Algebra Mantu

7/28/2019 Algebra Mantu

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ALGEBRA IS A METHOD OF WRITTEN CALCULATIONS that

help us reason about numbers. At the very outset, the student should

realize that algebra is a skill. And like any skill -- driving a car, bakingcookies, playing the guitar -- it requires practice. A lot of practice.

Written practice. That said, let us begin.

The first thing to note is that in algebra we use letters as well as

numbers. But the letters represent numbers. We imitate the rules of

arithmetic with letters, because we mean that the rule will be true for any

numbers.

The letters a and b mean Whatever numbers are in the numerators. The

letter c means Whatever number is in the denominator. The rule means:

"Whatever those numbers are, add the numerators

and write their sum over the common denominator."

Algebra is telling us how to do any problem that looks like that. That is

one reason why we use letters.

(The symbols for numbers, after all, are nothing but written marks. And

so are letters As the student will see, algebra depends only on the

patterns that the symbols make.)

The numbers are the numerical symbols, while the letters are called

literal symbols.

Question 1. What are the four operations of arithmetic, and

what are their operation signs?.

1) Addition: a + b. The operation sign is + , and is called the plus

sign. Read a + b as "a plus b."


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For example, if a represents 3, and b represents 4, then a + b

represents 7.

2) Subtraction: a b. The operation sign is , and is called the

minus sign. Read a b as "a minus b."

3) Multiplication: a b. Read a b as "a times b."

The multiplication sign in algebra is a centered dot. We do not use the

multiplication cross , because we do not want to confuse it with the

letter x.

Do not confuse the centered dot -- 25, which in the United States means

multiplication -- with the decimal point: 2.5.

However, we often omit the multiplication dot and simply write ab.

Read "a, b." In other words, when there is no operation sign between

two letters or between a letter and a number, it always means

multiplication. 2x means 2 times x.

4) Division: a/b . Read a/b as "a divided by b."

Note: In algebra we call a + b a "sum" even though we do not name an

answer. As the student will see, we name something in algebra simply

by how it looks. In fact, you will see that you do algebra with your eyes,

and then what you write on the paper, follows.

Similarly, we call a b a difference, ab a product, and a/b. a quotient.

This sign = of course is the equal sign, and we read this --

a = b

-- as "a equals (or is equal to) b."

That means that the number on the left that a represents, is equal to the

number on the right that b represents. If we write


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a + b = c,

and if a represents 5, and b represents 6, then c must represent 11.

Question 2. What is the function of parentheses () in algebra?

Parentheses signify that we should treat what they enclose

as one number.

Problem 1. In algebra, how do we write

a) 5 times 6? 5 6

b) x times y? xy

c) x divided by y? x / y

d) x plus 5 plus x minus 2? (x + 5) + (x 2)

e) x plus 5 times x minus 2? (x + 5)(x 2)

There is a common misconception that parentheses always signify

multiplication. In Lesson 3, in fact, we will see that we use parentheses

to separate the operation sign from the algebraic sign. 8 + (2).

Question 3. Terms versus factors.

When numbers are added or subtracted, they are called terms.

When numbers are multiplied, they are called factors.

Here is a sum of four terms: a b + c d.

In algebra we speak of a "sum" of several terms, even though there are

subtractions. In other words, anything that looks like what you see

above, we call a sum.

Here is a product of four factors: a.b.c.d.


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The word factor always signifies multiplication.

And again, we speak of the "product" abcd, even though we do not name

an answer.

Problem 3. In the following expression, how many terms are there?

And each term has how many factors?

2a + 4ab + 5a(b + c)

There are three terms. 2a is the first term. It has two factors:

2 and a.

4ab is the second term. It has three factors: 4, a, and b.

And 5a(b + c) is all one term. It also has three factors: 5, a, and

(b + c). The parentheses mean that we should treat whatever is enclosed

as one number.

Question 4. When there are several operations,

what is the order of operations?

Before answering, let us note that since skill in science is the reason

students are required to learn algebra; and since orders of operations

appear only in certain forms, then in these pages we present only those

forms that the student is ever likely to encounter in the actual practice of

algebra. The division sign is never used in scientific formulas, only

the division bar. And the multiplication cross is used only in scientific

notation -- therefore the student will never see the following:

Such a problem would be purely academic, which is to say, it is an

exercise for its own sake, and is of no practical value. It never even

comes up in arithmetic


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The order of operations is as follows:

(1) Evaluate the parentheses, if there are any, and if they require

evaluation.

(2) Evaluate the powers, that is, the exponents.

(3) Multiply or divide -- it does not matter.

(4) Add or subtract.

Question 5. What do we mean by the value of a letter?

The value of a letter is a number. It is the number that will replace the

letter when we do the order of operations.

Question 6. What does it mean to evaluate an expression?

It means to replace each letter with its value, and then do the order of

operations.

Example 6. Let x = 10, y = 4, z = 2. Evaluate the following. a) x +yz = 10 + 4 2 b) (x + y)z = (10 + 4)2

In each case, copy the pattern. Copy the + signs and copy the

parentheses ( ). When you come to x, replace it with 10. When you

come to y, replace it with 4. And when you come to z, replace it with 2.

Problem 6. Let x = 10, y = 4, z = 2, and evaluate the following. a)

x + 2(y + z) = b) (x + 2)(y + z) =Question 7. Why is a literal symbol also called a variable?

Because its value may vary.


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A variable, such as x, is a kind of blank or empty symbol. It is therefore

available to take any value we might give it: a positive number or, as we

shall see, a negative number; a whole number or a fraction.

Problem 7. Two variables. Let the value of the variable y depend

on the value of the variable x as follows:

y = 2x + 4.

Calculate the value of y that corresponds to each value of x:

When x = 0, y = 2 0 + 4 = 0 + 4 = 4.

When x = 1, y = 2 1 + 4 = 2 + 4 = 6.

Algebraic expressions

Real problems in science or in business occur in ordinary language. To

do such problems, we typically have to translate them in to algebraic

language.

Problem 8. Write an algebraic expression that will symbolize each of

the following.

a) Six times a certain number. 6n, or 6x, or 6m. Any letter will do.

b) Six more than a certain number. x + 6

c) Six less than a certain number. x 6

d) A certain number less than 6. 6 x

e) A number repeated as a factor three times. x x x = x3

f) A number repeated as a term three times. x + x + x

g) The sum of three consecutive whole numbers. The idea, for

example,


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g) of 6 + 7 + 8. [Hint: Let x be the first number.]

g) x + (x + 1) + (x + 2)

h) Eight less than twice a certain number. 2x 8

i) One more than three times a certain number. 3x + 1

Now an algebraic expression is not a sentence, it does not have a verb,

which is typically the equal sign = . An algebraic statement has an equal

sign.

Problem 9. Write each statement algebraically.

a) The sum of two numbers is twenty. x + y = 20.

b) The difference of two numbers is twenty. x y = 20.

c) The product of two numbers is twenty. xy = 20.

d) Twice the product of two numbers is twenty. 2xy = 20.

e) The quotient of two numbers is equal to the sum of those numbers. x

y = x + y.

Formulas

Example 7. Here is the formula for the area A of a rectangle whose

base is b and whose height is h.

A = bh.


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"The area of a rectangle is equal to the base times the height."

And here is the formula for its perimeter P -- that is, its boundary:

P = 2b + 2h.

"The perimeter of a rectangle is equal to two times the base

plus two times the height."

For, in a rectangle the opposite sides are equal.

Problem 10. Evaluate the formulas for A and P when b = 10 in, and h

= 6 in.

A = bh = 10 6 = 60 in2.

P = 2b + 2h = 2 10 + 2 6 = 20 + 12 = 32 in.

Problem 11. The area A of trapezoid is given by this formula,

A = (a + b)h.

Find A when a = 2 cm, b = 5 cm, and h = 4 cm.

A = (2 + 5)4 = 7 4 = 7 2 = 14 cm2.

When 1 cm is the unit of length, then 1 cm ("1 square centimeter") is

the unit of area.

SIGNED NUMBERS Positive and negative

IN ARITHMETIC we cannot subtract a larger number from a smaller:

2 3.

But in algebra we can. And to do it, we invent "negative" numbers.

2 3 = 1 ("Minus 1" or "Negative 1").


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That is, since 2 2 = 0,

then 2 3 is one "less" than 0. We call it 1. 1 is a signed number.

Its algebraic sign is ("minus")

1. What are the two parts of a signed number?

Its algebraic sign, + or , and its absolute value, which is simply the

arithmetical value, that is, the number without its sign.

The algebraic sign of +3 ("plus 3" or "positive 3") is + , and its absolute

value is 3.

The algebraic sign of 3 ("negative 3" or "minus 3") is . The absolute

value of 3 is also 3.

For better or for worse, the minus sign '' is not only the sign of a

negative number. It is also the sign for the operation of subtraction.

Those are two completely different concepts.

As for the algebraic sign + , normally we do not write it. The algebraic

sign of 2, for example, is understood to be + .

As for 0, it is useful to say that it has both signs: 0 = +0 = 0.

2. How do we subtract a larger number from a smaller?

5 8

1. What will be the sign of the answer?2. It would not be wrong to say that we cannot take 8 from 5. We

can however take 5 from 8 -- and that is what we do -- but we

report the answer with a minus sign


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5 8 = 3.

Even in algebra we can only do ordinary arithmetic. But then we must

choose the correct sign.

We may say that this is the first rule of signed numbers:

To subtract a larger number from a smaller,

subtract the smaller from the larger, but report the answer as negative.

1 5 = 4.

We actually do 5 1.

It was in order to subtract a larger number from a smaller that negative

numbers were invented.

3. What is the only difference between 8 5 and 5 8 ?

The algebraic signs. They have the same absolute value.

Problem 2. Subtract. a) 3 5 = 2 b) 1 8 = 7

Problem 3. You have 20 dollars in the bank and you write a check for

25 dollars. Now what is your balance?

20.00 25.00 = 5.00

he number line

The number line is a kind of "ruler" centered on 0. The negative

numbers fall to the left of 0; the positive numbers fall to the right.

We imagine every number to be on the number line. And so the fraction

will fall between 0 and 1; the fraction is between 0 and 1; and so

on.


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4. What is an integer?

Any positive or negative whole number, including 0.

0, 1, 1, 2, 2, 3, 3, etc.

On the number line, we typically place the integers.

And it is on the number line that we begin to see the practical uses for

signed numbers. In general, they show the "direction" of some quantity.

That quantity might be temperture: more than or less than a certain

temperature designated as 0. Or it might be the position or "address" of

some object: left or right of some fixed position chosen as 0. Or it

might be time: before or after a certain moment that again is chosen as

0. Or, as we all know, negative numbers can indicate a balance in a

checking account

Problem 4. A rocket is scheduled to launch at precisely 9:16 AM,

which is designated t (for time) = 0, and t will be measured in minutes.

a) What time is it at t = 10? 9:06 AM.

b) What time is it at t = 1? 9:15 AM.

c) What time is it at t = +5? 9:21 AM.

d) What is the value of t at 9:00 AM? t = 16.

e) What is the value of t at 9:30 AM? t = 14.

The negative of any number

Every number will have a negative. The negative of 3, for example, will

be found at the same distance from 0, but on the other side.


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It is 3.

Now, what number is the negative of 3?

The negative of 3 will be the same distance from 0 on the other side. It

is 3

(3) = 3.

"The negative of 3 is 3."

This will be true for any number a:

(a) = a

"The negative of a is a."

What is in the box is called a formal rule . This means that whenever we

see something that looks like this --

(a)

-- something that has that form, then we may rewrite it in this form:

a

For example,

(12) = 12.

To learn algebra is to learn its formal rules. For, what are calculations

but writing things in a different form? In arithmetic, we rewrite 1 + 1

as 2. In algebra, we rewrite (a) as a.

Problem 5. Evaluate the following.

a) (10) = 10 b) (2 6) =4 c) (1 + 4 7) = 2

The algebraic definition of the negative of a number


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Algebra Mantu