Math 150 Chapter 1 - Texas A&M University · Math(Symbols: ∈ ∀ ∃ iff ©Fry Texas A&M University Math 150 Chapter 1 !Fall 2013 ... ©Fry Texas A&M University Math 150 Chapter

Chapter 1A -‐-‐ Real Numbers

Types of Real Numbers

Name(s) for the set Symbol(s) for the set

1, 2, 3, 4, …Natural NumbersPositive Integers

…, -‐3, -‐2, -‐1 Negative integers

0, 1, 2, 3, 4, … Non-‐negative integers

…, -‐3, -‐2, -‐1, 0, 1, 2, 3, … Integers

Note: _________________________________ is the German word for number.

45

, −1713

, 129 Rational Numbers

Note: This is the Hirst letter of ____________________________

1. Circle One:

a) 5 is a rational number TRUE FALSE

b) π4 is a rational number TRUE FALSE

c) 165 is a rational number TRUE FALSE

d) 175

is a rational number TRUE FALSE

e) 0 is a rational number TRUE FALSE

©Fry Texas A&M University Math 150 Chapter 1 ! Fall 2013

1

De8inition: If the numbers a and b are both integers and b ≠ 0 , then ab is a rational number.

Decimal Expansions of Rational Numbers: Fact: The decimal expansions of rational numbers either

___________________________________ or ___________________________________________

Examples 14 = ________________________ ( This is a terminating decimal.)

49 = _____________________ (This is a repeating decimal.)

You know how to express a terminating decimal as a fraction. For example .25 = 25100

, but

how do you express a repeating decimal as a fraction?

2. Express these repeating decimals as the quotient of two integers:

a) .373737… or .37

Let x = .37373737... Then 100x =

b) .123

Let x = .123123123...


2

There are a lot of rational numbers, but there are even more irrational numbers. Irrational numbers cannot be expressed as the quotient of two integers. Their decimal expansions never terminate nor do they repeat.

Examples of irrational numbers are ___________________________________________________

The union of the rational numbers and the irrational numbers is called the

___________________________________ or simply the _______________________. It is denoted by _______.

Math Symbols:

∈

∀

∃

iff


3

Properties of Real Numbers


4

4. W

hat properties are being used?

Nam

e of

Pro

perty

a) (−40)+8(x+5)

=8(x+5)+(−40)

a+b=b+a

∀

a,b

∈

b) 8(x+5)+(−40)=8x

+40

+(−40)

a(b+c)=ab

+ac

∀

a,b

,c∈

c) 8x

+40

+(−40)=8x

+(40+(−40))

(a+b)

+c=

a+

(b+c)

∀

a,b

,c∈

d) 0+5=5+0=5

0+a=a+0=a

∀

a∈

e) 7+(−7)

=(−7)

+7=0

For e

ach

a∈, ∃

an

othe

r ele

men

t of

de

note

d by

−a

suc

h th

at

a+(−a)

=(−a)

+a=0

f) 6⋅1=1⋅6=6

1⋅a=a⋅1=

a

∀ a

∈

g) 4⋅

1 4⎛ ⎝⎜⎞ ⎠⎟=1 4⎛ ⎝⎜⎞ ⎠⎟⋅4=1

For e

ach

a∈, a≠0

∃

anot

her

elem

ent o

f de

note

d by

1 a s

uch

that

a⋅1 a⎛ ⎝⎜⎞ ⎠⎟=1 a⎛ ⎝⎜⎞ ⎠⎟⋅a=1

Absolute Value

4 = −9 =

Formal deHinition for absolute value

x =⎧⎨⎪

⎩⎪

It will help if you learn to think of absolute value as ________________________________________

Notice that -4 is 4 units away from the origin. This interpretation is important and useful because it extends to the complex numbers that we will study soon.Distance between two points:

3.) Find the distance between

a) 4 and 1

b) -1 and 5

c) 4 and -2

d) -3 and -4

e) 1 and x

f) x and y

Notice that x − y = ______________________________________________________

Notice that −5 + 4 = !! whereas −5 + 4 = ! ! ! So, in general x + y ≠


5

Chapter 1B - Exponents and Radicals

Multiplication is shorthand notation for repeated addition 4 + 4 + 4 + 4 + 4 = 5 × 4

Exponents are shorthand notation for repeated multiplication. So

2 × 2 × 2 × 2 × 2 = _______ xixixix = __________

You probably know that 3−1 = _________ and x−5 = ____________ this is because

by definition x−1 = __________ . In general if m ∈ Ν, then x−m =__________

Properties of exponents

Example In general

a4a3 xmxn

b8

b5xm

xn

y3

y3 x0 for x ≠ 0

Note: 00 ≠ 0 and 00 ≠1 00 is an

More Properties of Exponents

Example In general

(xy)3 (xy)m

xy

⎛

⎝⎜

⎞

⎠⎟2

xy

⎛

⎝⎜

⎞

⎠⎟m


6

1. Simplify

a) (−8)2 b) −82 c) 30

Dividing is the same as _____________________________________________

d) 1y−6

⎛⎝⎜

⎞⎠⎟

e) ab

⎛⎝⎜

⎞⎠⎟−1

!

e) x−3

x2 f) y4

y−5⎛⎝⎜

⎞⎠⎟

−1

g) 12a−3b2

4a4b−1

⎛

⎝⎜

⎞

⎠⎟−2

h)

13

⎛⎝⎜

⎞⎠⎟−1005

− 27334

9502 + 9501


7

Roots or Radicals

TRUE or FALSE: 4 = ±2 .

Important distinction: x2 = 25 has ______ solutions: x = _____ and x = _____

25 represents exactly ____ number. 25 = ________.

Recall that 25 = 32 . Now suppose you have x5 = 32 with the instructions: “Solve for x .”

How do you express x in terms of 32? The vocabulary word is __________________

In this case x equals the _______________________________.

The notation is x = ____________________ or x = ____________________

This second notation is called a fractional exponent.

Examples: ! 814 = 81( )14

⎛⎝⎜

⎞⎠⎟ = ____________ because 3 = _________.

! ! ! 125( )13

⎛⎝⎜

⎞⎠⎟ = ____________ because 5 = _________.

In general if an = b then ____________________

and if ____________________________________________________________

This kind of statement can be condensed using the term ____________________

This is often abbreviated __________ or _____________________

So in shorthand __________________________________


8

Properties of Roots (just like properties of exponents!)

Example In general

27i83

273 83

xym

364

364

⎛

⎝⎜

⎞

⎠⎟

xm

ym

Now consider 823 . Notice that 82( )

13 = _______________________________

Also 813

⎛⎝⎜

⎞⎠⎟⎛

⎝⎜⎞

⎠⎟

2

= _____________________________________________________

So it would seem that 823 = ________________________________________

In fact, more generally, it is true that xmn

⎛⎝⎜

⎞⎠⎟ = __________________________

Similarly, consider 643 = 6412

⎛⎝⎜

⎞⎠⎟⎛

⎝⎜⎞

⎠⎟

13

= _______________________________

Whereas, 643 = 6413

⎛⎝⎜

⎞⎠⎟⎛

⎝⎜⎞

⎠⎟

12

= _____________________________________

So it would seem that 643 _______________________________________

In general xnm _______________________________________________


9

Our next goal is to define ann . Let’s do several examples to find a pattern.

a) 255 = ________________ ! ! ! ! b) 533 = ______________

c) −5( )33 = ________________! ! ! ! d) −2( )55 =_____________

e) 244 = ________________! ! ! ! f) 322 =_______________

g) −2( )44 = ________________!! ! ! h) −3( )22 = _______________

ann =⎧⎨⎪

⎩⎪

Simplifying Radicals A radical expression is simplified when the following conditions hold: 1. All possible factors (“perfect roots”) have been removed from the radical. 2. The index of the radical is as small as possible. 3. No radicals appear in the denominator.

2. Simplify

a) −325

b) (−216)13

⎛⎝⎜

⎞⎠⎟

c) −912

⎛⎝⎜

⎞⎠⎟ d) −64


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e) (−13)66 f) 16x44

g) x53 h) 648x4y63

i) 16x4y128 j) −12564

⎛⎝⎜

⎞⎠⎟

−23

Rationalizing the Denominator

“Rationalizing the denominator” is the term given to the techniques used for eliminating radicals from the denominator of an expression without changing the value of the expression. It involves multiplying the expression by a 1 in a “helpful” form.

3. Simplify

a) 12! ! ! ! ! ! b) 1

23


11

c) 14x3

! ! ! ! ! ! d) 9x( )3 y−450x8y−5

Notice: x + 3( ) x − 3( ) = __________________________________________.

To simplify 1x − 5

we multiply it by 1 in the form of ____________________.

So 1x − 5

=

and 412 + 3 5

=


12

Adding and Subtracting Radical Expressions

Terms must be alike to combine them with addition or subtraction. Radical terms are alike if they have the same index and the same radicand. (The radicand is the expression under the radical sign.)

4. Simplify

a) 5 + 2 7 − 3 5 − 7 b) 16x43 − 3x 18x − x 250x3

c) 13− 1

27

Please notice 9 + 16 = ______________________! Whereas 25 = ___________

In other words ____________________________

In general __________________________


13

Chapter 1C - Polynomials

Which of these are examples of polynomials?

Polynomial Not a polynomial Polynomials should not have

x2 + 2x +1

x7 + π

2( ) x2 + 2x +14 x −1

1x+ 3x2

5x3 + 3x2 − 2( )14

5x3 + 3x2 − 2( )2

x2 + sin x

5

Polynomials are expressions of the form anxn + an−1x

n−1 ++ a1x + a0

where ai ∈ and n∈

n is called ___________________________________________

an is called ___________________________________________

a0 is called ___________________________________________

If n = 2, the polynomial is called a __________________________________

If n = 3, the polynomial is called a __________________________________

A polynomial with 2 terms is called a __________________________________

A polynomial with 3 terms is called a __________________________________


14

Techniques for Factoring Polynomials

Common Factors: 12x4 + 4x2 + 2x

Factor by Grouping: 3x3 + x2 −12x − 4

Factor using special patterns:

Look what happens when you multiply

(x − 3)(x + 3)= _________________________________

(4 − y)(4 + y) = __________________________________

Difference of Squares a2 − b2 =

Look what happens when you expand

(x + 5)2 = __________________________________________________

(3y − z)2 = __________________________________________________

Square of a Binomial a2 + 2ab + b2 =

a2 − 2ab + b2

Look what happens when you multiply

(a + b)(a2 − ab + b2 )

Sum of Cubes a3 + b3

Difference of Cubes a3 − b3


15

1. Factor

a) x2 −144 b) 16x4 − y6

c) 27 − y3 d) x6 + 64

e) a2 +10a + 25 f) x4 + 8x2 +16

Factoring quadratics, lead coefficient 1:

2. Factor : x2 +10x +16 = _______________________________________

Find 2 numbers whose product is _______ and whose sum is ___________

3. Factor: x2 − 2x − 48 = _______________________________________


Factoring quadratics with lead coefficient is not 1

Use the Blankety-Blank Method!

4. Factor: 10x2 +11x + 3 = _______________________________________


5. Factor: 3x2 −14x − 5 = ________________________________________



16

Important Fact: a2 + b2 does not factor over the real numbers.

So x2 + 64 cannot be factored nor can y2 +100

Dividing Polynomials (This is a lot like long division of integers!)

6. 2x3 +11x2 + 7x − 20

x + 4

7. x5 − 3x3 + x2 + 9

x2 + 2x


17

Chapter 1D - Rational Expressions

A rational expression is ___________________________________________

A rational expression is undefined when its ___________________________

For what values of x is the rational expression x3 +1x2 −1

defined ?

Simplifying rational expressions means reducing it to lowest terms. This is achieved by canceling factors common to the numerator and the denominator.

1. x3 +11x2 + 30xx2 +10x + 25

2.

8y+ 2

⎛

⎝⎜

⎞

⎠⎟iy2 − y − 2y2 −1

⎛

⎝⎜

⎞

⎠⎟

3. v2 − 4v − 21v+ 4

÷ (v2 + 7v+12)


18

4. x3 −1x +1x −1

x2 + 2x +1

5. 1x + 2

+ 1x − 3

6. u − 4u +1

− u2 − 8u +16u2 −1


19

A compound or complex fraction is an expression containing fractions with the numerator and/or the denominator. To simplify a compound fraction, first simplify the numerator, then simplify the denominator, and then perform the necessary division.

7. 5t+ 42t −15t− 3

8. 3 x + h( )−2 − 3x−2h


20

Chapter 1E - Complex Numbers

−16 exists!

So far the largest (most inclusive) number set we have discussed and the one we have the most experience with has been named the real numbers.

And ∀ x ∈ , x2 ≥ 0

But there exists a number (that is not an element of ) named i and i2 = _____

Since i2 =_____, −1 = __________ , so −16 = ________________________

i is not used in ordinary life, and humankind existed for 1000’s of years without considering i . i is, however, a legitimate number. i , products of i , and numbers like 2 + i are solutions to many problems in engineering. So it is unfortunate that it was

termed “imaginary!”

i and numbers like 4i and −3i are called _____________________________

Numbers like 2 + i and 173− 5i are called ____________________________

More formally is the set of all numbers _______________________________

When a complex number has been simplified into this _________ form, it is called

_________________________________

1. Put the following complex numbers into standard form:

a) 16 + −9( ) + −13− −100( ) = ________________________________________

b) a + bi( ) + (c+ di) = _______________________________________________

c) 2 + 3i( )(4 + 5i) = _______________________________________________

d) 5 + 4i( )(3− 2i) = _______________________________________________

e) 7 + −9( ) 6 + −5( ) = ______________________________________________


21

Graphing Complex Numbers

When we graph elements of , we use ________________________________

When we graph elements of , we use the complex plane which will seem a lot like the Cartesian plane.

In the Cartesian plane, a point represents ________________________________

In the complex plane, a point represents ___________________________________

The horizontal axis is the ______________________________________________________

The vertical axis is the ______________________________________________________

2. Graph and label the following points on the complex plane:

A 3+ 4i C 32− 3i E −4

B −2 + i D 3i F −1− 2i


22

Absolute Value of a Complex Number:

If z ∈ then z is defined as its _______________________________________

Calculate 3+ 4i

Consider a + bi , an arbitrary element of .

What is a + bi ?

In general z = _____________________________

3. Calculate −5 + i


23

Complex Conjugate

If z = 3+ 4i , then z = ________________ . If z = −1− 2i , then z = __________ .

Graphically the complex conjugate is the

________________________ of the number

through the ________________________

More generally if z = a + bi ,

then z = ______________________

4. Put the following complex numbers into standard form.

a) 2 − 3i = ________________! ! ! b) 4i = ___________________

c) 5 = ____________________

Finally notice that 3+ 4i( ) 3− 4i( ) = ______________________________________

More generally a + bi( ) a − bi( ) = ________________________________________

in other words ziz = ________________________________________________


24

Distance Between Two Complex Numbers

Plot and label two points in the complex plane z1 = 3+ 5i and z2 =1+ 2i

The distance between z1 and z2 is

What if the 2 points were arbitrary? z1 = a + bi and z2 = c+ di

The distance between z1 and z2 is

Notice that z1 − z2 = ____________________________________________________

So z1 − z2 = _________________________________________________________

So we have shown that the distance between two points z1 and z2 is _______________


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Math 150 Chapter 1 - Texas A&M University · Math(Symbols: ∈ ∀ ∃ iff ©Fry Texas A&M University Math 150 Chapter 1 !Fall 2013 ... ©Fry Texas A&M University Math 150 Chapter