would satisfy this compound inequality, then, are , , and . Some values that would not satisfy it are , , and .

A **coordinate plane**, or the Cartesian coordinate plane, is made up of one horizontal number line, called the **horizontal axis**, and one vertical number line, called the **vertical axis**. Each axis represents a different variable. By convention, the variable is usually shown on the horizontal axis, and the variable is usually shown on the vertical axis. Every point on the plane containing these axes can be defined by a number indicating its horizontal position and a number indicating its vertical position. These numbers, called the point’s **coordinates** are normally written with ordered pair notation. If a point has a horizontal coordinate of and a vertical coordinate of , it can be identified by the ordered pair .

The **origin** is the point on a coordinate plane where the axes meet one another, giving it the coordinates .

The

-axis is the horizontal number line the Cartesian coordinate plane. It represents values that the variable in an equation might equal. The origin represents an -value of , positive values lie to the right, and negative values lie to the left.The

-axis is the horizontal number line the Cartesian coordinate plane. It represents values that the variable in an equation might equal. The origin represents a -value of , and values increase going up the axis and decrease moving down.The x-intercept is the point at which a line (or graph) crosses the **(**. , 0)

The y-intercept is the point at which a line (or graph) crosses the **(0, **. )

The **slope** of a line is a number that tells us how steep the line is. We can find this number by calculating how many units the line rises vertically for every unit it moves to the right or to the left. This is sometimes referred to as “rise over run”. Two find the slope of the line on which the points and lie, we can use the following formula, in which represents the slope:

One way to write the equation of a line is to use **slope-intercept form**. If we call the slope of a line and the -coordinate of its -intercept , then the equation of that line in slope-intercept form will be .

If a line passes through the point **point-slope form** as follows:

The standard form for the equation of a line can be written as follows:
**A, B, and C must be integers, and **. Although the equation of a vertical line cannot be written in either slope-intercept form or point-slope form, it can be written in the form and are not both zero . Some sources write the general form of a line as .

Two lines are called **parallel** if their slopes are equal. Parallel lines that are not identical will never intersect one another. The symbol is sometimes used to show that two lines are parallel to one another.

Two lines are called **perpendicular** if they form a angle. This will happen if their slopes are the negative reciprocals of one another. Put more mathematically, if a line has a slope of , then every line perpendicular to it will have a slope of .

Horizontal lines have no rise. Therefore the numerator of the fraction that makes up the slope will be zero. This gives horizontal lines a **slope of zero**. Horizontal lines are of the form , where is a constant.

Vertical lines have no run. Therefore the denominator of the fraction that makes up the slope will be zero. Since the denominator of a fraction cannot equal zero, **vertical lines have no slope or have a slope that is undefined**. Vertical lines are of the form , where is a constant. Vertical lines do not represent functions.

The solution set of a linear inequality consists of a boundary line whose points may or may not be included in the solution of all points on one side of the boundary line.

The line that is graphed for a linear inequality because it splits the graph into two regions.

- A
**dashed or dotted boundary line**is used for < or > linear inequalities. - A
**solid boundary line**is used for or linear inequalities.

In an equation, an **independent variable** is a variable whose value does not depend on the value of other variables. In algebra, this is often the variable . The independent variable generally appears as part of an expression on one side of the equation, as with .

In an equation, a **dependent variable** is a variable whose value depends on the the values of one or more other variables. In algebra, this is usually the variable . The dependent variable is often isolated on one side of an equation, as with .

A **function** relates an independent variable, the *input* of the function, to a dependent variable, the **output** of the function so that for each value of the independent variable, there is only one value of the dependent variable. If we denote a function as , then is the dependent variable, is the independent variable, is the name of the function, and is the value of the function at .

There are many types of functions - polynomial functions (which include a wide variety of functions), rational functions, absolute value functions, square root functions, and more.

The **domain** of a function is the set of all inputs, or values of the independent variable, for which the function has an output. If some value is in the domain of a function, then the function is said to be *defined* at . If is not in the domain, then the function is said to be *undefined* at .

The **range** of a function is the set of all values that the function can take on. In other words, it is the set of all outputs that the function can produce.

We can use the **vertical line test** to determine whether or not a graph represents a function. Since a function can only have one output for each input, or one value of the dependent variable for each value of the independent variable, its graph can only cross each horizontal position on the coordinate plane at most one time. To perform the vertical line test on a graph, take a straight object, like a pencil or a ruler, position it so that it is parallel with the vertical axis of the graph, and then move it horizontally from one side of the graph to the other. If at any position the vertical line intersects the graph more than once, the graph does not represent a function.
Please note that use of the vertical line test assumes that the independent variable, , is represented on the horizontal axis and that the dependent variable, , is represented on the vertical axis.

A point of intersection is a point at which two or more lines or curves meet. In other words, it is a point shared by these graphs. The coordinates of this point will satisfy the equation of every line going through the point.

When a person decides to invest money in a bank account, the amount of money he or she initially invests is called the **principal**. Once this money is invested, a certain percent of that money will be added into the account at designated times. The money that is added is known as **interest**, and the total amount of money in the account at any given time is called the **balance**. The percent of the account balance that is earned in interest is called the **interest rate**.

A polynomial is a mathematical expression in which the variables have only whole number exponents (0, 1, 2, 3, ...). The expression is made up of variables and/or constants that are combined with addition, subtraction, and multiplication. This excludes any expressions that have variables in the denominator and any variables appearing within a radical. Examples:

Polynomials that have only one term like

.Polynomials that have exactly two terms like

.Polynomials that have exactly three terms like

To find the degree of a polynomial, we first need to find the degree of each of its terms. The **degree of a term** is the sum of the powers of all of the variables in that term. So, for instance, has a degree of 2, and has a degree of 4. The **degree of a polynomial** is equal to the degree of its highest-degree term. Examples: has a degree of 4 or, in other words, is a fourth degree polynomial.

A polynomial is said to be in **standard form** when it is written with its highest-degree term first, with the rest of the terms written in order of descending degree. For example, is written in standard form, whereas is not.

When we **factor** a polynomial, we break it down so that it is written as a product of factors. In this course, we will say that a polynomial is **fully factored** or **completely factored** when it is written as the product of factors with integer coefficients and none of its factors can be factored any further. For example, is fully factored, whereas is not because we could instead write it as .

A **linear factor** is a factor of degree 1. In other words, it can be written in the form , where and are constants and is not equal to . For the quadratic polynomial , the linear factors are and .

The **greatest common factor**, or **GCF**, of two or more numbers is the largest positive integer that divides evenly into all of the numbers (i.e. leaves a remainder of 0). For example, the GCF of 60 and 18 is 6.

The GCF of two or more polynomials is the largest polynomial of the highest degree that can be factored out of all of the polynomials in question. For example, the GCF of and is .

When we have a **difference of two squares**, like or , the expression can be factored according to a certain pattern. In general, . For the two examples already mentioned, then, we could factor to have and .

**Factoring by grouping** is a technique that can be used to factor certain polynomials that can be written with more than 3 terms. Let’s look at a couple of examples:
If we start out with a polynomial that only has three terms, we can often expand the middle term and then factor by grouping.

A **quadratic equation** is an equation of the form , where , , and are constants and is not equal to .

The **plus or minus sign** is used to indicate that the value to its can be either added or subtracted to the value to its left. For example, is a way of writing or using just one expression. To write this symbol with MathQuill, type \pm and then a space.

**Completing the square** is a technique used to convert an expression of the form , where , , and are constants, to an expression of the form , where and are both constants.

The **quadratic formula** is a formula that can be used to find solutions to quadratic equations. If a quadratic equation is written in the form , then the quadratic formula states that its solutions will be:

A **root** of a polynomial, also called a **zero** of a polynomial, is a value of that makes the polynomial equal to 0. For a polynomial equation, if a root is a real number, then this tells us the -coordinate of one of the graphs’ -intercepts. Solutions to the quadratic formula, for example, tell us the roots of quadratic equations.
Let’s say that we have the equation . We can factor this to be . If we set , then the values that solve the quadratic equation are the roots of the polynomial equation, and .

A **rational expression** is the ratio of one polynomial to another. Thus, a rational expression is anything that can be written as , where and are both polynomials and is not .

When solving a polynomial inequality or a rational inequality, **critical numbers** or **critical values** are values that divide the number line into **test intervals**, intervals in which we know the inequality will be either true for every value contained in the interval or not true for any value in it.

We can find the critical numbers for a polynomial inequality by simply finding the -coordinates of the -intercepts. If we rearrange a rational inequality so that one side of the inequality is equal to and the other side is just a rational expression, the critical numbers will be values of that make either the numerator or the denominator of the rational expression equal to .

If we have some number

, then a square root of is any number such that . In other words, multiplying by itself gives . We can denote the square root of a quantity by using a radical sign over : . By convention, denotes the positive square root of . For example, , and as well, but so . To indicate the negative square root, in this case, we must write .The

th root of a number is some number that, when raised to the th power, equals : If , then is an th root of . For example, since , is a fourth root of . We can denote the th root of a quantity using a radical sign with aNote: For example,

An **imaginary number** is any number that can be written in the form , where is a real number and . We can find imaginary numbers by taking the square roots of negative numbers. Since the imaginary number is defined by the equation , we can write all square roots of negative numbers in terms of . For example, , and . The set of imaginary numbers lies completely separate from the set of real numbers.

A **complex number** is any number that can be written in the form , where and are real numbers and . If , this will give us an imaginary number, and if , this will give us a real number. The set of complex numbers full encompasses the real numbers and the complex numbers, but it also has members that are neither real nor imaginary. In other words, all real numbers are complex, and all imaginary numbers are complex. Some complex numbers, however, have both imaginary and real parts, preventing them from falling into the imaginary category or the real category. has a real part and an imaginary part , so it is a complex number.
Although all imaginary numbers and all real numbers are complex numbers, when a number is specifically referred to as a complex number, this sometimes means that it has both real and imaginary parts (i.e. it is neither real nor imaginary).

We can use the **complex plane** to visualize complex numbers, just as we can use the number line to visualize real numbers. The complex plane is formed by two perpendicular axes. The horizontal one represents the real numbers, just like a real number line, and the vertical one represents the imaginary numbers. Any complex number of the form can be plotted on the complex plane by letting its horizontal coordinate be equal to its real part, , and its vertical coordinate be equal to the coefficient of its imaginary part , . We can find the number at the point , and we can find the number at the point .

**Complex conjugates** are complex numbers whose real parts are equal and whose imaginary parts have opposite signs. So if one complex number is , then its complex conjugate is .

Examples:

- The complex conjugate of is .
- The complex conjugate of is . (Remember, all imaginary numbers are complex numbers!)
- The complex conjugate of is . (Remember, all real numbers are complex numbers!)

In a problem involving direct variation, as one quantity *increases*, the other quantity also *increases*. Also, if one quantity *decreases*, then the other quantity *decreases*.

In a problem involving indirect variation, as one quantity *increases*, the other quantity *decreases*. Also, if one quantity *decreases*, then the other quantity *increases*.

A *right triangle* is a triangle with one right angle, or one angle that measures .

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

The **Pythagorean Theorem** states that if a right triangle’s sides have lengths , , and , where is the longest side (the hypotenuse), then , , and are related to one another according to the equation .

We can use the *distance formula* to find the distance between any two points on the coordinate plane. If our points are and and we call the distance between them , then we can say that
.
The distance formula treats the straight line connecting these two points as the hypotenuse of a right triangle, and it uses the Pythagorean Theorem to find the length of this line.

An equation of the form

where is the center of the circle and is the radius of the circle. Note: The equation of a circle will always have equal coefficients for the and terms.An equation of the form

This equation differs from a circle in that there are different coefficients or amounts of the and terms.An equation of the form

is a hyperbola that opens to the left and to the right. An equation of the form is a hyperbola that opens to the left and to the right. In both equations, the centers of the hyperbola are . This equation differs from a circle in that there are different coefficients or amounts of the and terms. This equation differs from a circle and an ellipse in that there is a subtraction sign between the and terms.Parabolas have either an

Parabolas of the form or that have an term open up or down. This includes parabolas of the form .
Parabolas of the form or that have a term open left or right. This includes parabolas of the form .

The **vertex** of a parabola is the point at which the parabola itself intersects its axis of symmetry, the line that divides it in half. For parabolas that open upward, the vertex is the lowest point on the curve, the minimum. For parabolas that open downward, the vertex is the highest point on the curve, the maximum.

If the coordinates of the vertex of a parabola are **vertex form** as:

where

is a constant. If , the parabola opens upward, and if , it opens downward.Given two functions, we can create a composite function by using the output of one of the functions as the input of the other function. If we have two functions named

and , then we could combine them to create the composite functions and . These can also be denoted as and , respectively. For example, if and , then , and .We can think of the **inverse function** of a function as the function, denoted that does the opposite operations that does. In other words, undoes what does to the independent variable (usually ).

More formally, a function **inverse** of a function if for all values in the domain of and for all values in the domain of . If all of that is true, then , and .

We can find the inverse of a function

graphically by reflecting the graph of in the line . This is because we can find the inverse of a function by switching its - and -coordinates.A **logarithmic function** is a function that can be written in the form

Here,

is the base of the logarithmic function, and is positive number not equal to . The domain of a function of this form is . The equation is true if and only if .A **logarithm** is an expression of the form , and it represents the number that would need to be taken to the power of in order to equal . For example, is equal to .

The **product property of logarithms** can be stated as the following:

The **quotient property of logarithms** can be stated as the following:

The **power property of logarithms** can be stated as the following:

if and only if

if and only if