Add: = /10 + 0/ / /10 + 7/100

9 -1 DECIM ALS — Their PLACE in the W ORLD M ath 2 1 0 ^F8 Definition: DECIM AL NUM BERS

1 2 3 4

If x is any w hole num ber, x . d d d d .. .

1 2 3 4 5

represent s t he num ber: x + d / 1 0 + d / 1 0 0 + d / 1 0 0 0 + d / 1 0 0 0 0 + d / 1 0 0 0 0 0 . ..

1 2 3 4 5

( w hic h is also k now n as: x + d x1 0^!1 + d x1 0^!2 + d x1 0^!3 + d x1 0^!4 + d x1 0^!5 . .. ) so, e. g. : 3 2 1 .5 0 6 7 8 = 3 2 1 + 5 / 1 0 + 0 / 1 0 0 + 6 / 1 0 0 0 + 7 / 1 0 0 0 0 + 8 / 1 0 0 0 0 0 f ully ex panded: 3^x10 + 2^{2 x}10 + 1^{1 x}10 + 5^{0 x}1 0^{– 1} + 0^x1 0^{– 2} + 6^x1 0^{– 3} + 7^x1 0^{– 4} + 8^x1 0^{– 5} The " . " (in 3 2 1 . 5 0 6 7 8 , e.g. ) is c alled t he decimal point.

The plac e v alues of t he digit s t o t he right of t he dec im al point are t ent hs, hundredt hs, t housandt hs, t en- t housandt hs, hundred-t housandt hs, m illiont hs, et c . (Not to be conf used w it h tens, hundreds, t housands, et c.)

In general: the place value of t he digit in t he nth decimal place (n places after the decimal point) is 1 0^{– n}.

Not ic e t his is not sy m m et ric .. . t he f irst plac e t o t he lef t of t he dec im al point is 1 0 , not 1 0 . .. !^{0 1} Read and ex pand int o long form (w ords): 7 8 6 .3 0 0 7

OPERATIONS ON DECIM AL NUM BERS

W e k now how t o add, subt rac t , m ult iply and div ide dec im als; w e now look at w hy .

A dd: 2 0 3 . 6 0 2 + 3 . 4 7 2 0 3 . 6 0 2 = 2 0 0 + 3 + 6 /1 0 + 0 /1 0 0 + 2 /1 0 0 0 + 3 . 4 7

+

CHIP M ODEL M A Y HELP!

Subt rac t 2 3 .5 ! 3 .4 7 2 3 . 5 2 3 . 5 2 3 + 5 /1 0 + 0 /1 0 0

– 3 . 4 7 – 3 . 4 7 — ( 3 + 4 /1 0 + 7 /1 0 0 )

.. .W hat ' s w rong

w it h t his pic t ure?

M ult iplic at ion & Div ision by 1 0 ' s:

2 . 1 7 x 1 0 = ( 2 + 1 / 1 0 + 7 /1 0 0 ) x 1 0 = 2 x 1 0 + (1 /1 0 )x 1 0 + (7 /1 0 0 )x 1 0

= 2 x 1 0 + 1 + 7 /1 0 = 2 1 . 7

Sim ilarly , 2 . 1 7 x 1 0 = ( 2 + 1/10 + 7/100 )² x 100 = 2 @1 0 0 + (1 /1 0 )@1 0 0 + (7 /1 0 0 )@1 0 0 = 2 0 0 + 1 0 + 7 = 2 1 7 2 1 . 7 x 1 0 = 2 1 . 7 x 1 /1 0 = 2 1 . 7 ÷ 1 0 ⁾¹

= ( 2 0 + 1 + 7 /1 0 ) ÷ 1 0 = 2 0 ÷1 0 + 1 ÷1 0 + (7 /1 0 ) ÷1 0

= 2 + . 1 + . 0 7 = 2 . 1 7 Sim ilarly , 2 1 . 7 x 1 0^{– 2} = 2 1 . 7 x 1 /1 0 0 = 2 1 . 7 ÷ 1 0 0 = . 2 1 7

M ULTIPLICA TION BY

10

(

).

DIVISION BY

10

.

M ak ing sense out of t he m ult iplic at ion and div ision algorit hm s:

2 3 . 5 (2 3 5 x 1 / 1 0 ) 1 5 .2 Ž 4 6 8 0 . 0 8 4 6 8 0 . 0 8 1 0 4 6 8 0 0 . 8

x 2 . 1 7 x (2 1 7 x 1 /1 0 0 ) 1 5 . 2 1 0^x 1 5 2

1 6 4 5 2 3 5

4 7 0 = (2 1 7 x 2 3 5 ) x (1 / 1 0 ) x (1 / 1 0 0 ) 5 0 9 9 5 = (2 1 7 x 2 3 5 ) x (1 / 1 0 0 0 )

= (2 1 7 x 2 3 5 ) / (1 0 ) ³ Estimate t he t ot al!

9 -1 A DECIM ALS AROUND the W ORLD M ath 2 1 0 f8 Rounding decimals:

Round 4 .3 6 7 t o t he nearest hundredt h. The nearest t w o dec im als in hundredt hs are 4 . 3 6 and 4 . 3 7 . The nearer is 4 . 3 7 , sinc e it is .0 0 3 great er t han 4 . 3 6 7 , as opposed t o 4 . 3 6 , w hic h is . 0 0 7 less t han 4 . 3 6 7 . This is apparent f rom a num ber line v iew :

* * * * * * * * * * *

4 . 3 6 0 4 .3 6 1 4 .3 6 2 4 .3 6 3 4 .3 6 4 4 .3 6 5 4 .3 6 6 4 .3 6 7 4 . 3 6 8 4 .3 6 9 4 . 3 7 0

4 .3 6 4 .3 7

Now round 4 . 3 6 4 5 t o t he nearest hundredt h: 4 . 3 7

4 .3 6 4 5 eit her rounds dow n t o 4 .3 6 , or up t o 4 .3 7 : 4 .3 6 4 5 )))))) 4 . 3 6 5 — 4 .3 6

W hen a num ber is rounded, it has been replac ed w it h a nearby , but less prec ise, v alue.

The dif f erenc e bet w een t he v alue bef ore rounding and t he rounded-of f v alue is c alled t he rounding error.

The problems w ith rounding: How do y ou round 4 .3 6 5 t o t he nearest hundredt h? The nearest num bers t erm inat ing in t he hundredt hs' plac e are 4 .3 6 and 4 .3 7 , bot h .0 0 5 f rom 4 .3 6 5 . There are t hree " rounding rules" t hat m ay be adopt ed t o c ov er suc h inst anc es:

1 ) Round to nearest even: The m ost w idely adopt ed rule. Fav ored bec ause about half t he t im e y ou round

up, half t he t im e y ou round dow n; so a long series of rounding t ends t o av erage out t he errors t o zero.

2 ) Round to nearest odd: Sam e as abov e, but opposit e, f or c ont rary people.

3 ) Round up: This is t he sim plest bec ause any dec im al w it h " 5 " in t he nex t plac e (af t er t he one t o w hic h t he num ber is being rounded) get s rounded up. The disadv ant age t o t his rule is t hat by alw ay s rounding half w ay -point s up, t ot als are t oo high on t he av erage. If y ou' re t he t ax collec t or, y ou round up!

Eg. Round 1 9 5 3 9 .9 8 7 4 8 t o t he nearest :

t housandt h t ent h unit hundred t housand

1 9 5 3 9 . 9 8 7 1 9 5 4 0 . 0 1 9 5 4 0 . 1 9 5 0 0 2 0 0 0 0

Caution:

Don' t round " piecemeal" :

For ex am ple, Teddy rounded 4 . 9 7 4 4 8 t o 4 . 9 8 t his w ay : 4 .9 7 4 4 8 . 4 . 9 7 4 5 . 4 . 9 7 5 . 4 . 9 8

Help Teddy : If 4 . 9 7 4 4 8 is t o be rounded t o t he nearest hundredt h. . .

4 . 9 7 4 4 8 w ill eit her t runc at e t o 4 . 9 7 or round up t o 4 . 9 8 — w hic h of t hese is c loser t o 4 . 9 7 4 4 8 ?

W hat is t he half w ay point bet w een 4 . 9 7 and 4 . 9 8 ? .. . is 4 . 9 7 4 4 8 abov e or below t hat " half w ay point " ? Caution:

Don' t round too soon: .. . espec ially w hen using a c alc ulat or, don' t round unt il t he c om put at ion is c om plet e!

Calc ulat ors c arry ex t ra digit s " f or f ree" , in m ost inst anc es. This m et hod m inim izes error f rom rounding. ( )

For ex am ple, t o c om put e 3 . 5 0 x q2 , sinc e 3 . 5 0 ex presses 3 digit s of prec ision, w e w ish t o hav e t hree digit s in t he f inal result . If w e round t he square root of 2 t oo early , w e risk an erroneous answ er. q2 . 1 . 4 1 4 2 1 3 5 6 2 . If w e round t o 1 . 4 1 , t hen m ult iply by 3 . 5 0 , w e obt ain 4 . 9 3 5 , w hic h w e round t o 4 . 9 4 ; how ev er, if w e carry t he ex t ra dec im als and m ult iply by 3 . 5 0 , w e obt ain 4 . 9 4 9 7 4 6 8 , w hic h rounds t o 4 . 9 5 (ev en 1 . 4 1 4 x 3 . 5 0 w ill giv e t his result ). Conclusion: don' t round too soon.

Caution:

Express correctly! Round 8 , 4 9 8 . 9 8 5 0 6 to the nearest tenth. Round 8 , 4 9 8 . 9 8 5 0 6 to the nearest ten.

8 , 5 0 0 . 0 8 , 5 0 0

A num ber w hic h has been rounded t o 3 4 0 0 is bet w een 3 3 5 0 & 3 4 5 0 .

A num ber w hic h has been rounded t o 3 4 0 0 . or 3 4 0 0 is bet w een 3 3 9 9 . 5 & 3 4 0 0 . 5

9 -1 B COM PUTATIONS W ITH SCIENTIFIC NOTATION M ATH 2 1 0 f8 t his m at erial w as disc ussed w eek s ago, but it m ak es sense t o rev iew it now

540000000000000 is difficult to see. We w rit e, alt ernat ely, 540,000,000,000,000 to

help count the zeroes and figure the place value. Even so, most do not fully comprehend

the w ords " billion" and " trillion" . Thus, to many of us, reading or hearing the number above

is five hundred forty trillion is not illuminating. Very large numbers (as w ell as very small

numbers) w rit ten in our tradit ional numeration system are cumbersome to w rit e, read and

use. For purposes of expressing, mult iplying and dividing such numbers, alt ernate means

are often employed, most commonly scientific notation— w rite as x.ddd... x 10 .

(w here x may not be 0.)

Use of scientific notation requires:

• recognition of place value expressed in exponential form,

• ability to round decimal numbers,

• full command of the arit hmetic propert ies of mult iplication and division,

• understanding of significant digits (this relates to rounding).

Writ e in scientif ic notation form. (Check by mult iplying!)

Key idea: Focus on the

Examples: 7,000,000 = 7 x 10

leading digit . If it is in

7,650,000 = 7.65 x 10

the right place, the rest

.000007 = 7 x 10

w ill follow !

.0129 = 1.29 x 10

Try it: 540,000,000,000 = 5.4 x 10 ¹ Fill in t he

.00000713 = 7.13 x 10 ¹ exponents

.0001 = 10.02 x 10 =

... And be able to convert from Scientific Notation to Standard Form:

3.784 x 10 =

3.702 x 10

= 4.50 x 10

= 4.50 x 10 =

0 .0 0 3 7 0 2 0 .0 4 5 0 4 5 0 0

3 7 8 4

Arithmetic in Scientific Notation:

(3.12

10 )x(4.25

10

) = (3.12

4.25) x (10

10

) Ñ (

)

* (4.34

10 ) / (2.728

10

) = (4.34 /2.728) x (10 /10

)

= 29.965727þ

10

Ñ

¸ No knees!: * 5.67

10 + 3.33

10 Ñ

¸ Use your calculator to assist in computing:

¸ * (5.407

10 ) x (3.66

10

) ÷ (1.3311

10 ) =

45.369268x10 ⁴

¸ Express the result w ith correct number of significant digits in scientific notation.

( 45.369268x10 Ñ 4.54x10 )^{4 5}

9 -2 Rationals & Decimals M ath 2 1 0 f8 Not e t here are t hree f orm s of dec im als: t erm inat ing (.0 2 5 )

non-t erm inat ing repeat ing (. 3 3 3 3 þ) and

non-t erm inat ing non-repeat ing (. 4 1 7 4 1 1 7 4 1 1 1 7 4 1 1 1 1 7 þ)

Ú RA T ION A L N U M BERS M A Y BE EX PRESSED A S D ECIM A LS

Converting rational fraction to decimal form: Use t he long div ision algorit hm . E.g.

7 = 1 = 3 = 3 = 1 = 9 = 3 =

1 0 2 4 2 5 1 2 5 1 6 6 4

2 = 5 = 4 = 2 = 5 = 1 = 1 =

3 9 7 1 1 1 3 7 5 1 9

A rat ional f rac t ion in reduced f orm is equiv alent t o a t erm inat ing dec im al if , and only if , t he denom inat or (of t he reduc ed f rac t ion) has only 2 ' s and/ or 5 ' s as prim e f ac t ors. For ex am ple, 1 / 1 2 5 t erm inat es in t hree plac es bec ause

1 1 1 @2 @2 @2 8 .. . w hic h happens bec ause 1 0 is a m ult iple of 5 ))) = )))) = ))))))))) = )))) = . 0 0 8

1 2 5 5 @5 @5 5 @5 @5 @2 @2 @2 1 0 0 0 (.. . and, t hus, 1 0 0 0 is a m ult iple of 5 )³

If a rat ional f rac t ion in reduc ed f orm has denom inat or w it h prim e f ac t ors ot her t han 2 and 5 , t he dec im al does not t erm inat e.

For example: Can 1 be w ritten as x ? ... as x ? ... x ?

3 10 100 1000

Rat ionals w rit t en as dec im als m ay be T ERM IN A T IN G (e. g. 3 /5 = . 6 ) or N ON-T ERM IN A T IN G (e. g. 3 /1 1 = . 2 7 2 7 2 @@@ ).

Rat ionals w hic h, reduced, hav e denom inat ors w it h prim e f ac t ors 2 & 5 only are T ERM IN A T IN G dec im als bec ause:

Ot her rat ionals hav e N ON-T ERM IN A T IN G, REPEA T IN G dec im al ex pansions— bec ause:

Ú ADECIM A L N U M ERA L M A Y BE EX PRESSED A S RA T IO OF T W O IN T EGERS,IF.. .

’ T H E DECIM A L IS T ERM IN A T IN G:

A t erm inat ing dec im al has a f init e num ber of digit s t o t he right of t he dec im al point .

Such dec im als are easily ex pressed in rat ional f orm (rat io of t w o int egers); just read t hem aloud.

E. g. : 3 . 4 7 = 3 + 4 /1 0 + 7 /1 0 0 = 3 0 0 /1 0 0 + 4 0 /1 0 0 + 7 /1 0 0 = (3 0 0 + 4 0 + 7 )/ 1 0 0 = 3 4 7 /1 0 0 1 .6 =

You can m ak e ev en short er w ork of suc h c onv ersions: 5 2 . 0 3 0 2 = 5 2 0 3 0 2 /

If a dec im al num ber t erm inat es in t he nt h plac e af t er t he dec im al, it is equiv alent t o a f rac t ion w hose num erat or is t he num ber f orm ed by t he digit s w it hout t he dec im al point , and denom inat or 1 0 (w here n is num ber of digit s ⁿ

af t er t he dec im al point ) . ’ T H E DECIM A L IS NON-T ERM IN A T IN G REPEA T IN G:

Call t he repeat ing (" unk now n" ) num ber x .

M ult iply x by 1 0 w here n is t he lengt h of t he repet end (t he num ber of digit s t hat repeat ). ⁿ Subt rac t x f rom 1 0 x t o obt ain (1 0 -1 )x . ^{n n}

Solv e f or x ; m ult iply result ing f rac t ion by 1 0 / 1 0 if nec essary t o c lear a dec im al. ^{p p} For ex am ple: x = 5 2 . 1 9 0 9 0 9 0 9 0 9 0 9 0 . . .

1 0 0 x = 5 2 1 9 . 0 9 0 9 0 9 0 9 0 . . .

– x = 5 2 .1 9 0 9 0 9 0 9 0 .. . so x = 5 1 6 6 .9 = =

9 9 x = 9 9

By t he w ay , not ic e t he ex pression of a rat ional num ber as a dec im al is not unique. . 9& = Ev ery t erm inat ing dec im al has an alt ernat e f orm . EG 5 . 6 8 7 = 5 . 6 8 6 9 9 9 9 9 9 9 9 9 þ

T H E SET OF TERM INATING

&

¹ st u d en t s/ d isc iples o f t h e sc h o o l of Py t h ag o ras at Cro t o n a in So u t h ern It aly

² . . . o r so h e is c redit ed . Py t h ag o ras m ay h av e prov ed it , p erhap s o n e or som e of h is st u d en t f o llo w ers.

³ t h o u g h it w as k n o w n t o t h e Bab y lo n ian s & Eg y p t ian s at least c en t u ries earlier

9-3 REAL NUMBERS!!! 8

T he Pyt hagoreans believed that " Number rules the universe" ;

everything can be described in terms of numbers, and, further, that b c

all numbers are rational– integers, or ratios of integers.

Pythagoras proved w hat is called the Pythagorean Theorem :

a

a = b + c the sum of the squares of the sides of a right triangle is the square of the hypotenuse.

(and this happens only in a right triangle.)

Pythagoras' view of the describability of the universe in terms of rational numbers w as contradicted!

dest royed

!by the very theorem w hich now bears his name. For if w e construct a right triangle w ith sides

of equal length 1, then w e can demonstrate that the hypotenuse (w hose length ought to correspond to

some number), is %&

, and %&

cannot be expressed as a rational number— cannot be expressed as the

ratio of tw o integers. As w e argue below :

1 2 3 4 k

r r r r r

1 2 3 4 k

First w e not e t hat ev ery w hole num ber has a unique prim e f ac t orizat ion. w = p p p p @@@ p

1 2 3 4 k

2 r 2 r 2 r 2 r 2 r

1 2 3 4 k

T hen w m ust hav e a prim e f ac t orizat ion w it h t he sam e prim es t o ev en pow ers: w = p p p p @@@ p^{2 2}

Suppose %

&

w h ere p, q , Z, q … 0 [ af t er all, t hat ' s w hat it w ould m ean f or %

&

. . . an d w e m ay assu m e p/ q has been reduc ed t o low est t erm s (so p and q hav e NO c om m on f ac t ors).

M ult iply ing bot h sides by q and squaring giv es us 2 q = p . ^{2 2}

W hic h m eans 2 m ust be a f ac t or of p , w hic h in t urn im plies 2 m ust be a f ac t or of p. ²

So p has an odd pow er of 2 in it s prim e f ac t orizat ion. .. This is not possible. ²

Sinc e ev ery part of our argum ent is t ru e, t he only f law m ust be in t he assum pt ion w e m ade in t he f irst place, t hat %

&

be w rit t en in rat ional f orm . ]

Pythagoras and his follow ers w ere devastated. Even today, w e are troubled to discover there are

numbers

w hich cannot be w rit ten in the friendly form of a ratio of integers— but the irritation is diminished by the

comf orting thought that although such numbers exist, they are relatively few , and may be mostly

ignored. NOT!

Let' s make a small digression. We now know these irrationals exist, and %&

is one of them, and can

not be w ritten as a ratio of integers; but w hat is the nature of an irrational? What is the decimal

representation of an irrational number? We have previously seen that a rational number can be

expressed in decimal form using the division algorithm, and that the decimal form eit her terminates or

repeats.

Furthermore, any terminating or repeating decimal can be expressed as a ratio of tw o integers.

Therefore: The set of rational numbers is exactly the set of terminating and repeating decimals. ®

The Irrationals (reals not expressible as ratio of tw o integers) are non-terminating, non-repeating

decimals.

Consider, e.g.

... or

The sum of tw o rationals is rational. What about the sum of tw o irrationals?

This can be a little difficult to explore, since w e have no general arithmetic algorithms for irrationals.

What is %&

+ %&

? (Hint: square that!)

Is the sum of tw o irrationals alw ays irrational? ...How about %&

+ %&

? !3 %&

+ %&

&

?

What is the sum of a rational number and an irrational? Consider, for example, %&

+ ½.

Intuitively, w e have a non-terminating non-repeating decimal plus .5, the total of w hich should be

non-terminating non-repeating. How ever, w e look for proof....

⁴

1 8 4 5 -1 9 1 8 Can t o r' s t h eo ry o f t h e in f in it e so ro c k ed t h e sc ien t if ic an d , part ic u larly , m at h em at ic al w o rld o f h is t im e t h at so m e an t ag o n ist s w ere ab le t o b lo c k h is adv an c em en t ; h e w as v iew ed as sub v ersiv e by so m e, u n b alan c ed b y o t h ers. T h e bit t erness of h is lif e, c o u p led w it h in sec u rit y b red in Can t o r, and p erhap s so m e gen et ic p redisp o sit io n , led t o a nu m b er of b reak d o w n s f o r w h ic h Can t o r w as h o sp it alized . In t h e early t w en t iet h c en t u ry h is w o rk b eg an t o b e rec o g n ized as t h e prof o u n d ly real w o rk o f a gen iu s; t h is rec o g n it io n w as t o o lit t le and t o o lat e.

We can easily prove %&

+ ½ is irrational:

Suppose %&

+ ½ = a/b ... w here a/b is rational.

Then %&

= a/b – ½ w hich, since it is the difference of tw o rational numbers, is a rational number.

This contradicts the know n status of %&

– it’ s irrational.

The argument above show ing %&

is irrational may be used to show that the square root of any prime (or

of any composite number w hose prime factors are not all to an even pow er) is not rational.

Furthermore, there are ot her irrational numbers (pi and e for instance) that " occur" in numerous

circumstances. In addition, for every irrational, w e can construct a w hole family of irrationals by adding

any rational. (Irrational + rational is ...) Thus w e can easily see there are at least as many irrational

numbers as rationals. It w as not until the end of the nineteenth century that a means of tackling the

cardinality of infinite sets w as devised to settle this question: How does the size of the set of irrationals

compare with the size of the set of rationals?

G eorg Cantor w as primarily responsible for the development of theory of sets & classes, w ith particular

emphasis on the infinite. He devised the means of comparing cardinalit y of sets by mapping, or

establishing a 1-1 correspondence betw een tw o sets to show they are of the same cardinality. (Recall

our demonstrations that the cardinality of N is equivalent to that of Z.) Although the rationals are dense

in the number line, the cardinality of Q is the same as that of N! But the cardinality of the irrationals is

greater! Here is Cantor' s ingeniously simple proof:

Any set w hose cardinality is that of N can be listed. (The act of listing est ablishes a 1-1

correspondence w it h N.) Suppose a " list" of all irrationals is presented. We w ill show that the

list does not— CANNOT— contain all the irrationals, by constructing an irrational number that is not

in the list.

Suppose t he f irst f ew num bers

in t he alleged list are:

m.d d d d d ... e.g.

10.1450368... .3

1 2 3 4 5

n.e e e e e ... 4.2907863... 7

1 2 3 4 5

o.f f f f f ... .0006721... 2

1 2 3 4 5

p.g g g g g ... .0332737... 4

"

We select a digit different from d as the first decimal digit of our number; select a digit different

2 3

from e as the second digit of our number; select a digit different from f as the third digit, and so

on. The number so constructed cannot be the first number in the list because it differs in the first

decimal place; cannot be the second number in the list because it differs in the second decimal

place from that number; and so on. So it is not in the list at all! Thus the list is not complete.

Therefore, it is not possible to list all the irrationals, and thus they cannot be put into 1-1

correspondence w ith the natural numbers. The cardinality of the set of irrationals is greater than

the cardinality of N. Thus although the rationals are dense in the number line (no interval gaps),

there are other numbers (irrationals) in the number line, and they far outnumber the rationals!

Properties of Arithmetic Operations on REAL numbers:

Addition and multiplication on t he set of all real numbers have the follow ing properties:

C

; C

; A

; I

(0 for + ; 1 for

);

I

for + : for any decimal a, -a is the additive inverse.

I

for

:

for each decimal b other than 0, there is another decimal number, 1/b such that b

(1/b) = 1

Subtraction and Division have the closure propert y except for division by 0.

. Our new Universe: N   W   Z   Q   R

. N = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, @@@ } N

. W = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, @@@ } W

. Z = { @@@ , !5, !4, !3, !2, !1, 0, 1, 2, 3, 4, 5, @@@ } Z

. Q = { / | p,Z, q,Z, and q … 0 }

Q

1 1 2 2 3 3 3 1 1 3

= { 0, / , / , / , / , / , / , / , / , / , / , @@@ }

= The set of all terminating and repeating decimals

. R = The set of all decimals R

= The set containing all rationals and all irrationals

= Q c I

= Set

decimals that terminate or repeat c Set

nonterminating nonrepeating decimals

= Set of all decimals that can c Set of all decimals that cannot

9 be w ritten as a ratio of integers A 9 be w ritten as a ratio of integers A

Note:

If w e think of R as our new Universe, I, the set of irrationals, is the complement of Q, the set of rationals.

Every real number is rational or irrational– one or the other/

13-3 Some details & footnotes Math 210 .2

Trichotomy:

Given any tw o real numbers a and b, exactly one of the follow ing holds: a < b or a > b or a = b.

In particular, given any real number r, then either r is positive (r > 0), or r is negative (r < 0), or r is 0.

Some important details:

" The irrationals form a set disjoint (separate) from the rationals.

" Together the rationals and irrationals make up R, the set of real numbers.

" N f W f Z f Q f R. (Each of these is contained in — is part of — the next.)

" The set of real numbers is larger than the set of rationals, and is dense in the number line!

" The set of real numbers can be thought of as corresponding to the points (all points) on a line.

(The set of real numbers may also be thought of as the set of all decimal numbers, including those

w hich terminate or repeat (rationals) and those w hich neither terminate nor repeat ( irrationals). )

Radicals & Fractional exponents:

%, called radical, stands for the principle square root— the posit ive root of a positive number.

To indicate the negative of the square root, w e w rite !% . For instance, %16 = 4. !%16 = -4.

%&

may be w ritten in the form x

.

Notice that

%&

%&

= x (provided %&

exists)... And this can also be w rit ten

x

x

= x

(STRICTLY Optional !!) More Radicals & Fractional exponents:

The %, or radical, symbol is used to denote other roots of numbers, such as cube root: %&

&

= 3

We define: % denotes the nth root of a number. For instance, %64 = 2 because 2 = 64.

Just as %x may be w ritten in the form x

, %&

is x

and in general: x

= %&

.

...and more generally: x = (x

) = ( %x) .

E.g. 8

= %&

= 2 32

= (32

) = (2) = 4

8

= 1/ %8 = 1/2

32

= 1/(32

) = 1/(2) = 1/4

QUIZ YOURSELF! (1-15 simplify; 16-20 answ er rational, irrational, unpredictable)(* = optional !)

1*. 16

= 2*. 32

= 3*. 27

= 4*. 64

=

5. 4

= 6. 36@36

= 7. 7

@7

= 8*. 4

=

9*. ( / )

= 10*. (16/9)

= 11. %x =

12. %72a b =

13. %72 + %18 = 14. %36@2 @x y =

15. { 64x y @(x+ y) }

=

16. The sum of a rational and an irrational is: 17. The product of a rational and irrational is:

18. The sum (difference) of tw o rationals is: 19. The sum (difference) of tw o irrationals is:

20. The product (quot ient ) of tw o rationals is: 21. The product (quot ient ) of tw o irrationals is:

¿ Answ ers?:

a %a

1. 2 2. 2 3. 3 4. 4 5. 1/2 6. 6 7. 7 8. 8 9. ( / ) or / 10. 3/4 11. %(x ) = x

12. %8@9a b

%4@9a b 2b

2@3a b %2b 13. %9@4@2 + %9@2

6%2 + 3%2

9%2

14. %2 @3 @2 @x y

%2 @3 @(x ) @y

4@3x y%2y

12x y%2y

15. { 64x y @(x+ y) }

{ 2 x y @(x+ y) }

8x y (x+ y) { x}

16. irrational (see prior discussion)

17. if the rational is zero, the product is rational because it' s 0; otherw ise, irrational.

(Consider, eg, 2@%2, and.... If q , Q and q@b

p , Q then b

p/q must be rational unless q is zero.)

18. rational (+ or !) 19. unpredictable (+ or !) (Consider %2 + %2

2%2; (9 ! %2) + (%2 ! 5)

4)

20. rational (@ or ÷) 21. unpredictable (@ or ÷) (Consider %2@%2

2; %2@%3

%6)