Algebra-Mathematics_Resource_Part_I-Chu.pdf

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Mathematics Resource

Part I of III: Algebra

TABLE OF CONTENTS

T HE N ATURE OF N UMBERS

II. E XPRESSIONS , E QUATIONS , AND P OLYNOMIALS (EEP!)

III. A B IT O N I NEQUALITIES

IV. M OMMY , W HERE DO L INES C OME F ROM ?

V. S YSTEMS OF E QUATIONS

VI. B IGGER S YSTEMS OF E QUATIONS

VII. T HE P OLYNOMIALS ’ F RIEND , THE R ATIONAL E XPRESSION

VIII. I RRATIONAL N UMBERS

IX. Q UADRATIC E QUATIONS

X. A PPENDIX

XI. A BOUT THE A UTHOR

CRAIG CHU

CALIFORNIA INSTITUTE OF TECHNOLOGY

REVIEWED BY

LEAH SLOAN

PROOFREADER EXTRAORDINAIRE

FOR

MY TWO COACHES ( MRS . REEDER , MS . MARBLE , AND MRS . STRINGHAM )

FOR THEIR PATIENCE , UNDERSTANDING , KNOWLEDGE , AND

PERSPECTIVE

ALGEBRA RESOURCE

D EMI D EC

R ESOURCES AND E XAMS

ALGEBRA

A LITTLE ON THE NATURE OF NUMBERS

Real Number

Additive Inverse

Multiplicative Inverse Subtraction

Division

Cancellation Law

Negative

Distributive Property

Commutative

Property

Associative Property Additive Identity

Multiplicative Identity

When you think about math, what comes to your mind? Numbers. Numbers make the world go

round; they can be used to express distances and amo unts; they make up your pho ne number and

zip code, and they mark the passage of time. The concepts connected with numbers have been

around for ages. A real number is any number than can exist on the number line. At this point in

your schooling, you are likely to have already come across the number line; it is usually drawn as a

horizontal line with a mark representing zero. Any point on the line can represent a specific real

number. 1

One of the easiest and most obvious ways to classify numbers is as either positive or negativ e .

What does it mean for a number to be negative? Well, first of all, it is graphed to the left of the zero

mark on a horizontal number line, but there’s more. A negative signifies the opposite of whatever is

negated. For example, to say that I walked east 50 miles would be mathe matically equivalent to

saying that I walked west negative 50 miles. 2 I could also say that having a bank bala nce of -$41.90

is the same as being $41.90 in debt. The negative in mathematics represents a logical opposite.

When two numbers are added , their values combine. When two numbers are multiplied, we perform

repeated (or multiple) additions.

Examples:

3 + 5 = 8

-11 + 9 = -2 19 – 2 = 17 -3 + 91 = 88 12 – 15 = -3

3 × 5 = 5 + 5 + 5 = 15

4 × 2 = 2 + 2 + 2 + 2 = 8

5 × 1 = 1 + 1 + 1 + 1 + 1 = 5

2 × 4 = 4 + 4 = 8

Here, I’m just rehashing things with which most of you readers are probably already acquainted. 3 I

know of very few high school students (and even fewer decathletes) who have trouble with basic

addition and multiplication of real numbers. Sometimes, negatives complicate the fray a bit, but for a

brief review, you should know the negation rules for multiplication and division.

1 It’s possible you haven’t yet come across non-real numbers. I wouldn’t worry about it. Non-real

numbers enter the picture when you take the square root of negatives, and they shouldn’t be your

concern this decathlon season.

2 Um… I wouldn’t recommend actually saying something like this on a regular basis to ordinary people.

I just wouldn’t. Trust me on this one.

3 In fact, this resource is going to operate under the assumption that decathletes already have experience

with much of this year’s algebra curriculum. I’m not going to go into detail about the mechanics of

arithmetic. I’m also, rather presumptuously, going to use × , • , and ( ) interchangeably to indicate

multiplication.

ALGEBRA RESOURCE

o Ne gative × Negative = Positive

o Ne gative × Positive = Negative

o Positive × Negative = Negative

o Ne gative ÷ Negative = Positive

o Ne gative ÷ Positive = Negative

o Positive ÷ Negative = Negative

Make a note that these sign patterns are the same for both multiplication and division; we’ll talk more

about that in just a quick sec. Also, notice that I didn’t list addition and subtraction properties of

negative numbers. When something is negative, it means we go leftward on the number line, while

positives take us rightward. When you add and subtract positives with negatives, the sign of the

answer will have the same sign as the “bigger” number.

Note a few more examples here:

5 + -5 = 0

2 × 1 = 1

3 + -3 = 0

5 × 1 = 1

In the examples above, we see two instances of two numbers adding to 0 and two instances of two

numbers multiplying to a product of 1. If you look closely, there is consistency here. The additive

inv erse (or the opposite ) of any number “x” is denoted by “-x.” The multiplicative inverse (or the

reciprocal ) of any number “y” is written “ 1 .” A number and its additive inverse sum to zero; a

number and its multiplicative inverse multiply to one.

Additive Inverse of a:

-a

a + (-a) = 0

Multiplicative inverse of a:

1 = 1

a ×

Note a few more examples here:

-3 + 0 = -3 12 × 1 = 12

9 + 0 = 9

-8 × 1 = -8

In these four examples, we see two instances of the addition of 0 and two instances of multiplication

by 1. The operations “adding 0” and “multiplying by 1” produce results identical to the original

numbers, and thus we can name two mathematical identities.

The Additive Identity :

a + 0 = a

The Multip licativ e Identity :

a × 1 = a

0 is known as the “additive identity element,” and 1 is known as the “multiplicative identity element.”

With identities and inverses in mind, we can continue with our discussion of algebra. To say “x + -x

= 0” is the same as “x – x = 0.” This may sound weird to say at first, but it is one of the closely

guarded secrets of mathematics that subtraction and division, as separate operations, do not really

exist. Youngsters are trained to perform simple procedures that they call subtract and divide, but

from a mature, sophisticated, mathematical point of view, those operations are nothing more than

special cases of addition and multi plication.

ALGEBRA RESOURCE

Definition of Subtraction:

x – y = x + (-y)

Definition of Division:

x ÷ y = x •

In addition to knowing these formal definitions for subtraction and division, the astute decathlete

should be (and probably already is) familiar with several properties of the real numbers. These

include the Commutative Properties , the Associative Properties , and the Distributive

Properties .

Commutative Property of Addition:

m + n = n + m

Commutative Property of Multiplication:

m • n = m • n

Associative Property of Addition:

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

Associative Property of Multiplication:

a • (b • c) = (a • b) • c

Distributive Property:

a • (b + c) = a • b + a • c

Example:

Arbitrarily pick some real numbers and verify the distributive property.

Solution :

I’ll choose 3, 5, and 7, for a, b, and c, respectively.

Distributive Property : 3 • (5 + 7) = 3 • 5 + 3 • 7. Can we verify this? The left side of the

equation gives 3 • (5 + 7) = 3 • 12 = 36. The right side of the equation gives 3 • 5 + 3 • 7 =

15 + 21 = 36. The Distributive Property holds.

Be wary; sometimes, confused students have conceptual pro ble ms with the Distributive Property. I

have on occasion seen people write that a + (b • c) = a + b • a + c. Such a thing is wrong.

Remember that multiplication distributes over addition, not vice versa.

This is also a good time to discuss the algebraic order of operations. The example above assumes

an elementary knowledge that operations grouped in parentheses are performed first. The official

mathe matical order of operations is Parentheses/Groupings, Expo nents 4 , Multiplication/Division,

Addition/Subtraction. In many pre-algebra and algebra classes, a common mnemonic device for this

is “ P lease e xcuse m y d ear A unt S ally.”

A brief example is now obligatory to expand on the order of operations.

Example:

−

(

−

(

−

)

Solution:

This may seem a little extreme as a first example, but it is fairly simple if approached

systematically. Remember, the top and bottom (that’s numerator and denominator for you

terminology buffs) of a fraction should generally be evaluated separately and first; a giant

fraction bar is a form of parentheses, a grouping symbol. On the top, we find two sets of

parentheses, and start with the inside one, so -2 is our starting point. The exponent co mes

first, so we evaluate (-2) 4 = (-2)(-2)(-2)(-2) = 16. Then, substituting gives -4 + 16 = 12. We

4 An exponent, if you do not know, is a small superscript that indicates “the number that I’m above is

multiplied by itself a number of times equal to me.” If it helps, imagine the exponent saying this in a cute

pair of sunglasses. For example, 3 4 = 3 × 3 × 3 × 3 = 81. 4 is the exponent. Come to think of it,

exponents look a lot like footnote references. - Craig

ALGEBRA RESOURCE

are not done, but the whole expression reduces to the considerably simpler

−

3 2

−

The first bit in the numerator will cause the most misery in this expression, as many people

make this very common error: -3 2 = (-3)(-3) = 9. DON’T DO THIS! By our standard order of

operations, the exponent must be evaluated first. It is often convenient to think of a negative

sign as a •

( , rather than a subtraction. By order of operations, negatives are evaluated

with multiplication. The correct evaluation of the numerator is -21:

− 1

-3 2 – 12 =

(-1) × 3 2 – 12 =

(-1) × 9 – 12 =

-9 – 12 =

-21

Once this is done, we turn our attention to the fairly straightforward denominator. We take

order of operations into account here.

1 + 2 × 3 = 1 + 6 = 7. Now we put everything back into the original expression, and matters

seem far simpler:

−

. I guess you could say we did all of that work to

get nothing for our answer. Hah! Never forget the difference between the forms (-x) y and

–x y !

The last of the algebra basics to be discussed is the cancellation law . The cancellation law in its

abstract form can look quite intimidating.

Cancellation Law:

ab = as long as

≠ a and

≠ c

This little formula can be quite intimidating, but the cancellation law in layman’s terms says that

anything divided by itself is 1 and can be “cancelled out.” You’ve probably been using this law for

quite some time, possibly without even realizing it, to simplify fractions. You know of course that

8 = , but you may have become so familiar with the practice that you’re not even aware of the

cancellation law operating “behind the scenes”. Observe:

⋅

The numerator and denominator are written as products, and a common factor of 4 is “cancelled

out.” Don’t think that simplifying fractions is the only application for the cancellation law, though. It’s

that very law, albeit applied in reverse, that allows us to produce common denominators.

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