Monday, May 7, 2012

Here's part of chapter 1 of the first book:

Chapter 1

Algebra Basics

What is Algebra?

Algebra is a type of mathematics where symbols are used to represent amounts that are unknown. These unknown quantities are generally combined with mathematical operations (addition, subtraction, multiplication, division, square root, cube root, exponents, etc) to form statements that describe the relationship of things that change over a period of time. These statements are expressed using equations, expressions and terms. Problems can be solved by translating words into algebraic equations. The description of a problem using an equation and other mathematical concepts is known as a mathematical model. A mathematical model can be used to solve numerous types of problems in every day life.

Suppose you are in a store looking for some new shirts. On one rack a brand is selling for \$17.95 per shirt and on another rack you see some shirts selling for \$20 for one and \$15 for each additional shirt. You decide that you need 4 new shirts, all of the first brand or all of the second brand. What is the total cost and which is the better deal?

You can use a mathematical model to figure out the total cost of the shirts. The procedure to solve such a problem can be thought of in terms of mathematical symbols and variables. A variable is an unknown quantity and is typically represented by a letter. In word problems, many times the variable is a letter which relates to the unknown quantity. For example, if you want to solve for the number of objects, use n. If you want the length or width of a two dimension figure, use l and w, respectively.

The model used for the shirts on the first rack is “c equals 17.95 times n” (c = 17.95n), where c is the total cost of shirts and n is the number of shirts. For the first brand of shirts, you will pay \$17.95 times 4 for a total of \$71.80.

The model used for the shirts on the second rack is “c equals 20 plus 15 times a” (c = 20 + 15a), where c is the total cost of shirts and a is the number of additional shirts purchased. The first shirt on the second rack is \$20 plus \$15 times 3 additional shirts for a total of \$65. Therefore the second brand is the better buy when you want to buy 4 shirts.

An algebraic expression is a combination of variables and/or numbers and mathematical operations (addition, subtraction, multiplication, division, raising to a power and finding a root). In the example above example, an equation is used to represent each model because they contain an = sign. It's easy to remember the difference between an expression and an equation. The word "equation" contains the word equate, which means equals. An equation can be solved for the given variable or variables.

Words and phrases can represent different mathematical operations. It's important to be able to translate words and phrases into equations when solving word problems.

Here are some common words and phrases and their associated mathematical operation.

minus, decreased, less than, reduced by: subtraction

twice, product, multiplied by, times, of: multiplication

quotient, divided by, ratio, into: division

Types of Numbers

In algebra you will be working with a wide variety of types of numbers. The types of numbers are as follows:

Integers, natural numbers, whole numbers, rational numbers, irrational numbers.

The set of all of these numbers form what is known as the real numbers.

Natural numbers are the easiest to understand. When you count, you start at 1 and count up to 2, 3, 4, 5, etc. assuming you are counting by 1. The natural way to count is counting by 1. If you think of natural numbers in that sense, it's easy to remember that the natural numbers are the set of numbers {1,2,3,4,5,6,7,....}

When thinking about whole numbers think of the word whole and what it means. When thinking of a whole, you think of the entire thing. The statement I ate the whole pie means that there is no portion of the pie left over, no fraction of the pie. The set of whole numbers are basically the set of natural numbers with 0 added to the set {0,1,2,3,4,5,6,7,.....}

Integers include all the whole numbers and also the negative of the whole numbers. So the set of integers are {...., -4, -3, -2, -1, 0, 1, 2, 3, 4, ....}. Even integers are integers that are divisible by 2, whereas odd integers are not divisible by 2.

Rational numbers are all numbers that can be represented as a ratio or a fraction. The easiest way to remember this is notice the word "ratio" is the first 5 letters in rational.

Some examples of rational numbers are as follows:

4 1/2, 16/5, 7/2 , 4.35, -12.2, -35

Terminating and repeating decimals are also rational because every terminating and repeating decimal can be written as a fraction.

For example 2/3 = .666..... (repeating)

2/11 = .18181... (repeating)

45% = 0.45 (terminating)

120.5 = 120 1/2 = 241/2 (terminating)

Irrational numbers are decimals that do not terminate and do not repeat. A very common irrational number is Pi (π) = 3.1415926...... many square roots are also irrational numbers, although not all of them. Those that can be simplified such as √9 = 3, √16 = 4, etc are rational.

Fractions and Decimals

Fractions and decimals represent the same thing in different ways. If you have half of a pie, you can represent that as the fraction ½ or the decimal 0.50. A quarter is ¼ or 0.25. Converting decimals to fractions can be done simply by moving the decimal point over to the right until you are past the last digit. That number becomes the numerator of the fraction (the number above the bar in a fraction). Then count how many decimal places you moved and that number represents a power of 10 which goes in the denominator of the fraction (the number under the bar in a fraction). For example, take 0.25 and move the decimal point over 2 places to the right to get 25. The 25 becomes the numerator of the fraction. Since you moved the decimal 2 places, that is the 2nd power of 10 or 102 = 10 times 10, which is 100. The fraction becomes 25/100, which is simplified to ¼ since 25 divides evenly into both 25 and 100.

Other examples:

0.001 changed to a fraction becomes 1/1000 since the decimal point is moved 3 places to the right, 103 = 1000.

0.37 changed to a fraction becomes 37/100 since the decimal point is moved 2 places to the right, 102 = 100.

2.426 changed to a fraction becomes 2426/1000, which can be simplified to 1213/500 since 2 divides evenly into both 2426 and 1000.

To change a fraction to a decimal we must divide the numerator by the denominator. For practice, try long division with fractions and check your work on a calculator.

Sometimes quantities will be represented as percentages, which can be converted to decimals by moving the decimal point over 2 places to the left. An easy way to remember to move the decimal 2 places is to think of the word cent. That is part of percent. A cent is 1/100 of a dollar. One percent (represented as 1%) is 1/100 and 100 is 102. Recall that the power corresponds to the number of decimal places you move.

Example:

Change 35% to a decimal.

Take 35 and move the decimal point over two places to the left to get 0.35.

Note that the decimal point is to the right of the last number in a whole number.

From this point you can change 0.35 into a fraction.

35/100 simplifies to 7/20 since 5 divides evenly into both 35 and 100.

Note that all numbers ending in 5 are divisible by 5.

Example:

Change 524% to a decimal.

Take 524 and move the decimal point over two places to the left to get 5.24.

From this point you can change 5.24 into a fraction.

524/100 = 131/25 since 4 divides evenly into both 524 and 100.

Note if you don't know that 4 divides evenly, you can start by dividing the numerator and denominator by 2 since all even numbers are divisible by 2.

Fractions with a numerator larger than the denominator are known as improper fractions. These fractions can be changed into a mixed number, which is a combination of a whole number and a proper fraction (numerator is smaller than the denominator).
Example:

18/5, 32/7, and 9/4 are improper fractions.

2 ½, 5 7/8 and 11 2/3 are mixed numbers.

To change 18/5 into a mixed number you divide 5 into 18, which divides into 3 times since 5 times 3 is 15. The remainder is 3, which turns into 3/5 as a fraction (the remainder over the denominator). The mixed number is then 3 3/5. To change a mixed number to an improper fraction, multiply the denominator of the fraction by the whole number then add the numerator. Take that result and put it over the denominator.

Example:

Change 4 3/7 to an improper fraction.

7(4) = 28 (denominator times whole number)

28 + 3 = 31 (add the numerator)

31/7 (result over the denominator)

You can also think of a fraction as a ratio between two numbers. The fraction 1/3 can also be written as 1 to 3 or 1:3. To find an equivalent fraction you can simply multiply the numerator and denominator by the same number.

Example: (1/3)(2/2) = 2/6

(1/3)(3/3) = 3/9

(1/3)(4/4) = 4/12

(1/3)(5/5) = 5/15

These are all equivalent fractions. This technique is used when you find a common denominator when adding or subtracting fractions.

Example: 2/3 + 5/6

The common denominator is 6, so we have to get 2/3 into an equivalent fraction with a denominator of 6. We have to multiply the denominator by 2 to get 6, so we must multiply the numerator by 2 as well. Therefore,

2/3 + 5/6

(2/3)(2/2) + 5/6

4/6 + 5/6

Now we can add the fractions. Finding common denominators for adding and subtracting fractions will be discussed in further detail in a later chapter.

A proportion is two or more equivalent fractions equal to each other.

Examples of proportions include

2/3 = x/6, 4/9 = 12/27, -1/3 = 2/6, x/8 = 6/16 and -3/2 = -12/8.

Prime Numbers, Composite Numbers and Absolute Value

A prime number is a natural number that is only divisible by 1 and itself. Note that 1 is not considered a prime number. The first several prime numbers are 2,3,5,7,11,13,17,19,23,29 and 31.

A composite number is a natural number that is not a prime number, therefore it is divisible by more than just itself and 1.

The first several composite numbers are 4,6,8,9,10,12,14,15,16,18,20,21,22,24,25,26,27,28 and 30.

Absolute value is essentially the distance between a number and 0 and since distance is always a positive number, the absolute value of any number is always positive.

For example:

Take two numbers -4 and 4. The distance from -4 to 0 is 4 and the distance from 4 to 0 is also 4, so the absolute value of -4 and 4 are both 4.

Absolute value is noted as follows:

Absolute value of 4... |4| = 4

Absolute value of -4... |-4| = 4

Numbers with the same absolute value are known as opposites. Notice in the following examples that opposites add to 0.

Examples of opposites:

5 and – 5, 5 + (-5) = 0

14 and -14, 14 + (-14) = 0

-4.3 and 4.3, -4.3 + 4.3 = 0

Review Problems: Set 1

1. The cost of a general admission ticket to a baseball game is \$8. Groups of 20 or more receive a \$20 discount. Write a mathematical model to describe the total cost of the tickets for groups of 20 or more.

2. Determine whether each of the following is an algebraic expression or an equation.

a. 3x + 6 = 14

b. (14x – 32)/16

c. 6a +14b - 9c + 1

d. y = 13x – 14

e. F = 1.8C + 32

3. Use variables to create a formula for the following scenarios:

a. The perimeter of a rectangle is two times the length plus two times the width.

b. The area of a triangle is one half the length of the base times the height.

c. The sales price is the difference between the regular price and the amount of the discount.

Classify each as rational or irrational
4. 1/3
5. √14
6. 3 2/9
7. -14.8
8.
9. 15%

Change each fraction to a decimal and determine whether the decimal is terminating or repeating.

10. 8/9
11. 10/4
12. -4/22
13. . 24/40

Change each percent to a decimal.

14. 14%

15. 435%

16. 2.7%

Change each decimal to a fraction.

17. 0.85

18. 0.096

19. 0.146

Change each improper fraction to a mixed number.

20. 23/4

21. 42/5

22. 19/6

Change each mixed number to an improper fraction.

23. 4 4/7

24. 10 1/8

25. 7 2/9

Real Numbers: Operations and Properties

Order of Operations

A mathematical operation is a calculation by mathematical methods which include addition, subtraction, multiplication, division, raising a number to an exponent and taking absolute value. Oftentimes there is more than one operation in an expression. When this occurs there is a specific order in which the expression must be evaluated. First you perform all calculations inside parentheses, then evaluate exponents and roots, multiplication and division from left to right, then addition and subtraction from left to right.

The order of operations is often difficult to remember, but I like to use a system I learned when I was in school. Remember the simple sentence “Please excuse my dear aunt Sally”. The first letter in each word is the first letter in each operation in the correct order (PEMDAS).

Here's a few examples on how evaluating the expression ignoring the order of operations will give a different and wrong answer than using the order of operations.

Example: Evaluate 4+ 3[-1 + 6(5-3)2 +2]

Ignoring the order of operations you could take 4+3 to get 7, then multiply by -1 to get -7. From there add 6 to get -1. Next you would multiply by 5 to get -5, subtract 3 to get --8 then square it to get 64. Finally add 2 to get 66.

Using the correct order of operations you get the following

4 + 3[-1 + 6(2)2 +2] (evaluate the 5-3 inside the parentheses)

4 + 3[-1 + 6(4) +2] (evaluate 22)

4+ 3[-1 + 24 + 2] (multiply 6 and 4)

4+ 3(25) (add the numbers inside the brackets)

4 + 75 (multiply 3 and 25)

Example: Evaluate 5(6 + 4)2 - 3(2 - 32) + 4

Ignoring the order of operations, you would take 5 times 6 to get 30, then add 42 to get 30 + 16 = 46. From there subtract 3 to get 43 then multiply by 2 to get 86. Next, subtract 32 which would be 86 – 9 = 77. Finally add 4 to get 81.

Using the correct order of operations you get the following
5(10)2 - 3(2- 32) + 4 (evaluate the 6 + 4 inside the parentheses)
5(10)2 - 3(2 - 9) + 4 (evaluate 32 inside the parentheses)

5(10)2 -3(-7) + 4 (evaluate 2 – 9 inside the parentheses)

5(100) - 3(-7) + 4 (evaluate 102)

500 - 3(-7) + 4 (multiply 5 and 100)

500 + 21 + 4 (multiply -3 and -7)

525 (add 500, 21 and 4)

Note if you have absolute value symbols in the problem, perform them with the parentheses and other grouping symbols first.

Properties

There are many different properties of real numbers.

The commutative property of addition states that changing the order when adding doesn't affect the sum.

Think of the word commute and how one might commute to work or to school. This means that a person travels to and from work or school. The distance is the same both directions, assuming the same route is taken to and from. So for any two numbers, a and b, a + b = b + a demonstrates the commutative property.

The commutative property of multiplication states that changing the order when multiplying doesn't affect the outcome. For any two numbers a and b, a · b = b · a.

Note that a · b is often written as ab.

The associative property of addition states that changing the grouping does not affect the outcome when adding. Think of the meaning of the word associate. When one associates with someone, he or she is grouped with that person. Same can be applied here. For any numbers a,b and c, a +(b + c) = (a + b)+c.

The associative property of multiplication follows the same principle and states that changing the grouping does not affect the outcome when multiplying. Therefore, (ab)c= a(bc).

The distributive property of multiplication shows how multiplication distributes over addition. For numbers a,b and c, a · (b + c) = ab + ac.

The identity properties show how when you add 0 to a number or multiply 1 to a number you still get that number. Identity property of addition is (a + 0 = a) and the Identity property of multiplication is a(1)= a.

The inverse property of addition shows how when you add a number to its inverse (or opposite), the result is 0. For example -4 + 4= 0.

The inverse property of multiplication shows how when you multiply a number by its inverse, the result is 1. For example 2(½) = 1.

Division properties:

0/a = 0 for all numbers of a, except when a is 0

0/a is undefined if a = 0

0/0 is indeterminate

Note that for all practical purposes, it's important to understand how to use the property more than knowing the name of the property.

Examples:

1 + 2 = 2 + 1

3 = 3

Commutative property of multiplication

2· 3 = 3 · 2

6 = 6

It doesn't matter what order we add or multiply numbers together, the result will be the same.

1 + (2 + 3) = (1 + 2) + 3

1 + 5 = 3 + 3 (from the order of operations, we add what's inside the parentheses first)

6 = 6

Associative property of multiplication

2 · (3 · 4) = (2 · 3) · 4

2 · 12 = 6 · 4 (from the order of operations, we multiply what's inside the parentheses first)

24 = 24

Grouping the numbers differently does not affect the answer when adding or multiplying numbers.

Distributive property of multiplication

2(4 + 5) = 2(4) + 2(5) (multiplied the 2 by 4 and then 2 by 5)

2(9) = 2(4) + 2(5) (added numbers inside the parentheses on left side of the = sign)

18 = 8 + 10 (multiplied on both sides of the equation)

18 = 18

Identity property of addition and identity property of multiplication

2 + 0 = 2, 2(1) = 2

When you add 0 to a number, the result is that number. When you multiply 1 to a number, the result is that number.

Inverse property of addition and inverse property of multiplication

2 + (-2) = 0, 2(1/2) = 1

When you add a number to its opposite (the negative of that number), the result is 0. When you multiply a number by its inverse (switch numerator and denominator, ex: 1/3 is inverse of 3/1), the result is 1.

Using Operations With Real Numbers

When adding two positive numbers, the result is a positive number. When adding two negative numbers, the result is a negative number.

Examples:

4 + 9 = 13, -4 + -9 = -13.

When subtracting two positive numbers the result will be positive if the smaller number is being subtracted from the larger number and negative if the larger number is being subtracted from the smaller number. The result will be zero is both numbers have the same value.

Examples:

10 - 15 = -5, 15 - 10 = 5, 10 - 10 = 0.

If you have trouble with this concept consider a real life situation involving money. If you have \$15 and give someone \$10, you still have \$5 left. If you have \$10 and owe someone \$15, you can pay \$10 and still owe \$5, this having a \$5 deficit or -\$5.

When subtracting a negative number, you add the second number to the first. The double negative turns into a positive, or an addition. For example, -4 - (-6) turns into -4 + 6, which is 2. Think of a number line and this might make more sense. Typically when subtracting you move to the left on the number line, the the double negative makes you move right on the number line. It's as if you have two magnets with the negatives against each other, it repels, pushing the magnets in opposite directions.

Examples:

-3 - (-13) = -3 + 13 = 10, -15 - (-7) = -15 + 7 = -8

Multiplying and dividing numbers with like signs results in a positive number, whereas multiplying and dividing numbers of opposite signs yields a negative number.

Examples:

24/6 = 4, -24/6 = -4, (-2)(-7) = 14, (-2)(7) = -14

If you have a number raised to a positive integer exponent, you multiply that number by itself the number of times indicated in the exponent. If you have a fraction raised to a positive integer exponent, you multiply the fraction by itself the number of times indicated in the exponent.

Examples:

53 = 5 (5)(5)

24 = 2(2)(2)(2)

(1/3)3 = (1/3)(1/3)(1/3).

(3/5)3 = (3/5)(3/5)(3/5)

When multiplying fractions, multiply the numerators and the denominators separately.

Examples:

(4/7)(2/5) = (4)(2)/(7)(5) = 28/10

(2/9)(5/6) = (2)(5)/(9 )(6) = 10/54

When dividing fractions, multiply the first fraction by the reciprocal of the second.

Example:

(4/7) / (2/5) = (4/7)(5/2) = 20/14

(2/9) / (5/6) = (2/9)(6/5) = 12/45

Evaluating Expressions

Sometimes you will encounter problems where you have to evaluate an expression given the values of the variables in the expression.

Example: Evaluate the expression for x = 4, y = 2 and z = -1

(2xy – 3)2 + 5z - 14

In a problem like this, you substitute the values for the variables into the expression. Sometimes people get confused and put the numbers in the expression but fail to remove the variable. If you think of a substitute, it's a replacement. A substitute teacher is in for the regular teacher. Both teachers are not present at the same time. In a sporting event, when a player is put in as a substitute, the other player leaves the game. So in the previous problem, evaluating would be as follows:

[2(4)(2) - 3]2+5(-1) -14

Following the order of operations, you get

(16-3)2 +5(-1) -14 (parentheses)

132 + 5(-1) - 14 (parentheses)

169 + 5(-1) -14 (exponent)

169 - 5 -14 (multiplication)

150 (subtraction)