Video Math Tutor: Basic Math: Lesson 1 - Numbers

8/14/2019 Video Math Tutor: Basic Math: Lesson 1 - Numbers

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BA S I CMA T HASelf-Tutorial

by

LuisAnthonyAstProfessionalMathematics Tutor

LESSON1:

NUMBERS

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E-mail may be sent to: [email protected]


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LASSIFICATIONOF UMBERS

One of th e best ways t o visua lize th e differen t class ificat ion of num bers

is by using t he Real Nu mber Line. We can do th is since every nu mber can

be represent ed by a point an d every point can be represent ed by somenumber:

FNATURALNUMBERS : These a re a lso called th e C o u n t i n g

N u m bers an d a re t he ones you can coun t with : 1 2 6 4 5 (useyour fingers and toes if you h ave t o).

T h e M a t h S y m b o l t h a t r e p r e s e n t s t h e N a t u r a l N u m b e r s i s :N

On th e num ber line: dddddd

1 2 3 4 5 6

Smallest na tu ra l nu mber: 1 Largest na tu ra l nu mber: None.

( To type natu ra l numbers, just use th e num berkeys: , , , , and so on .

F : These include th e na tu ra l num bers, but a lso

include Zero. Zero is th e fixed point of reference on t he n um ber line a nd is

also ca lled th e Orig in .

Mat h S y m b o l: W

On th e num ber line: ddddddd

1 2 3 4 5 6

Sma llest whole nu mber: 0 Lar gest whole nu mber: None


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( To type whole nu mbers, just use t he nu mberkeys: , , , , , , , , , .

F The whole nu mbers, plus t heir r espectiven e g a t i v e values , mak e up this collection of num bers .

Mat h S y m b o l: Z

On th e num ber line: ddddddd

0 1 2 3

Sma llest integer: None Lar gest integer: None

( To type integers, use the number keys: , , ,, , etc., but in addition to this, you can u se the negation key:

=Mak e su re you dont mix up th e nega tion (nega tive) key with

th e subt ra ction key , oth erwise you get th is err or:

Pr essing will ret ur n you to where you were work ing. The

calculat or will blink over t he m istyped cha ra cter . Corr ect it a nd

then press .

F : If you divide one int eger by an oth er (notzero!) you get a

ratio th at is called a ra tiona l number. All na tu ra l numbers, whole

nu mbers, an d int egers are Ra t i o n a l since we can divide them by 1.

Oth er exam ples: all typical fra ctions like , , , decima l nu mber sth at stop like: 0.25, 0.12, 56.10823667, decimals t ha t n ever en d, but use

a consist ent , repeat ing nu mber of digits like 0.12121212or 34.6238888

These decimals can be re-writt en by placing a horizont al bar over th e

repeat ing par t: or .


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HOT

TIP!

You should t ry t o a lways place a zero in front of a d ecima l

th a t is bet ween 1 a nd 1. Its for clar ity. Youd be su rp rised

how often I used t o write a decimal nu mber, then th ink t he

decima l point was a n egat ive sign, or wa sn t even t her e! (I

wrote t oo small a dot ). With th e leading zero, this does not

ha ppen a nymore. I also tr y to ma ke th e decimal point lar ger

th an norma l, again, for clar ity.Mat h S y m b o l: Q or

On t he n um ber line, it would be impossible to point out all rational

nu mbers, so her e ar e a few exam ples:

ddddd

2 .4 0 1.65 3.33333

Smallest ra tiona l nu mber: None Largest ra tiona l nu mber: None

( To type rat iona l num bers, use th e nu mber keys:, , , , , etc., the negat ion key: , the decimal poin t : to enter

nu mbers with decimals in th em, an d th e division key: to type in

fractions.

L:

Some ca lcula tor m odels have a specific key to inpu t

fra ctions. It ma y look like:= or .

FIRRATIONALNUMBERS: Any number on th e number line tha t is not

a r at iona l num ber is by defau lt an irra tiona l one. Some examples include:

, , , e.

A decimal nu mber t ha t never ends an d does notha ve a consist ent ,

repeat ing pat ter n of digits is irra tiona l. Ex: 1.7320508075

Mat h S y m b o l: J, or

J


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Again, on th e nu mber line, it would be impossible to point out all

irra tiona l num bers, so her e are a few examples:

ddddd

e

( To type irra tiona l num bers, use an y of th eprevious ly ment ioned k eys, and the following:

To get

t h i s :Typ e t h e follow in g: E xa m p le :

y


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( To type rea l nu mbers , use any of th e previouslymen tioned keys.

If you wan t your calculat or to only perform operat ions with r eal num bers,

P r es s t h e k ey, t h en m ake sur e t h a t t h e wor d is h igh ligh t ed

(whit e letter s on black backgroun d). If it isn t, use th e down a rr ow keyto get the cu r sor to blink over the word . P ress , t hen .

This does not a pply to a TI-82.

FOTHER NUMBERS: Any nu mber th at can not be represent ed on areal nu mber line is not a real nu mber.Im a gin a ry N u m bers are used to

help represent these nu mbers. The symbol is used to mean an imaginary

nu mber. It is defined to be . Squa re roots of oth er negat ive nu mbers

ar e also not rea l (th ey are ima gina ry too!). I ment ion t hese n um bers, since

somet imes on t ests you will be asked t o determ ine what is theclass ificat ion of a nu mber . If you see, for exam ple, or 3 , then youwill sta te t hey ar e ima gina ry (or t ha t t hey are n ot r eal). A Com p lex

N u m ber is a num ber tha t can be written in t he form : a + b where a and bar e real nu mbers an d is th e imagina ry nu mber defined to be .

2 + 5 is an example of a complex num ber. It is NOT r eal. It cann ot berepresented on a real nu mber line.

( To type complex nu mber s, use a ny of th eprevious ly ment ioned keys, but also put th e calculat or in complex mode.To do t h is , pr ess t h e k ey, t h en m ake sur e t h a t t h e is

highlight ed (whit e lett ers on black backgroun d). If it isn t, u se the down

ar row key , th en B t o get t he cu r sor t o blin k over t he . P r es s

, th en . This does not apply to a TI-82. Then pressy


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{SETSOFNUMBERS}A S et is a collect ion of objects or it ems. The definit ions above can be u sed

to describe num bers as m ember s of set s. Braces { } a re used t o enclose

th e members of a set. Capital lett ers a re usu ally used to nam e a set. The

mem bers/items of th e set a re called th e El em en ts of th e set. The ma thsymbol: means is an element of. Ther e is no order to the element s

with in a set.

For example: th e set Hof nu mbers t ha t can be coun ted with one han d is:

H= {1, 2, 3, 4, 5}

To sa y: 1 is an element of set H, we write: 1 H

A S u b se t is a set t ha t is inside an oth er set. The set F= {1, 2} is a su bset of

Hdefined a bove. The set G = {4, 6} is NOT a su bset ofFsince it ha s an

elemen t, 6, th at is not in F. By th e way, Fcan also be writt en as {2, 1}

..If you n eed to lear n m ore, my Fin ite Math Lesson: Set s sh ould

help.

We can now discus s th e class ificat ion or types of nu mber s as set s of

numbers:

Y The Set of Nat ura l Numbers: N = {1, 2, 3, 4, 5}

Y The Set of Whole Nu mber s: W= {0, 1, 2, 3 , 4, 5}Y The Set of Int egers: Z = {3, 2, 1, 0, 1, 2 , 3}

Y The Set of Posit ive Int egers = {1, 2, 3} N

YThe Set of Nega t ive Int egers = {3, 2, 1}

Y The Set of Rat iona l Num bers: Q = { | a and b ar e integers ( )}

|

Y The Set of Irr at iona l Nu mbers: J = {All nu mbers th at ha ve a non-

repeating, non-terminating decimal representation}

Y The Set of Real Num bers: R = {All nu mbers th at ar e eith er r at iona l or

irrational}


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Sample test ques tions from t his Lesson can be done a coup le of ways. One

is to give you a nu mber, t hen you d ecide what type of nu mber it is

(In teger , Ra tiona l, Real) or wha t set it is in: (Z, Q, R). Another way is for

th e instr uctor t o give you a list of nu mbers, t hen you decide which ar e

na tu ra l, whole, int egers, etc. I will give examples of both of th ese ways.

P ROBL E M S E T : I d e n t i fy t h e s e t (s ) t h a t t h e n u m b e r i s a ne le m e n t o r m e m b e r o f.

YWhat set(s) is the n um ber 9 an elemen t of?

SOLUTION: 9 is a element of the following set s: N, W, Z, Q, R

[This quest ion can also be asked a s: Wha t t ype of nu mber is 9?

SOLUTION: 9 is a na tu ra l, whole, integer, rat iona l, an d real nu mber.]

GGGGGGGGGGGGGGGGGGGYWhat set(s) is the n um ber 1 an element of?

SOLUTION: 1 is a element of the following set s: Z, Q, R

[Alter na te Version: Wha t t ype of nu mber is 1?

SOLUTION: 1 is an integer, ra tiona l, an d real n umber.]

GGGGGGGGGGGGGGGGGGG

YWhat set(s) is the n um ber 0 a m ember of?

SOLUTION: 0 is a mem ber of th e following set s: W, Z, Q, R

[Altern at e Version: What type of num ber is 0?

SOLUTION: 0 is a wh ole, integer, ra tiona l, an d rea l num ber.]

GGGGGGGGGGGGGGGGGGG

YWhat set(s) is th e nu mber an elemen t of?

SOLUTION: is a mem ber of th e following set s: Q, R

[Altern at e Version: What t ype of nu mber is ?

SOLUTION: is a rat iona l and a r eal number.]

GGGGGGGGGGGGGGGGGGG


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SOLUTION: 1.7320508075688772935 is a mem ber of th e

following set s: J, R

[Altern at e Version: What type of nu mber is ?

SOLUTION: is an irr at iona l an d a real num ber.]

GGGGGGGGGGGGGGGGGGG

Now lets do some t r ickier ones:

P ROBL E M S E T : I d e n t i fy t h e s e t (s ) t h a t t h e n u m b e r i s a n

e le m e n t o r m e m b e r o f.


SOLUTION: is an element of th e following set s: N, W, Z, Q, R

(It is a n at ur al, whole, integer, rat iona l, an d rea l num ber).

N o t e : is really just 2. Many stu dents see th e

ra dical symbol , an d au tomat ically as sum e it is an irra tiona l nu mber.

In t his case, it is not a mem ber of th e J set.

GGGGGGGGGGGGGGGGGGG


SOLUTION: is a element of the following set s: J, R (It is irra tiona l

and r eal)

L is th e value you would get if you d ivided t he len gth ofth e circum feren ce of a circle by th e length of its

diameter. It is an irrat iona l num ber becau se it is a

decimal with n o repea ting digits an d never en ds. Most

students have the approximate value of mem orized to

a couple of decima l places: 3.14. J ust for k icks, her e is

approximated to 99 decimal places:

3.141592653589793238462643383279502884197169399375105820974944

592307816406286208998628034825342117068

GGGGGGGGGGGGGGGGGGG


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SOLUTION: is a m ember of NONE of th e sets of nu mbers we have

been discuss ing. It is a ctu ally th e imaginary nu mber . It is not a real

nu mber, since it can not be represented as a point on th e real nu mber line.

GGGGGGGGGGGGGGGGGG


SOLUTION: = 5, so its a member of the following se ts: N, W, Z, Q, R.

In other words, its a n at ur al nu mber, whole, integer, ra tiona l, an d rea l

number.

N o t e : Man y stu dent s just n otice it is a fraction, so aut oma tically

assum e it is a ra tiona l num ber a nd r eal, but since th e fra ction can be

redu ced to a n at ur al nu mber, it is a m ember of th e oth er set s t oo.GGGGGGGGGGGGGGGGGGG

YWhat set(s) is th e nu mber a mem ber of?

SOLUTION: is a m ember of NONE of th e sets of nu mbers we have

been discussin g. It is a ctu ally UNDEFINE D. Since division by zero is not

allowed.

GGGGGGGGGGGGGGGGGGG

Here is a special diagram th at shows th e relationsh ipbetween th e differen t set s of nu mber s (with exam ples). Regions inside of

oth er r egions mean th ey are a subset of th e nu mbers in t he larger r egion.

Irrational

Numbers

Rational

Numbers

Real Numbers

Integers

Whole Number s

Natural

Numbers

1, 2, 3, 4,

0

3

8

9 .867

5

5

1.24


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LESSON 1 QUIZ1Write t he letter n am e to each set of nu mbers:

a) Nat ur al Nu mbers: ___b) Whole Numbers: ___

c) In teger s: ___

d)Rational Numbers: ___

e) Irr at iona l Num bers: ___

f) Real Nu mber s: ___

2The nu mber 57 is an elemen t of which set of nu mbers?

3Which of th e following nu mber s is irr at iona l?

3.14

4Fr om th e following set :

List t hose tha t are:

a) N a tura l Number s: _________________________________

b) Whole Numbers: _________________________________

c) Integers: _________________________________

d) Rational Numbers: _________________________________

e) Ir ra t iona l Number s: _________________________________

f) Real: _________________________________

ANSWERSONNEXTPAGE


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ANSWERS1Write t he letter na me t o each set of nu mbers:

Natu ral Numbers: N

Whole Num bers: WIntegers: Z

Rationa l Numbers: Q

Irra tiona l Num bers: J

Real Numbers: R

2The nu mber 57 is an elemen t of which set of nu mbers?

57 is an e l em en t o f N, W, Z, Q,a n d R (n a t u r a l , w h o le ,i n t e g e r , r a t i on a l a n d r e a l )

3Which of th e following nu mber s is irr at iona l?i s t h e o n l y on e t h a t i s ir r a t i o n a l .

3.14 is NOT . Its a ra tiona l nu mber approximation to . 0 is whole, an

integer, ra tiona l an d rea l, but NOT irra tiona l. is imagina ry.

4Fr om th e following set :

List t hose tha t are:

a) N a t u r a l N u m b e r s : (It s r ea lly just 4!)

b) Wh o le N u m b er s : (Its rea lly just zero!),

c) I n t e g e r s : , ,

d) R a t i o n a l N u m b e r s : , , , , ,

e) I r r a t i o n a l N u m b e r s : ,

f) Rea l : , , , , , , ,

END OFLESSON 1

Video Math Tutor: Basic Math: Lesson 1 - Numbers

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