irrationalitydexterityscfsqualityresistance how to be ... ·...

Irrationality Dexterity SCFs Quality Resistance

≡ How To Be Irrational ≡Mathπath 2017 • Mt. Holyoke College

Sam Vandervelde • Proof School • June 30, 2017

Sam Vandervelde How To Be Irrational

Tricky Math Ahead

Use care while attempting to comprehend!

POP QUIZ

Which of these numbers are irrational?

(A)√144

(B)√1444

(C) log6 7776

(D)( 3√6

3√3)3√9(E) π−e

e−π

None of them!

POP QUIZ

(A)√144

(B)√1444

(C) log6 7776

(D)( 3√6

3√3)3√9(E) π−e

e−π

None of them!

POP QUIZ

(A)√144 = 12

(B)√1444 = ??

(C) log6 7776

(D)( 3√6

3√3)3√9(E) π−e

e−π

POP QUIZ

(A)√144 = 12

(B)√1444 = 38

(C) log6 7776 = ??

(D)( 3√6

3√3)3√9(E) π−e

e−π

POP QUIZ

(A)√144 = 12

(B)√1444 = 38

(C) log6 7776 = 5

(D)( 3√6

3√3)3√9= ??

(E) π−ee−π

POP QUIZ

(A)√144 = 12

(B)√1444 = 38

(C) log6 7776 = 5

(D)( 3√6

3√3)3√9= 6

(E) π−ee−π = ??

POP QUIZ

(A)√144 = 12

(B)√1444 = 38

(C) log6 7776 = 5

(D)( 3√6

3√3)3√9= 6

(E) π−ee−π = −1

POP QUIZ

Which of these numbers is most irrational?

(A)√2 ≈ 1.414213562

(B)√3 ≈ 1.732050808

(C) e ≈ 2.718281828

(D) π ≈ 3.141592654

What Did You Say!?

POP QUIZ

Clearly (A)-(B) and (C)-(D) go together.

Let’s vote on which pair is more irrational.

(A)√2 ≈ 1.414213562

(B)√3 ≈ 1.732050808

(C) e ≈ 2.718281828

(D) π ≈ 3.141592654

}Now vote here}

Finally vote here

POP QUIZ

(A)√2 ≈ 1.414213562

(B)√3 ≈ 1.732050808

(C) e ≈ 2.718281828

(D) π ≈ 3.141592654

}Now vote here

}Finally vote here

POP QUIZ

(A)√2 ≈ 1.414213562

(B)√3 ≈ 1.732050808

(C) e ≈ 2.718281828

(D) π ≈ 3.141592654

}Now vote here}

Finally vote here

The Results Are In

It appears collectively we agree that. . .

Most irrational: π

Very irrational: e

Quite irrational:√3

Pretty irrational:√2

Unreal: i

How on earth

can we quantify

irrationality?

The Results Are In

Most irrational: π

Very irrational: e

Unreal: i

How on earth

can we quantify

irrationality?

The Results Are In

Most irrational: π

Very irrational: e

Unreal: i

How on earth

can we quantify

irrationality?

The Results Are In

Most irrational: π

Very irrational: e

Unreal: i

How on earth

can we quantify

irrationality?

For The Record

As we all know, an irrational number is. . .

a number that cannot be written as (is not

equal to) a rational pq for integers p and q.

IDEA: measure the irrationality of

a number α according to how well

we can approximate α by fractions.

For The Record

How Close Can You Go?

The most obvious measure of how well a

fraction pq approximates a given irrational

number α is proximity.

Here are some approximations to π:

−1617,

Which fraction scores worst for proximity?

−1617,

Which fraction scores best for proximity?

−1617,

Which fraction scores best for proximity from

among all rational numbers?

VERY GOOD

−1617,

Which fraction scores best for proximity from

among all rational numbers? VERY GOOD

Since any irrational may be approximated

arbitrarily closely by rationals, we conclude

Proximity is not a use-

ful yardstick by which to

measure irrationality.

Fortunately, some approximations are more

dexterious than others.

Since any irrational may be approximated

arbitrarily closely by rationals, we conclude

Proximity is not a use-

ful yardstick by which to

measure irrationality.

Fortunately, some approximations are more

dexterious than others.

Opinionated People

Which of the following approximations to π

are commonly used?

3, 3.1, 3.14, 3.142, 3.1416

Why do we prefer some over others?

DISCUSS

We like fractions that approximate π with

greater dexterity.

Opinionated People

are commonly used?

3, 3.1, 3.14, 3.142, 3.1416

DISCUSS

greater dexterity.

Opinionated People

are commonly used?

3, 3.1, 3.14, 3.142, 3.1416

DISCUSS

greater dexterity.

Opinionated Fractions

How could we define dexterity via lengths?

You’ve probably heard of 227 ; it’s an anti-

quated approximation to π. What’s the big

deal over 227 , anyway?

Check it out:

Maybe π isn’t so resistant to being approxi-

mated by fractions after all. . .

deal over 227 , anyway? Check it out:

Formulation

Let pq be an approximation to an irrational

number α. Write a formula that measures

the dexterity of the approximation, in terms

of p, q, and α. (Using absolute value.)

q∣∣∣pq − α∣∣∣ or |p− qα|

The dexterity of 3141610000 is 0.0735, while the

dexterity of 227 is 0.0089.

Formulation

q∣∣∣pq − α∣∣∣ or |p− qα|

Formulation

q∣∣∣pq − α∣∣∣ or |p− qα|

Finding Fractions

For a change of pace, let’s search for more

and more dexterious approximations to√3.

p q dexterity

2 1 0.267955 3 0.196157 4 0.0718019 11 0.0525626 15 0.0192471 41 0.0140897 56 0.00515265 153 0.00377362 209 0.00138989 571 0.00101

What do you notice here?

Finding Fractions

p q dexterity

2 1 0.267955 3 0.196157 4 0.0718019 11 0.0525626 15 0.0192471 41 0.0140897 56 0.00515265 153 0.00377362 209 0.00138989 571 0.00101

Finding Fractions

p q dexterity

2 1 0.267955 3 0.196157 4 0.0718019 11 0.0525626 15 0.0192471 41 0.0140897 56 0.00515265 153 0.00377362 209 0.00138989 571 0.00101

Dexterity is Doomed

For starters, the dexterity values seem to be

getting closer and closer to 0. To confirm,

the dexterity of 99730815757961 is about 0.000001.

FACT: for any irrational number

α, there exist rational approxi-

mations pq with arbitrarily small

dexterity value. Uh-oh.

Dexterity is Doomed

FACT: for any irrational number

α, there exist rational approxi-

mations pq with arbitrarily small

dexterity value. Uh-oh.

Dexterity is Doomed

Dexterity is also not a

useful yardstick by which

to measure irrationality.

(Although we might measure how quickly the

dexterity values approach 0.)Sam Vandervelde How To Be Irrational

Finding Fractions

There is a clever method to find dexterious

fractions that provides the key to everything.

p q dexterity

2 1 0.267955 3 0.196157 4 0.0718019 11 0.0525626 15 0.0192471 41 0.0140897 56 0.00515265 153 0.00377362 209 0.00138989 571 0.00101

Simple Continued Fractions

Suppose we were to write√3 in the form

� +1

� + · · ·

How do

we start?

� +1

� + · · ·

value left:

0.732050808

1x → 1.3660254. . .

� +1

� + · · ·

value left:

0.7320508081x → 1.3660254. . .

� +1

� + · · ·

value left:

0.366025404

1x → 2.7320508. . .

� +1

� + · · ·

value left:

0.3660254041x → 2.7320508. . .

� +1

� + · · ·

value left:

0.7320508

1x → 1.3660254. . .

� +1

� + · · ·

value left:

0.73205081x → 1.3660254. . .

Simple Contained Factions

2 + · · ·

the SCF

repeats!

The partial convergents approximate√3:

Those sure

look familiar:21, 5

3, 74, 19

11, 2615, 71

Those sure

look familiar:21, 5

3, 74, 19

11, 2615, 71

Those sure

look familiar:21, 5

3, 74, 19

11, 2615, 71

Those sure

look familiar:21, 5

3, 74, 19

11, 2615, 71

Those sure

look familiar:21, 5

3, 74, 19

11, 2615, 71

Those sure

look familiar:21, 5

3, 74, 19

11, 2615, 71

Those sure

look familiar:21, 5

3, 74, 19

11, 2615, 71

We Love Shortcuts

Let an be the nth “box value”, and let pn/qnbe the corresponding nth partial convergent.

How do the an values generate the fractions?

an 1 1 2 1 2 1 2

pn 1 2 5 7 19 26 71

qn 1 1 3 4 11 15 41

For n ≥ 1, we have pn+1 = an+1pn + pn−1.

We Love Shortcuts

an 1 1 2 1 2 1 2

pn 1 2 5 7 19 26 71

qn 1 1 3 4 11 15 41

We Love Shortcuts

an 1 1 2 1 2 1 2

pn 1 2 5 7 19 26 71

qn 1 1 3 4 11 15 41

For n ≥ 1, we have pn+1 =

an+1pn + pn−1.

We Love Shortcuts

an 1 1 2 1 2 1 2

pn 1 2 5 7 19 26 71

qn 1 1 3 4 11 15 41

We Love Shortcuts

an 1 1 2 1 2 1 2

pn 1 2 5 7 19 26 71

qn 1 1 3 4 11 15 41

For n ≥ 1, we have qn+1 = an+1qn + qn−1.

We Love Shortcuts

an - 1 1 2 1 2 1 2

pn 1 1 2 5 7 19 26 71

qn 0 1 1 3 4 11 15 41

For n ≥ 0, we have qn+1 = an+1qn + qn−1.

We Love Shortcuts

Here are some an values for the SCF for π.

Compute the partial convergents:

an - 3 7 15 1 292

We Love Shortcuts

an - 3 7 15 1 292

pn 1 3

qn 0 1

We Love Shortcuts

an - 3 7 15 1 292

pn 1 3 22

qn 0 1 7

We Love Shortcuts

an - 3 7 15 1 292

pn 1 3 22 333

qn 0 1 7 106

We Love Shortcuts

an - 3 7 15 1 292

pn 1 3 22 333 355

qn 0 1 7 106 113

We Love Shortcuts

an - 3 7 15 1 292

pn 1 3 22 333 355 BIG

qn 0 1 7 106 113 BIG

That 15 has a lot to do with why 227 is such

a dexterious approximation to π.

Imagine

how dexterious the fraction 355113 is!

We Love Shortcuts

an - 3 7 15 1 292

pn 1 3 22 333 355 BIG

qn 0 1 7 106 113 BIG

That 15 has a lot to do with why 227 is such

a dexterious approximation to π. Imagine

how dexterious the fraction 355113 is!

We Love Shortcuts

Actually, it comes to about 0.000030144.

Compare that to the best we can do for√3

using three digit-numbers, which is 989571, with

a dexterity value of 0.001011122.

It would seem that√3 is more resistant to

being approximated by rationals than π.

We Love Shortcuts

Mind Your ps and qs

Here are the partial convergents for√3.

Can you spot the pattern relating various

pn and qn values?

an - 1 1 2 1 2 1 2

pn 1 1 2 5 7 19 26 71

qn 0 1 1 3 4 11 15 41

We’ll write this as |pnqn+1 − pn+1qn| = 1.

There’s an lovely proof by induction.

Mind Your ps and qs

Here are the partial convergents for√3.

Can you spot the pattern relating various

pn and qn values?

an - 1 1 2 1 2 1 2

pn 1 1 2 5 7 19 26 71

qn 0 1 1 3 4 11 15 41

We’ll write this as |pnqn+1 − pn+1qn| = 1.

There’s an lovely proof by induction.

Mind Your ps and qs

Now we can explain small dexterity values!

∣∣∣∣pnqn − α∣∣∣∣

∣∣∣∣pnqn − pn+1

∣∣∣∣= qn

∣∣∣∣pnqn+1 − qnpn+1

qnqn+1

∣∣∣∣= qn

qnqn+1=

Dexterity values become arbitrarily small.

Mind Your ps and qs

∣∣∣∣pnqn − α∣∣∣∣ < qn

∣∣∣∣

qnqn+1

∣∣∣∣= qn

qnqn+1=

Mind Your ps and qs

∣∣∣∣pnqn − α∣∣∣∣ < qn

∣∣∣∣= qn

qnqn+1

∣∣∣∣

qnqn+1=

Mind Your ps and qs

∣∣∣∣pnqn − α∣∣∣∣ < qn

∣∣∣∣= qn

qnqn+1

∣∣∣∣= qn

qnqn+1=

Mind Your ps and qs

∣∣∣∣pnqn − α∣∣∣∣ < qn

∣∣∣∣= qn

qnqn+1

∣∣∣∣= qn

qnqn+1=

Mind Your ps and qs

In fact, we can go further:

∣∣∣∣pnqn − α∣∣∣∣ < 1

an+1qn + qn−1<

an+1qn

Therefore q2n

∣∣∣∣pnqn − α∣∣∣∣ < 1

This suggests examining an even more

stringent measure for how well a fraction

approximates an irrational. . .

Mind Your ps and qs

∣∣∣∣pnqn − α∣∣∣∣ < 1

an+1qn + qn−1

an+1qn

Therefore q2n

∣∣∣∣pnqn − α∣∣∣∣ < 1

Mind Your ps and qs

∣∣∣∣pnqn − α∣∣∣∣ < 1

an+1qn + qn−1<

an+1qn

Therefore q2n

∣∣∣∣pnqn − α∣∣∣∣ < 1

Mind Your ps and qs

∣∣∣∣pnqn − α∣∣∣∣ < 1

an+1qn + qn−1<

an+1qn

Therefore q2n

∣∣∣∣pnqn − α∣∣∣∣ < 1

Mind Your ps and qs

∣∣∣∣pnqn − α∣∣∣∣ < 1

an+1qn + qn−1<

an+1qn

Therefore q2n

∣∣∣∣pnqn − α∣∣∣∣ < 1

The Suspense Is Intense

quantity formula

proximity∣∣∣pq − α∣∣∣

dexterity q∣∣∣pq − α∣∣∣

quality q2∣∣∣pq − α∣∣∣

We have just shown that quality < 1an+1

partial convergent pnqn

, so it is less than 1

infinitely often for any irrational number α.

quantity formula

Looking for Low Quality

Let’s take a look at quality values for

approximations to π,√3 with q < 100 000.

We’ll only display quality values less than 1.

As you survey these values, think about how

to define the resistance of an irrational α to

being approximated by fractions.

Values under 0.1 are highlighted in orange.

p q quality for π

3 1 0.141592653596 2 0.56637061435919 6 0.90266447076722 7 0.061959974100144 14 0.24783989640166 21 0.55763976690188 28 0.991359585602333 106 0.935055734916355 113 0.00340631193291710 226 0.01362524773161065 339 0.03065680741541420 452 0.05450099092651775 565 0.08515779829052130 678 0.122627229662

p q quality for π

2485 791 0.1669092847572840 904 0.2180039637063195 1017 0.2759112665083550 1130 0.3406311931623905 1243 0.412163743674260 1356 0.4905089186474615 1469 0.5756667169124970 1582 0.6676371390295325 1695 0.7664201855680 1808 0.8720158548246035 1921 0.984424148501

103993 33102 0.63321929943104348 33215 0.365864174164208341 66317 0.538117528835312689 99532 0.288714520168

Looking For Low Quality

p q quality for√3

2 1 0.2679491924313 2 0.9282032302765 3 0.588457268127 4 0.28718707889812 7 0.87048957087519 11 0.57814771583426 15 0.28856829700345 26 0.86634591656171 41 0.57740752328397 56 0.288667464001168 97 0.866048415566265 153 0.57735437985362 209 0.288674583873627 362 0.866027055935

Looking For Low Quality

p q quality for√3

989 571 0.5773505643161351 780 0.2886750952032340 1351 0.8660255224413691 2131 0.5773502895465042 2911 0.2886751327868733 5042 0.86602540625213775 7953 0.57735025835618817 10864 0.28867513535132592 18817 0.86602533927451409 29681 0.57735016395670226 40545 0.288675282063121635 70226 0.866025410098

Using quality, define the resistance of√3.

Join the Resistance

Here is one of several possible definitions.

Definition:The resistance of an irrational number α to

being approximated by fractions is equal to

the smallest value of the quantity q2|pq − α|in the long run, i.e. in the limit as q gets

larger and larger.

What is the resistance of e and√2?

Join the Resistance

p q quality for e

3 1 0.2817181715415 2 0.8731273138368 3 0.46453645613111 4 0.50749074465519 7 0.19580959449338 14 0.78323837797387 32 0.479407657938106 39 0.506661086208193 71 0.141302737955386 142 0.5652109518191264 465 0.4883585570391457 536 0.5038110301722721 1001 0.1103977914625442 2002 0.441591165847

Join the Resistance

p q quality for e

8163 3003 0.99358012588723225 8544 0.49252671457425946 9545 0.50246393913849171 18089 0.090523341423298342 36178 0.362093365693147513 54267 0.814710862498517656 190435 0.494802226895566827 208524 0.5017237309371084483 398959 0.07663415162822168966 797918 0.306536606513

It appears that the resistance of e is steadily

marching down to 0. Whaddayaknow.

Join the Resistance

p q quality for e

8163 3003 0.99358012588723225 8544 0.49252671457425946 9545 0.50246393913849171 18089 0.090523341423298342 36178 0.362093365693147513 54267 0.814710862498517656 190435 0.494802226895566827 208524 0.5017237309371084483 398959 0.07663415162822168966 797918 0.306536606513

It appears that the resistance of e is steadily

marching down to 0. Whaddayaknow.

Join the Resistance

p q quality for√2

1 1 0.4142135623733 2 0.3431457505084 3 0.7279220613587 5 0.35533905932710 7 0.70353544371817 12 0.35324701827424 17 0.70771952582541 29 0.35360595577358 41 0.70700165082799 70 0.353544371834140 99 0.707124818707239 169 0.353554937972338 239 0.707103686437577 408 0.353553125106

Join the Resistance

p q quality for√2

816 577 0.7071073122251393 985 0.353553436251970 1393 0.7071066897783363 2378 0.3535533819964756 3363 0.7071067967588119 5741 0.35355339372211482 8119 0.70710676592219601 13860 0.35355335196127720 19601 0.70710682423347321 33461 0.35355355604466922 47321 0.707106588219114243 80782 0.353553249734

Time for the final reckoning.

Join the Resistance

p q quality for√2

816 577 0.7071073122251393 985 0.353553436251970 1393 0.7071066897783363 2378 0.3535533819964756 3363 0.7071067967588119 5741 0.35355339372211482 8119 0.70710676592219601 13860 0.35355335196127720 19601 0.70710682423347321 33461 0.35355355604466922 47321 0.707106588219114243 80782 0.353553249734

Time for the final reckoning.

Putting up Resistance

rank num res Simple Cont Fraction

4. π 0 [3; 7, 15, 1, 292, 1, 1, . . . ]

3. e 0 [2; 1, 2, 1, 1, 4, 1, 1, 6, . . . ]

2.√3 0.289 [1; 1, 2, 1, 2, 1, 2 . . . ]

1.√2 0.354 [1; 2, 2, 2, 2, 2, 2 . . . ]

Precisely the opposite of what we expected.

Can you guess the number that holds the

record for being most irrational?

4. π 0 [3; 7, 15, 1, 292, 1, 1, . . . ]

3. e 0 [2; 1, 2, 1, 1, 4, 1, 1, 6, . . . ]

2.√3 0.289 [1; 1, 2, 1, 2, 1, 2 . . . ]

1.√2 0.354 [1; 2, 2, 2, 2, 2, 2 . . . ]

4. π 0 [3; 7, 15, 1, 292, 1, 1, . . . ]

3. e 0 [2; 1, 2, 1, 1, 4, 1, 1, 6, . . . ]

2.√3 0.289 [1; 1, 2, 1, 2, 1, 2 . . . ]

1.√2 0.354 [1; 2, 2, 2, 2, 2, 2 . . . ]

4. π 0 [3; 7, 15, 1, 292, 1, 1, . . . ]

3. e 0 [2; 1, 2, 1, 1, 4, 1, 1, 6, . . . ]

2.√3 0.289 [1; 1, 2, 1, 2, 1, 2 . . . ]

1.√2 0.354 [1; 2, 2, 2, 2, 2, 2 . . . ]

Tip: It’s the Golden Ratio

I told you it was the Golden Ratio

1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661319928290267880675208766892501711696207032221043216269548626296313614438149758701220340805887954454749246185695364864449241044320771344947049565846788509874339442212544877066478091588460749988712400765217057517978834166256249407589069704000281210427621771117778053153171410117046665991466979873176135600670874807101317952368942752194843530567830022878569978297783478458782289110976250030269615617002504643382437764861028383126833037242926752631165339247316711121158818638513316203840052221657912866752946549068113171599343235973494985090409476213222981017261070596116456299098162905552085247903524060201727997471753427775927786256194320827505131218156285512224809394712341451702237358057727861600868838295230459264787801788992199027077690389532196819861514378031499741106926088674296226757560523172777520353613936210767389376455606060592165894667595519004005559089502295309423124823552122124154440064703405657347976639

4. π 0 [3; 7, 15, 1, 292, 1, 1, . . . ]

3. e 0 [2; 1, 2, 1, 1, 4, 1, 1, 6, . . . ]

2.√3 0.289 [1; 1, 2, 1, 2, 1, 2 . . . ]

1.√2 0.354 [1; 2, 2, 2, 2, 2, 2 . . . ]

0. ϕ 0.447 [1; 1, 1, 1, 1, 1, 1 . . . ]

What are those positive resistance values,

and where do they come from?

4. π 0 [3; 7, 15, 1, 292, 1, 1, . . . ]

3. e 0 [2; 1, 2, 1, 1, 4, 1, 1, 6, . . . ]

2.√3 0.289 [1; 1, 2, 1, 2, 1, 2 . . . ]

1.√2 0.354 [1; 2, 2, 2, 2, 2, 2 . . . ]

0. ϕ 0.447 [1; 1, 1, 1, 1, 1, 1 . . . ]

What are those positive resistance values,

and where do they come from?

A Neat Trick

Can you think of a perfect square that is

exactly three times a perfect square?

02 = 3(02) (√42)2 = 3(

√14)2

More precisely, can you think of positive

integers x and y such that x2 = 3y2?

There are none.

A Neat Trick

02 = 3(02)

(√42)2 = 3(

√14)2

There are none.

A Neat Trick

02 = 3(02) (√42)2 = 3(

√14)2

There are none.

A Neat Trick

02 = 3(02) (√42)2 = 3(

√14)2

There are none.

A Neat Trick

02 = 3(02) (√42)2 = 3(

√14)2

There are none.

Deja Vu All Over Again

Can you find positive integers x and y such

that |x2 − 3y2| is small, say equal to 1 or 2?

x y |x2 − 3y2|2 1 1

19 11 2

26 15 1

71 41 2

x y |x2 − 3y2|2 1 1

19 11 2

26 15 1

71 41 2

x y |x2 − 3y2|2 1 1

19 11 2

26 15 1

71 41 2

x y |x2 − 3y2|2 1 1

19 11 2

26 15 1

71 41 2

x y |x2 − 3y2|2 1 1

19 11 2

26 15 1

71 41 2

x y |x2 − 3y2|2 1 1

19 11 2

26 15 1

71 41 2

x y |x2 − 3y2|2 1 1

19 11 2

26 15 1

71 41 2

Didn’t See That Coming

It is a very nifty and handy fact that if pq is a

partial convergent approximation to√3,

then p2 − 3q2 is a small positive integer.

This makes sense, because if pq ≈√3, then

q2≈ 3, meaning that p2 ≈ 3q2.

Our fact applies to any value√N . (And

with some care, to numbers like ϕ also.)

It All Falls Into Place

Suppose that p2 − 3q2 = 1. What is the

quality of the approximation pq to√3?

p2 − 3q2 = 1

(p + q√3)(p− q

√3) = 1

q2(pq −√3)(pq +

√3) = 1

q2(pq −√3) =

1pq +√3

Hence quality → 12√3≈ 0.289.

p2 − 3q2 = 1

(p + q√3)(p− q

√3) = 1

q2(pq −√3)(pq +

√3) = 1

q2(pq −√3) =

1pq +√3

p2 − 3q2 = 1

(p + q√3)(p− q

√3) = 1

q2(pq −√3)(pq +

√3) = 1

q2(pq −√3) =

1pq +√3

p2 − 3q2 = 1

(p + q√3)(p− q

√3) = 1

q2(pq −√3)(pq +

√3) = 1

q2(pq −√3) =

1pq +√3

p2 − 3q2 = 1

(p + q√3)(p− q

√3) = 1

q2(pq −√3)(pq +

√3) = 1

q2(pq −√3) =

1pq +√3

Hence quality →

12√3≈ 0.289.

p2 − 3q2 = 1

(p + q√3)(p− q

√3) = 1

q2(pq −√3)(pq +

√3) = 1

q2(pq −√3) =

1pq +√3

What do you suppose happens if we were to

start with p2 − 3q2 = 2 instead?

In this case quality → 22√3≈ 0.577.

What are the lowest quality values we can

achieve for√2? Quality → 1

2√2≈ 0.354.

How does the golden ratio do better?

G O O D L U C K

2√2≈ 0.354.

G O O D L U C K

achieve for√2?

Quality → 12√2≈ 0.354.

G O O D L U C K

2√2≈ 0.354.

G O O D L U C K

2√2≈ 0.354.

G O O D L U C K

2√2≈ 0.354.

G O O D L U C K

See You Tomorrow

Thanks for being such a great audience!

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