Algebra 1 Notes

The Language of Algebra

Rational Number

A number that can be expressed as the quotient of two integers. Fractions, mixed numbers, decimals, and integers are all rational numbers, because they may be expressed as a quotient of two integers.

Irrational Number

Some numbers cannot be written as a quotient of two integers, and these are called irrational numbers.

The decimal form of an irrational number does not terminate or repeat.

Real Number

The numbers we use to measure real-world quantities, such as length, temperature, or volume, are called real numbers. All the rational and irrational numbers make up the set of real numbers. The numberline is a model of the set of real numbers.

Variable

A letter that stands for a number in a mathematical expression is called a variable, because its value can vary.

In the expression 2x = 3 x is a variable

Expression

A mathematical phrase made up of variables and/or numbers and operations is called an expression.

3ab + 7ab -2a is an example

Terms

In an expression, the terms are the elements seperated by the plus or minus signs. In the expression 3ab + 7ab -2a the terms are 3ab, 7ab, and 2a.

Coefficient

A number that appears before a letter in a term. For example, in the term 3ab, 3 is the coefficient.

Constant

A term that has only one number and no variables is called a constant, because its value does not vary. In the expression 3ab + 7ab + 5 the number 5 is a constant.

Definitions

Learning algebra is a little like learning another language. In fact, algebra is a simple language, used to create mathematical models of real-world situations and to handle problems that we can't solve using just arithmetic. Rather than using words, algebra uses symbols to make statements about things. In algebra, we often use letters to represent numbers.

Since algebra uses the same symbols as arithmetic for adding, subtracting, multiplying and dividing, you're already familiar with the basic vocabulary.

In this lesson, you'll learn some important new vocabulary words, and you'll see how to translate from plain English to the "language" of algebra.

The first step in learning to "speak algebra" is learning the definitions of the most commonly used words.

Algebraic Expressions | Variables | Coefficients | Constants | Real Numbers | Rational Numbers | Irrational Numbers | Translating Words into Expressions

Algebraic Expressions
An algebraic expression is one or more algebraic terms in a phrase. It can include variables, constants, and operating symbols, such as plus and minus signs. It's only a phrase, not the whole sentence, so it doesn't include an equal sign.

Algebraic expression:

3x² + 2y + 7xy + 5

In an algebraic expression, terms are the elements separated by the plus or minus signs. This example has four terms, 3x², 2y, 7xy, and 5. Terms may consist of variables and coefficients, or constants.

Variables
In algebraic expressions, letters represent variables. These letters are actually numbers in disguise. In this expression, the variables are x and y. We call these letters "variables" because the numbers they represent can vary—that is, we can substitute one or more numbers for the letters in the expression.

Coefficients
Coefficients are the number part of the terms with variables. In 3x² + 2y + 7xy + 5, the coefficient of the first term is 3. The coefficient of the second term is 2, and the coefficient of the third term is 7.

If a term consists of only variables, its coefficient is 1.

Constants
Constants are the terms in the algebraic expression that contain only numbers. That is, they're the terms without variables. We call them constants because their value never changes, since there are no variables in the term that can change its value. In the expression 7x² + 3xy + 8 the constant term is "8."

Real Numbers
In algebra, we work with the set of real numbers, which we can model using a number line.

Real numbers describe real-world quantities such as amounts, distances, age, temperature, and so on. A real number can be an integer, a fraction, or a decimal. They can also be either rational or irrational. Numbers that are not "real" are called imaginary. Imaginary numbers are used by mathematicians to describe numbers that cannot be found on the number line. They are a more complex subject than we will work with here.

Rational Numbers
We call the set of real integers and fractions "rational numbers." Rational comes from the word "ratio" because a rational number can always be written as the ratio, or quotient, of two integers.

Examples of rational numbers
The fraction ½ is the ratio of 1 to 2.

Since three can be expressed as three over one, or the ratio of 3 to one, it is also a rational number.

The number "0.57" is also a rational number, as it can be written as a fraction.

Irrational Numbers
Some real numbers can't be expressed as a quotient of two integers. We call these numbers "irrational numbers". The decimal form of an irrational number is a non-repeating and non-terminating decimal number. For example, you are probably familiar with the number called "pi". This irrational number is so important that we give it a name and a special symbol!

Pi cannot be written as a quotient of two integers, and its decimal form goes on forever and never repeats.

Translating Words into Algebra Language
Here are some statements in English. Just below each statement is its translation in algebra.

the sum of three times a number and eight
3x + 8

The words "the sum of" tell us we need a plus sign because we're going to add three times a number to eight. The words "three times" tell us the first term is a number multiplied by three.

In this expression, we don't need a multiplication sign or parenthesis. Phrases like "a number" or "the number" tell us our expression has an unknown quantity, called a variable. In algebra, we use letters to represent variables.

the product of a number and the same number less 3
x(x – 3)

The words "the product of" tell us we're going to multiply a number times the number less 3. In this case, we'll use parentheses to represent the multiplication. The words "less 3" tell us to subtract three from the unknown number.

a number divided by the same number less five

The words "divided by" tell us we're going to divide a number by the difference of the number and 5. In this case, we'll use a fraction to represent the division. The words "less 5" tell us we need a minus sign because we're going to subtract five.

Order of Operations

When an expression contains more than one operation, you can get different answers depending on the order in which you solve the expression. Mathematicians have agreed on a certain order for evaluating expressions, so we all arrive at the same answers. We often use grouping symbols, like parentheses, to help us organize complicated expressions into simpler ones. Here's the order we use:

1. First, do all operations that lie inside parentheses.
2. Next, do any work with exponents or roots.
3. Working from left to right, do all multiplication and division.
4. Finally, working from left to right, do all addition and subtraction.

In Example 1, without any parentheses, the problem is solved by working from left to right and performing all the addition and subtraction. When parentheses are used, you first perform the operations inside the parentheses, and you'll get a different answer!

Example 1 - Parenthesis

Example 2

Writing Equations

In the Language of Algebra, an equation is the basic number "sentence". An equation is a mathematical expression that contains an equals sign.

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An equation is a mathematical sentence containing an equals sign. It tells us that two expressions mean the same thing, or represent the same number. An equation can contain variables and constants. Using equations, we can express math facts in short, easy-to-remember forms and solve problems quickly.

Here are several examples of equations. You can think of the letters as containers, or boxes, that can hold different numbers.

Example 1

3z + 2 = 14       x - 9 = 20       p + 2p = 3

The most important skill to develop in algebra is the ability to translate a word problem into the correct equation, so that you can solve the problem easily. Let's try a few examples:

Example 2

A number n times 3 is equal to 120.

This is an easy one. The word "times" tells you that you must multiply the variable n by 3, and that the result is equal to 120. Here's how to write this equation:

3n = 120

Here's one that's a little trickier.

Example 3.

Ten dollars was two-thirds of the total money spent.

What are we trying to find in this statement? The unknown amount is the total money spent. Let's call this m. We know that ten dollars is equal to two-thirds of m, so we can write the equation like this:

Example 4.

Tim worked for 7 hours on Saturday and mowed 3 lawns. How much time, on average, did he spend on each lawn?

Let the letter "t" represent the average time per lawn, the unknown value. Then, 3t  would represent the time to mow all three lawns, and we know that this is equal to 7 hours. We can write the equation like this:

3t = 7 hours

There are several tips to help you keep track of algebraic expressions.

What was that variable?
It's a good idea to write down what your variables mean. For example, write this equation: Today a boy is twice as old as he was 4 years ago.

Let b = the boy's age today

b = 2 (b - 4 years)
b = 2b - 8 years
b = 8 years

The boy is 8 years old today. A common mistake would be to get confused whether the answer was the age today or 4 years ago. Of course, this is a simple example, but as your equations get more complicated, it is very important to keep track of what your variables represent.

Units
Remember to include units. If you have a real-world problem that uses units, keeping the units in the equation will help you figure it out.

For example, write this expression - The cost was $2.00 less than half of $5.00

Let x = the cost

x = 1/2 ($5.00) - $2.00
x = $2.50 - $2.00
x = $0.50 or fifty cents

Without the units, It would be easy to lose track of what the answer of .5 meant. Again, this is a simple example, but as things get more complicated, it's important to keep units straight.

Writing Inequalities

Another type of sentence used in algebra is called an inequality.

An inequality is used when we don't know exactly what an expression is equal to. Instead of an equals sign, we use one of these symbols:

> greater than
< less than
< less than or equal to
> greater than or equal to

Another type of number sentence used in algebra is called an inequality. An inquality is used when we don't know exactly what an expression is equal to. Instead of an equals sign, we can use one of these symbols:

> greater than
< less than
< less than or equal to
> greater than or equal to

It takes practice to translate a word problem into an inequality, just as it does to translate a problem into an equation. Let's practice now.

Example 1.

A number minus 4 is greater than 2.

The words "a number" tell us that we need a variable in our inequality, and that the result of the variable less "4" is more than 2. We can write it like this:

n - 4 > 2

Let's try one more example.

Example 2.

The sum of x and 5 is less than or equal to -2.

The words "the sum of" give us a clue that our inequality will involve addition. We can write the inequality like this:

x + 5 < -2

The Basics of Algebra

Commutative Properties

Addition
a + b = b + a

Multiplication
ab = ba

Associative Properties

Addition
(a + b) + c = a + (b + c)

Multiplication
(ab)c = a(bc)

Distributive Property

a(b + c) = ab + ac

Density Property

Between any two real numbers, there is always another real number.

Identity Properties

Addition
a + 0 = a

Multiplication
a x 1 = a

Properties of Real Numbers

In this lesson we look at some properties that apply to all real numbers. If you learn these properties, they will help you solve problems in algebra. Let's look at each property in detail, and apply it to an algebraic expression.

#1. Commutative properties
The commutative property of addition says that we can add numbers in any order. The commutative property of multiplication is very similar. It says that we can multiply numbers in any order we want without changing the result.

addition
5a + 4 = 4 + 5a

multiplication
3 x 8 x 5b = 5b x 3 x 8

#2. Associative properties
Both addition and multiplication can actually be done with two numbers at a time. So if there are more numbers in the expression, how do we decide which two to "associate" first? The associative property of addition tells us that we can group numbers in a sum in any way we want and still get the same answer. The associative property of multiplication tells us that we can group numbers in a product in any way we want and still get the same answer.

addition
(4x + 2x) + 7x = 4x + (2x + 7x)

multiplication
2x²(3y) = 3y(2x²)

#3. Distributive property
The distributive property comes into play when an expression involves both addition and multiplication. A longer name for it is, "the distributive property of multiplication over addition." It tells us that if a term is multiplied by terms in parenthesis, we need to "distribute" the multiplication over all the terms inside.

2x(5 + y) = 10x + 2xy

Even though order of operations says that you must add the terms inside the parenthesis first, the distributive property allows you to simplify the expression by multiplying every term inside the parenthesis by the multiplier. This simplifies the expression.

#4. Density property
The density property tells us that we can always find another real number that lies between any two real numbers. For example, between 5.61 and 5.62, there is 5.611, 5.612, 5.613 and so forth.

Between 5.612 and 5.613, there is 5.6121, 5.6122 ... and an endless list of other numbers!

#5. Identity property
The identity property for addition tells us that zero added to any number is the number itself. Zero is called the "additive identity." The identity property for multiplication tells us that the number 1 multiplied times any number gives the number itself. The number 1 is called the "multiplicative identity."

Addition
5y + 0 = 5y

Multiplication
2c × 1 = 2c

In algebra, you'll often be working with exponents. Here are some rules:

Any number raised to the zero power (except 0) equals 1.

Any number raised to the power of one equals itself.

To multiply terms with the same base, add the exponents.

To divide terms with the same base, subtract the exponents.

When a product has an exponent, each factor is raised to that power.

A number with a negative exponent equals its reciprocal with a positive exponent.

Exponents are used in many algebra problems, so it's important that you understand the rules for working with exponents. Let's go over each rule in detail, and see some examples.

Rules of 1

There are two simple "rules of 1" to remember.

First, any number raised to the power of "one" equals itself. This makes sense, because the power shows how many times the base is multiplied by itself. If it's only multiplied one time, then it's logical that it equals itself.

Secondly, one raised to any power is one. This, too, is logical, because one times one times one, as many times as you multiply it, is always equal to one.

Product Rule

The exponent "product rule" tells us that, when multiplying two powers that have the same base, you can add the exponents. In this example, you can see how it works. Adding the exponents is just a short cut!

Power Rule

The "power rule" tells us that to raise a power to a power, just multiply the exponents. Here you see that 5² raised to the 3rd power is equal to 5⁶.

Quotient Rule

The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. You can see why this works if you study the example shown.

Zero Rule

According to the "zero rule," any nonzero number raised to the power of zero equals 1.

Negative Exponents

The last rule in this lesson tells us that any nonzero number raised to a negative power equals its reciprocal raised to the opposite positive power.

Evaluating Expressions

We have learned that, in in an algebraic expression, letters can stand for numbers. When we substitute a specific value for each variable, and then perform the operations, it's called evaluating the expression.

Let's evaluate the expression 3y + 2y when 5 = y.

Step 1

Replace each letter with its assigned value.

The first step: 3(5) + 2(5)

(Use Parentheses!)

Step 2

Perform the operations, to find the value of the expression.

The second step:

3(5) + 2(5) = 15 + 10 = 25

We have learned that, in an algebraic expression, letters can stand for numbers. Here are the steps for evaluating an expression:

1. Replace each letter in the expression with the assigned value.
   First, replace each letter in the expression with the value that has been assigned to it. To make your calculations clear and avoid mistakes, always enclose the numbers you're substituting inside parentheses. The value that's given to a variable stays the same throughout the entire problem, even if the letter occurs more than once in the expression.

However, since variables "vary", the value assigned to a particular variable can change from problem to problem, just not within a single problem.

2. Perform the operations in the expression using the correct order of operations.
   Once you've substituted the value for the letter, do the operations to find the value of the expression. Don't forget to use the correct order of operations: first do any operations involving exponents, then do multiplication and division, and finally do addition and subtraction!

Here's an example. Let's evaluate the expression 2x³ – x² + y for x = 3 and y = –2.

Like Terms

Like terms are terms that contain the same variables raised to the same power. Only the numerical coefficients are different. In an expression, only like terms can be combined. We combine like terms to shorten and simplify algebraic expressions, so we can work with them more easily. To combine like terms, we add the coefficients and keep the variables the same. We can't combine unlike terms because that's like trying to add apples and oranges!

Look at these 10 terms. Let's find all the like terms that can be combined.

Be careful when combining!
Terms like x²yz and xy²z look a lot alike, but they aren't and cannot be combined. Write the terms carefully when working out problems.

Don't overlook terms that are alike!
Terms obey the associative property of multiplication - that is, xy and yx are like terms, as are xy^{2 and} y²x.

Simplifying

Whenever a problem can be simplified, you should simplify it before substituting numbers for the letters. This will make your job a lot easier! To simplify an algebraic expression:

1. Clear the parentheses.
   
   
   
2. Combine like terms by adding coefficients.
   
   
   
   
    
3. Combine the constants.

Before you evaluate an algebraic expression, you need to simplify it. This will make all your calculations much easier. Here are the basic steps to follow to simplify an algebraic expression:

1. remove parentheses by multiplying factors
2. use exponent rules to remove parentheses in terms with exponents
3. combine like terms by adding coefficients
4. combine the constants

Let's work through an example.

When simplifying an expression, the first thing to look for is whether you can clear any parentheses. Often, you can use the distributive property to clear parentheses, by multiplying the factors times the terms inside the parentheses. In this expression, we can use the distributive property to get rid of the first two sets of parentheses.

Now we can get rid of the parentheses in the term with the exponents by using the exponent rules we learned earlier. When a term with an exponent is raised to a power, we multiply the exponents, so (x²)² becomes x⁴.

The next step in simplifying is to look for like terms and combine them. The terms 5x and 15x are like terms, because they have the same variable raised to the same power -- namely, the first power, since the exponent is understood to be 1. We can combine these two terms to get 20x.

Finally, we look for any constants that we can combine. Here, we have the constants 10 and 12. We can combine them to get 22.

Now our expression is simplified. Just one more thing -- usually we write an algebraic expression in a certain order. We start with the terms that have the largest exponents and work our way down to the constants. Using the commutative property of addition, we can rearrange the terms and put this expression in correct order, like this.

Equations and Inequalities

Solving an equation is like solving a puzzle! It means finding a value for the variable that makes the equation true. Using the properties of real numbers that you've learned, you can rearrange the terms of an equation and use inverse operations to help you find the value of the variable. But be careful! You can think of an equation like a balance scale—whatever you do to one side of the scale, you must also do to the other side, to keep it in balance.

First, let's look at a simple addition equation, x + 15 = 30.

To solve the equation we must try to get the variable x alone on one side. We can use the inverse of adding 15 - or subtracting 15 - to get x alone on the left side. Now we have x alone on the left side, since 15 – 15 = 0, but the scale is not in balance. To balance the scale, we must also subtract 15 from the right side of the equation.

x + 15 - 15 = 30 - 15
         x = 15

30 – 15 = 15, so we find that x = 15.

We can check this solution by substituting the value 15 for x in the original equation. When we evaluate for x = 15 we get 30 = 30, which is a true statement. We know our solution is correct!

      x + 15 = 30
(15) + 15 = 30
                             30 = 30 Correct!

Now, let's look at a subtraction equation, y – 9 = 3

To solve this equation we must try to get the variable y alone on one side. We can use the inverse of subtracting 9, or adding 9, to get y alone on the left side.

y - 9 + 9 = 3 + 9
   y - 0 = 12
        y = 12

Now we have y alone on the left side, since –9 + 9 = 0, but the scale is not in balance. To balance the scale, we must also add 9 to the right side of the equation.

Now we have y alone on the left side of the equation. Three plus nine is twelve, so we find that y = 12.

We can check this solution by substituting the value 12 for y in the original equation. When we evaluate for y = 12, we get 3 = 3, which is a true statement. Our solution is correct!

      y - 9 = 3
(12) - 9 = 3
                          3 = 3 Correct!

Solve Multiplication equations

Mt. Everest in Nepal is the world's tallest mountain, about 29,000 ft. high. It is twice as high as Mount Whitney in California. How high is Mount Whitney?

We can write a multiplication equation to find the answer to problems like this. Our unknown number is the height of Mount Whitney. Let x represent this height. We know that 2x is the height of Mount Everest. We can write our equation like this:

2x = 29,000 ft

To solve this equation, we can use the inverse of multiplying by 2, which is dividing by 2.

If we divide the left side of the equation by 2, we will get x alone on the left. Remember, any operation done to one side must also be done to the other side, so we must also divide the right side by 2.

We divide, and find that x is equal to 14,500 ft. This is very close to the actual height of Mount Whitney, which is 14,494 ft.

Mental Math

You don't always need to use the calculator or pencil and paper to solve equations. You can solve many of them mentally. Use your mental muscle on these problems! Come up with an answer on your own, before looking at the answers below.

1.  10x = 350
2.  (12)(5)n = 0
3.  6 + 6 = 3x
4.  19y = 1900

Answers:

1.  x = 35 (Divide both sides by 10, an easy mental problem)
2.  n = 0 (Any number divided into 0 is 0.)
3.  x = 4 (6 + 6 = 12 and 12 ÷ 3 = 4)
4.  y = 100 (Divide both sides by 19, an easy mental problem)

The continental United States - the lower 48 states - has about 16,900 miles of shoreline. This is about ½ the length of the shoreline of Alaska. About how many miles of shoreline does Alaska have?

We can write a division equation to find the answer to problems like this. Our unknown number is the length of shoreline in Alaska. Let x represent this number. We know that x divided by 2 is the approximate length of shoreline in the lower 48 states. We can write the equation like this:

To solve this equation, we can use the inverse of dividing by 2, or multiplying by 2. If we multiply the left side of the equation by 2, we will get x alone on the left. Remember, any operation done to one side must also be done to the other side, so we must also multiply the right side by 2.

We multiply, and find that x, the length of the shoreline of Alaska, is equal to 33,800 miles.

To check our answer, we substitute the value of 33,800 into the original equation, like this.

Just like with equations, the solution to an inequality is a value that makes the inequality true. You can solve inequalities in the same way you can solve equations, by following these rules.

• You may add any positive or negative number to both sides of an inequality.

• You may multiply or divide both sides of an inequality by any positive number.

Watchout! If you multiply or divide both sides of an inequality by a negative number, reverse the direction of the inequality sign!

Solving an inequality is very similar to solving an equation. You follow the same steps, except for one very important difference. When you multiply or divide each side of the inequality by a negative number, you have to reverse the inequality symbol! Let's try an example:

-4x > 24

Since this inequality involves multiplication, we must use the inverse, or division, to solve it. We'll divide both sides by –4 in order to leave x alone on the left side.

When we simplify, because we're dividing by a negative number, we have to remember to reverse the symbol. This gives "x is less than –6," not "x is greater than –6."

x < -6

Why do we reverse the symbol? Let's see what happens if we don't. Think about the simple inequality –3 < 9. This is obviously a true statement.

-3 < 9

To demonstrate what happens when we divide by a negative number, let's divide both sides by –3. If we leave the inequality symbol the same, our answer is obviously not correct, since 1 is not less than –3.

We must reverse the symbol in order to find the correct answer, which is "1 is greater than –3."

Let's go back to the original problem and graph the solution x < –6 on a number line. To graph the solution for an inequality, you start at the defining point in the inequality. Here, it's –6. Then you graph all the points that are in the solution.

The red arrow shows that all the values on the number line less than –6 are in the solution. The open circle at –6 shows us that –6 is not in the solution. If the solution had been "x is less than or equal to –6," the circle would be a dark, or filled in, circle.

How can we check our answer? We can't use –6 to substitute in the inequality, because it lies outside our solution. To check, we can choose any value that lies in the solution. Let's use –7.

-4x > 24
-4(-7) > 24
28 > 24 Correct!

Our substitution gave a true result, so the solution is correct.

Formulas are equations that state a fact or a rule relating two or more variables. You can solve a formula for any of its variables using the rules for solving equations.

Here are two commonly used formulas. Click each one to see how to solve it.

A formula is a statement of a rule that relates different variables to each other. Formulas are used in many situations: banking, building bridges, buying a house, taking a trip, figuring sales tax, or designing a space shuttle. There's even a formula for calculating what the temperature is from how fast the crickets are chirping!

Formulas are often used in sports. For example, baseball fans use this formula to calculate batting averages of players. Here are the records for 3 baseball players for the first 20 games of the season. Let's use the formula to see which one has the best batting average.

Here are the records for 3 baseball players for the first 20 games of the season. Let's use the formula to see which one has the best batting average.

Homerun Harry had 9 hits out of 49 at bats. We can plug these numbers into the formula to calculate his batting average. A batting average is always rounded to three decimal places.

Slugger Sam had 8 hits out of 38 at bats. Here's his batting average.

Pinchhit Pete had 13 hits out of 64 at bats. Here's his batting average.

Slugger Sam has the best batting average of the three players.

By the way, in case you were wondering, here's the cricket formula:

where n stands for the no. of chirps per minute, and F stands for the temperature in degrees F.

What is the temperature if the crickets are chirping 80 times per minute?

It takes two steps to solve an equation or inequality that has more than one operation:

1. Simplify using the inverse of addition or subtraction.
2. Simplify further by using the inverse of multiplication or division.

Remember, when you multiply or divide an inequality by a negative number, you must reverse the inequality symbol.

Click on each equation to see how it is solved.

Now we'll solve some more complicated equations and inequalities - ones that have two-step solutions, because they involve two operations. Remember, solving equations is like solving a puzzle. Just keep working through the steps until you get the variable you're looking for alone on one side of the equation.

Here's a two-step equation. Let's start with the variable x, and describe, step by step, what is being done to x in an equation.

Solving an equation is like running the equation backwards to discover what number will work in the equation. Now let's work backwards and use inverse operations to undo all the steps. We can start with the result of 14.

Do you see how it's important when solving an equation to "undo" all the steps in the correct order? No matter how many steps are in the original equation, you can work backwards and apply the inverse operations, in order, to arrive at the solution!

You can solve two-step inequalities in exactly the same way. Just work backwards, using the inverse operations, to arrive at the solution. But watch out when multiplying or dividing by a negative number! Just as you learned in the last lesson, you must reverse the inequality symbol.

Graphing

Functions

Conic Sections

History

Conic sections are among the oldest curves, and is an oldest math subject studied systematically and thoroughly. The conics seems to have been discovered by Menaechmus (a Greek, c.375-325 BC), tutor to Alexander the Great. They were conceived in an attempt to solve the three famous problems of trisecting the angle, duplicating the cube, and squaring the circle. The conics were first defined as the intersection of: a right circular cone of varying vertex angle; a plane perpendicular to an element of the cone. (An element of a cone is any line that makes up the cone) Depending the angle is less than, equal to, or greater than 90 degrees, we get ellipse, parabola, or hyperbola respectively. Appollonius (c. 262-190 BC) (known as The Great Geometer) consolidated and extended previous results of conics into a monograph Conic Sections, consisting of eight books with 487 propositions. Quote from Morris Kline: “As an achievement it [Appollonius’ Conic Sections] is so monumental that it practically closed the subject to later thinkers, at least from the purely geometrical standpoint.” Book VIII of Conic Sections is lost to us. Appollonius’ Conic Sections and Euclid's Elements may represent the quintessence of Greek mathematics.

Appollonius was the first to base the theory of all three conics on sections of one circular cone, right or oblique. He is also the one to give the name ellipse, parabola, and hyperbola. A brief explanation of the naming can be found in Howard Eves, An Introduction to the History of Math. 6th ed. page 172. (also see J.H.Conway's newsgroup message, link at the bottom)

In Renaissance, Kepler's law of planetary motion, Descarte and Fermat's coordinate geometry, and the beginning of projective geometry started by Desargues, La Hire, Pascal pushed conics to a high level. Many later mathematicians have also made contribution to conics, espcially in the development of projective geometry where conics are fundamental objects as circles in Greek geometry. Among the contributors, we may find Newton, Dandelin, Gergonne, Poncelet, Brianchon, Dupin, Chasles, and Steiner. Conic sections is a rich classic topic that has spurred many developments in the history of mathematics.

The knowledge of conic sections can be traced back to Ancient Greece. Menaechmus is credited with the discovery of conic sections around the years 360-350 B.C.; it is reported that he used them in his two solutions to the problem of "doubling the cube". Following the work of Menaechmus, these curves were investigated by Aristaeus and of Euclid. The next major contribution to the growth of conic section theory was made by the great Archimedes. Though he obtained many theorems concerning the conics, it does not appear that he published any work devoted solely to them. Apollonius, on the other hand, is known as the "Great Geometer" on the basis of his text Conic Sections, an eight-"book" (or in modern terms, "chapter") series on the subject. The first four books have come down to us in the original Ancient Greek, but books V-VII are known only from an Arabic translation, while the eighth book has been lost entirely.

In the years following Apollonius the Greek geometric tradition started to decline, though there were developments in astronomy, trigonometry, and algebra (Eves, 1990, p. 182). Pappus, who lived about 300 A.D., furthered the study of conic sections somewhat in minor ways. After Pappus, however, conic sections were nearly forgotten for 12 centuries. It was not until the sixteenth century, in part as a consequence of the invention of printing and the resulting dissemination of Apollonius' work, that any significant progress in the theory or applications of conic sections occurred; but when it did occur, in the work of Kepler, it was as part of one of the major advances in the history of science.

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Two well-known conics are the circle and the ellipse. These arise when the intersection of cone and plane is a closed curve. The circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone. If the plane is parallel to a generator line of the cone, the conic is called a parabola. Finally, if the intersection is an open curve and the plane is not parallel to generator lines of the cone, the figure is a hyperbola. (In this case the plane will intersect both halves of the cone, producing two separate curves, though often one is ignored.)

Parabola

Ellipse

In mathematics, an ellipse (from the Greek ἔλλειψις, literally absence) is the locus of points on a plane where the sum of the distances from any point on the curve to two fixed points is constant. The two fixed points are called foci (plural of focus). An alternate definition would be that an ellipse is the path traced out by a point whose distance from a fixed point, called focus, maintains a constant ratio less than one with its distance from a straight line not passing through the focus, called the directrix.

Hyperbola

In this unit we'll be learning about equations in two variables. A coordinate plane is an important tool for working with these equations. It is formed by a horizontal number line, called the x-axis, and a vertical number line, called the y-axis. The two axes intersect at a point called the origin.

You can locate any point on the coordinate plane by an ordered pair of numbers (x,y), called the coordinates.

Move your mouse cursor over the coordinate plane to learn more.

The idea of graphing with coordinate axes dates all the way back to Apollonius in the second century B.C. Rene Descartes, who lived in the 1600s, gets the credit for coming up with the two-axis system we use today. The story goes that he lay in bed and watched flies crawling over tiles on the ceiling. He realized that he could describe a fly's position using the intersecting lines of the tiles. The system is often called the "Cartesian coordinate system" in his honor.

When working with equations that have two variables, the coordinate plane is an important tool. It's a way to draw pictures of equations that makes them easier to understand.

To create a coordinate plane, start with a sheet of graph or grid paper. Next, draw a horizontal line. This line is called the x-axis and is used to locate values of x. To show that the axis actually goes on forever in both directions, use small arrowheads at each end of the line. Mark off a number line with zero in the center, positive numbers to the right, and negative numbers to the left.

Next draw a vertical line that intersects the x axis at zero. This line is called the y-axis and is used to locate the values of y. Mark off a number line with zero in the center, positive numbers going upwards, and negative numbers going downwards. The point where the x and y axes intersect is called the origin. The origin is located at zero on the x axis and zero on the y axis.

Locating Points Using Ordered Pairs
We can locate any point on the coordinate plane using an ordered pair of numbers like the example shown here, the ordered pair 3 and 1 (point P). We call the ordered pair the coordinates of the point. The coordinates of a point are called an ordered pair because the order of the two numbers is important.

The first number in the ordered pair is the x coordinate. It describes the number of units to the left or right of the origin. The second number in the ordered pair is the y coordinate. It describes the number of units above or below the origin. To plot a point, start at the origin and count along the x axis until you reach the x coordinate, count right for positive numbers, left for negative. Then count up or down the number of the y coordinate (up for positive, down for negative.)

For example, to graph the point P above, with the ordered pair (4, 2) we count right along the x axis 4 units, and then count up 2 units. Be careful to always start with the x axis, the point (4,2) is very different than the point (2,4)!

Quadrants
To make it easy to talk about where on the coordinate plane a point is, we divide the coordinate plane into four sections called quadrants.

Points in Quadrant 1 have positive x and positive y coordinates.
Points in Quadrant 2 have negative x but positive y coordinates.
Points in Quadrant 3 have negative x and negative y coordinates.
Points in Quadrant 4 have positive x but negative y coordinates.

Every straight line can be represented by an equation: y = mx + b. The coordinates of every point on the line will solve the equation if you substitute them in the equation for x and y.

The slope m of this line - its steepness, or slant - can be calculated like this:
                                            m = change in y-value
                                                   change in x-value

The equation of any straight line, called a linear equation, can be written as: y = mx + b, where m is the slope of the line and b is the y-intercept.

The y-intercept of this line is the value of y at the point where the line crosses the y axis.

In the previous lesson, you learned how to graph points on the coordinate plane. We can connect two points with a straight line.

To graph the equation of a line, we plot at least two points whose coordinates satisfy the equation, and then connect the points with a line. We call these equations "linear" because the graph of these equations is a straight line.

There are two important things that can help you graph an equation, slope and y-intercept.

Slope
We're familiar with the word "slope" as it relates to mountains. Skiers and snowboarders refer to "hitting the slopes." On the coordinate plane, the steepness, or slant, of a line is called the slope. Slope is the ratio of the change in the y-value over the change in the x-value. Carpenters and builders call this ratio the "rise over the run." Using any two points on a line, you can calculate its slope using this formula.

Let's use these two points to calculate the slope m of this line.
A = (1,1) and B = (2,3)

Subtract the y value of point A from the y-value of point B to find the change in the y value, which is 2. Then subtract the x value of point A from the x value of point B to find the change in x, which is 1. The slope is 2 divided by 1, or 2.

When a line has positive slope, like this one, it rises from left to right.

WATCH OUT! Always use the same order in the numerator and denominator!

It doesn't really matter whether you subtract the values of point A from the values of point B, or the values of point B from the values of point A. Try it - you'll get the same answer both ways. But you must use the same order for both the numerator and denominator!

You can't subtract the y value of point A from the y value of point B, and the x value of point B from the x value of point A - your answer will be wrong.

Let's look at another line. This line has a negative slope, it falls from left to right. We can take any two points on this line and find the slope. Let's take C (0, -1) and D (2, -5).

Using these two points, we can calculate the slope of this line. We subtract the y value of point C from the y value of point D, and the x value of point C from the x value of point D, and divide the first value by the second value. The slope is -2.

Y-Intercept
There's another important value associated with graphing a line on the coordinate plane. It's called the "y intercept" and it's the y value of the point where the line intersects the y- axis. For this line, the y-intercept is "negative 1." You can find the y-intercept by looking at the graph and seeing which point crosses the y axis. This point will always have an x coordinate of zero. This is another way to find the y-intercept, if you know the equation, the y-intercept is the solution to the equation when x = 0.

Equations
Knowing how to find the slope and the y-intercept helps us to graph a line when we know its equation, and also helps us to find the equation of a line when we have its graph. The equation of a line can always be written in this form, where m is the slope and b is the y-intercept:

y = mx + b

Let's find the equation for this line. Pick any two points, in this diagram, A = (1, 1) and B = (2, 3).

We found that the slope m for this line is 2. By looking at the graph, we can see that it intersects the y-axis at the point (0, –1), so –1 is the value of b, the y-intercept. Substituting these values into the equation formula, we get:

y = 2x –1

The line shows the solution to the equation: that is, it shows all the values that satisfy the equation. If we substitute the x and y values of a point on the line into the equation, you will get a true statement. We'll try it with the point (2, 3).

Let's substitute x = 2 and y = 3 into the equation. We get "3 = 3", a true statement, so this point satisfies the equation of the line.

To graph a linear equation, we can use the slope and y-intercept.

1. Locate the y-intercept on the graph and plot the point.

2. From this point, use the slope to find a second point and plot it.

3. Draw the line that connects the two points.

Let's draw the graph of this equation.

One method we could use is to find the x and y values of two points that satisfy the equation, plot each point, and then draw a line through the points. We can start with any two x values we like, and then find y for each x by substituting the x values into the equation. Let's start with x = 1.

Let's plot these points and draw a line through them.

Graphing Using Slope and Y-Intercept
There's another way to graph an equation using your knowledge of slope and y-intercept. Look at the equation again.

We can find the slope and y-intercept of the line just by looking at the equation: m = 1/2 and y intercept = 2.

Just by looking at these values, we already know one point on the line! The y-intercept gives us the point where the line intersects the y-axis, so we know the coordinates of that point are (0, 2), since the x value of any point that lies on the y axis is zero.

To find the second point, we can use the slope of the line. The slope is ½ , which gives us the change in the y value over the change in the x value. The change in the x value, the denominator, is 2, so we move to the right 2 units.

The change in the y value, the numerator, is positive one. We move up one unit. This gives us the second point we need. Now we can draw the line through the points.

This is the exact same line we found using the first method. Do you see that it's quicker and easier to use the y-intercept and the slope? You can use either method to graph the line, depending on what information you have about the line and its equation.

Closure Property of Addition

Sum (or difference) of 2 real numbers equals a real number

Additive Identity

a + 0 = a

Additive Inverse

a + (-a) = 0

Associative of Addition

(a + b) + c = a + (b + c)

Commutative of Addition

a + b = b + a

Definition of Subtraction

a - b = a + (-b)

Closure Property of Multiplication

Product (or quotient if denominator 0) of 2 reals equals a real number

Multiplicative Identity

a * 1 = a

Multiplicative Inverse

a * (1/a) = 1     (a 0)

(Multiplication times 0)

a * 0 = 0

Associative of Multiplication

(a * b) * c = a * (b * c)

Commutative of Multiplication

a * b = b * a

Distributive Law

a(b + c) = ab + ac

Definition of Division

a / b = a(1/b)

By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola; or in the special case when the plane touches the vertex: a point, line or 2 intersecting lines.

The type of section can be found from the sign of: B² - 4AC

The Conic Sections. For any of the below with a center (j, k) instead of (0, 0), replace each x term with (x-j) and each y term with (y-k).

(a+b) ² = a ² + 2ab + b ²

(a+b)(c+d) = ac + ad + bc + bd

a ² - b ² = (a+b)(a-b) (Difference of squares)

a ³ b ³ = (a b)(a ² ab + b ²) (Sum and Difference of Cubes)

x ² + (a+b)x + AB = (x + a)(x + b)

if ax ² + bx + c = 0 then x = ( -b (b ² - 4ac) ) / 2a (Quadratic Formula)

Powers

x ^a x ^b = x ^{(a + b)}

x ^a y ^a = (xy) ^a

(x ^a) ^b = x ^(ab)

x ^(a/b) = b^th root of (x ^a) = ( b^th (x) ) ^a

x ^(-a) = 1 / x ^a

x ^{(a - b)} = x ^a / x ^b

Logarithms

y = log_b(x) if and only if x=b ^y

log_b(1) = 0

log_b(b) = 1

log_b(x*y) = log_b(x) + log_b(y)

log_b(x/y) = log_b(x) - log_b(y)

log_b(x ⁿ) = n log_b(x)

log_b(x) = log_b(c) * log_c(x) = log_c(x) / log_c(b)

Synonyms: correspondence, mapping, transformation

Definition: A function is a relation from a domain set to a range set, where each element of the domain set is related to exactly one element of the range set.

An equivalent definition: A function (f) is a relation from a set A to a set B (denoted f: A®B), such that for each element in the domain of A (Dom(A)), the f-relative set of A (f(A)) contains exactly one element.

Some common functions (with discussions)

• trig functions
  • sine, sin(x)
  • cosine, cos(x)

Riemann Hypothesis

zeta(s) = 1/1^s + 1/2^s + 1/3^s + ... (s = a + it) all 0's of zeta(s) in strip 0<=a<=1 lie on central line a=1/2

Twin Primes occur infinitely

Twin primes are primes that are 2 integers apart. Examples include 5 & 7, 17 & 19, 101 & 103

Goldbach's Postulate

Every even # > 2 can be expressed as the sum of 2 primes.
4=2+2, 6=3+3, 8=3+5, 10=5+5, 12=5+7, .. , 100=3+97, ...

Euclid's Parallel Postulate

Through a point, not on a line, there exists exactly 1 line parallel to the given line. (Then there's those non-Euclidean people...)

(k=1..) 1/kⁿ = ?

Although others have found that this expression equals PI² / 6 when n=2, PI⁴ / 90 when n = 4 and similar solutions for all possible even values of n, no one has discovered an exact value when n is an odd integer (3, 5, 7, ...) (note: when n=1, the sum does not converge, but it does has relations to the gamma constant).

Vector Notation: The lower case letters a-h, l-z denote scalars. Uppercase bold A-Z denote vectors. Lowercase bold i, j, k denote unit vectors. <a, b>denotes a vector with components a and b. <x₁, .., x_n>denotes vector with n components of which are x₁, x₂, x₃, ..,x_n. |R| denotes the magnitude of the vector R.

|<a, b>| = magnitude of vector = (a ²+ b ²)

|<x₁, .., x_n>| = (x₁²+ .. + x_n²)

<a, b> + <c, d> = <a+c, b+d>

<x₁, .., x_n> + <y₁, .., y_n>= < x₁+y₁, .., x_n+y_n>

k <a, b> = <ka, kb>

k <x₁, .., x_n> = <k x₁, .., k x₂>

<a, b> <c, d> = ac + bd

<x₁, .., x_n> <y₁, ..,y_n> = x₁ y₁ + .. + x_n y_n>

R S= |R| |S| cos ( = angle between them)

R S= S R

(a R) (bS) = (ab) R S

R (S + T)= R S+ R T

R R = |R| ²

|R x S| = |R| |S| sin ( = angle between both vectors). Direction of R x S is perpendicular to A & B and according to the right hand rule.

        | i  j  k |

R x S = | r1 r2 r3 | = / |r2 r3|   |r3 r1|   |r1 r2| \

        | s1 s2 s3 |   \ |s2 s3| , |s3 s1| , |s1 s2| /

S x R = - R x S

(a R) x S = R x (a S) = a (Rx S)

R x (S + T) = R x S + Rx T

R x R = 0

If a, b, c = angles between the unit vectors i, j,k and R Then the direction cosines are set by:

    COs a = (R i) / |R|; COs b = (R j) / |R|; COs c = (R k) / |R|

|R x S| = Area of parrallagram with sides Rand S.

Component of R in the direction of S = |R|COs = (R S) / |S|(scalar result)

Projection of R in the direction of S = |R|COs = (R S) S/ |S| ² (vector result)

Synonyms: correspondence, mapping, transformation

Definition: A function is a relation from a domain set to a range set, where each element of the domain set is related to exactly one element of the range set.

An equivalent definition: A function (f) is a relation from a set A to a set B (denoted f: A®B), such that for each element in the domain of A (Dom(A)), the f-relative set of A (f(A)) contains exactly one element.

Some common functions (with discussions)

• trig functions
  • sine, sin(x)
  • cosine, cos(x)

Prelude: A vector, as defined below, is a specific mathematical structure. It has numerous physical and geometric applications, which result mainly from its ability to represent magnitude and direction simultaneously. Wind, for example, had both a speed and a direction and, hence, is conveniently expressed as a vector. The same can be said of moving objects and forces. The location of a points on a cartesian coordinate plane is usually expressed as an ordered pair (x, y), which is a specific example of a vector. Being a vector, (x, y) has a a certain distance (magnitude) from and angle (direction) relative to the origin (0, 0). Vectors are quite useful in simplifying problems from three-dimensional geometry.

Definition:A scalar, generally speaking, is another name for "real number."

Definition: A vector of dimension n is an ordered collection of n elements, which are called components.

Notation: We often represent a vector by some letter, just as we use a letter to denote a scalar (real number) in algebra. In typewritten work, a vector is usually given a bold letter, such as A, to distinguish it from a scalar quantity, such as A. In handwritten work, writing bold letters is difficult, so we typically just place a right-handed arrow over the letter to denote a vector. An n-dimensional vector A has n elements denoted as A1, A2, ..., An. Symbolically, this can be written in multiple ways:

A = <A1, A2, ..., An>
A = (A1, A2, ..., An)

Example: (2,-5), (-1, 0, 2), (4.5), and (PI, a, b, 2/3) are all examples of vectors of dimension 2, 3, 1, and 4 respectively. The first vector has components 2 and -5.

Note: Alternately, an "unordered" collection of n elements {A1, A2, ..., An} is called a "set."

Definition: Two vectors are equal if their corresponding components are equal.

Example: If A = (-2, 1) and B = (-2, 1), then A = B since -2 = -2 and 1 = 1. However, (5, 3) not_equal (3, 5) because even though they have the same components, 3 and 5, the component do not occur in the same order. Contrast this with sets, where {5, 3} = {3, 5}.

Definition: The magnitude of a vector A of dimension n, denoted |A|, is defined as

|A| = sqrt(A1^2 + A2^2 + ... + An^2)

Geometrically speaking, magnitude is synonymous with "length," "distance", or "speed." In the two-dimensional case, the point represented by the vector A = (A1, A2) has a distance from the origin (0, 0) of sqrt(A1^2 + A2^2) according to the pythagorean theorem. In the three-dimension case, the point represented by the vector A = (A1, A2, A3) has a distance from the origin of sqrt(A1^2 + A2^2 + A3^2) according to the three-dimensional form of the Pythagorean theorem (A box with sides a, b, and c has a diagonal of length sqrt(a2+b2+c2) ). With vectors of dimension n greater than three, our geometric intuition fails, but the algebraic definition remains.

Definition: The sum of two vectors A = (A1, A2, ..., An) and B = (B1, B2, ..., Bn) is defined as

A + B = (A1 + B1, A2 + B2, ..., An + Bn)

Note: Addition of vectors is only defined if both vectors have the same dimension.

Example:

(2, -3) + (0, 1) = (2+0, -3+1) = (2, -2).
(0.1, 2) + (-1, PI) = (0.1 + -1, 2 + PI) = (-0.9, 2+PI)

Justification: Physical and geometric applications warrant such a definition. IF a train travels East at 5 meters/second relative to the ground, which will be denoted in vector notation as VT = (0, 5), and a person on the train walks South at 1 meter/second relative to the train, which will be denoted as VP = (-1, 0), THEN the direction and speed that the person is traveling relative to the ground is represented by the vector VG = VT + VP = (0, 5) + (-1, 0) = (0 + -1, 5 + 0) = (-1, 5). This vector has a magnitude of |VG| = sqrt((-1)^2 + 5^2) = sqrt(6) = 2.449..., which means that the person is traveling at about 2.449 meters/second relative to the ground and the net direction is mostly East but slightly South.

Definition: The scalar product of a scalar k by a vector A = (A1, A2, ..., An) is defined as

kA = (kA1, kA2, ..., kAn)

Example:

2(5, -4) = (2*5, 2*-4) = (10, -8)
-3(1, 2) = (-3*1, -3*2) = (-3, -6)
0(3, 1) = (0*3, 0*1) = (0, 0)
1(2, 3) = (1*2, 1*3) = (2, 3)

Note: In general, 0A = (0, 0, ..., 0) and 1A = A, just as in the algebra of scalars. The vector of any dimension n with all zero elements (0, 0, ..., 0) is called the zero vector and is denoted 0.