Maths

I cannot do'th without counters - Clown, The Winter's Tale, Shakespeare

Edmund Landau tulis satu buku Grundlagen der Analysis in 1930, and the book
was translated to English in 1950 as Foundations in Analysis.

Dalam buku ni, dia buat dua preface yang berbeza untuk cikgu dan untuk student.

Preface Untuk Student:-

1. Please don’t read preface for the teacher

2. I will ask of you only the ability to read English and to think logically -
ni high school mathematics and certainly no higher mathematics.

3. Please forget everything you learned in school; for you haven’t learned it.

4. The multiplication table will not occur in the book, not even the theorem 2 x 2 = 4.

but I would recommend, as an exercise, that you define

2 =1+1

4 = (((1+1) + 1) + 1) +1 and then prove the theorem.



Preface untuk teachers:-

I hope that I have written this book in such a way that a normal student can read it in two days.

And then (since he already knows the formal rules from school), he may forget its content.

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Biron : By yea and nay, sir, that I swore in jest. What is the end of study? let me know.

King : Why, that to know which else we should not know.

Biron : things hid and barred, you mean from common sense?

King : Ay, that is study’s god-like recompense.

Shakespeare - Love’s Labour’s Lost, Scene : Navarre.


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I thought of writing about the ideas of maths raised by pikir kool, sekali da sampai keluar Al Hujurat ayat 11, i guess something is wrong somewhere.

macam ni je la.

Let R be set of all sets of math knowledge.

R = {logic, finite state machine, mental calculation techniques, etc..}

so when it was written “tapi matematik ni bukan limited to kaedah…semata-mata”, then it meant that logic is subset of R and not the only set in R.

sebab if i insist logic is the only set in R, then it will defy the meaning of R which is to be set of all sets in math knowledge.

akan create something called Russell’s paradox there.

so tak perlu cakap pasal logic tu Maths semata2…aku da tulis “matematik ni bukan limited to kaedah..”

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Man usually counts, then calculates.

Apa beza counting dengan calculating?

Best explain in the fiction below.

Imagine zaman dahulu (5000 years before, hahaha)…where a village which prospers from cultivation tiba-tiba dikepung barbaric tribes yang nak plunder kampung tu.

King : Oh penduduk2, apa kah gerangan nya semua ini?

Penduduk : Kampung kita dikepung segerombolan penjahat, Tuanku.

Note that the word Gerombolan…it is to say a group of bandits..or a set of bandits is attacking.

I say that the act of counting has taken place, in the sense that mentally, the bandits are “grouped” into one set.

One set of things.
Satu bakul durian musang (actually kita tak tahu exactly brapa biji, we just group it into one bakul)
Satu cawan kopi.

english grammar ada separate measurable and immeasurable adjectives. This shows the concept of counting in English mindset.

(Note also Meleis are quite alike the serfs during zaman Gelap Europe, since they called their kings as Tuanku which literally means  my master, or my lord)

ok sambung citer kampung yang nak kena serang. Boleh dengar lagu Queensryche -Before The Storm untuk feel macam mana perasaan penduduk kampung yang nak diplunder.

King : Menteri, give me an assessment of the situation..

Menteri : Ninja spy kita dah pergi buat scouting Tuanku. (mcm citer naruto la plak)…based on the trails, they came from our southern border.

King : Apa strength of these bandits?

menteri : Ninja spy kita follows the trails, in between the way they notice the cooking pots left behind….lebih kurang ada 50 cooking pots tu. And they got no cavalry.

King : wah satu cooking pot boleh bagi makan 4 orang..eh 200 people je bandit tu…can we defend?

Menteri : No hal la Tuanku, kampung kita ni ada 1000 orang lebih, voluntary army je pun da 300..we take about 1 hour approx. to assemble our troops to fight those pricks!

The overall assessment is what I define as calculate.
To give in-depth meaning and analyse our action based on the counting taken.

Yes, zaman China dulu2,one of the way to assess army strength is by calculating the cooking pots (boleh cari kat Popular buku Romance of Three Kingdom… ni citer best)

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So as history progress, men use counting and calculation as tools.

Sampai la zaman Greeks.

now, the reason aku cerita panjang lebar previously pasal serangan kampung tu semua, is to stress out one point the Greeks had in mind about maths.

That is all calculation should be countable and commensurable.

http://en.wikipedia.org/wiki/Commensurability_%28mathematics%29
http://en.wikipedia.org/wiki/Countable_set

senang cerita la..semua calculation mesti dapat satu exact number.

exact number as in 1, 2, 3, etc..

dan boleh express as fractions.

1 and a half.
2 suku.
satu per lapan. (kalau dalam construction, orang panggil 1 hon as 1 per lapan inch)

The element of determinism tu mesti ada dalam calculation, the Greeks said.

Then they ran into problems.

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ikut cerita mitos dia, pada 428BC, orang2 Athens ni kena satu wabak.

so diorang pun pergi “sembahyang” ramai2 kat temple dia kat Delios.

temple kat Delios ni second important temple kat Greek la..1st temple kat Delphi.

nak dijadikan cerita, temple kat delios ni bentuk kiub (segiempat on all sides).

Then satu malam, paderi Greek tu bermimpi.

“kalau kau nak wabak ni hilang, kau kena doublekan size altar untuk Apollo”…gitu la mimpi Paderi Greek ni.

maknanya kena besarkan tempat sembahyang twice the volume.

Kalau isipadu temple tu 10 meter cube, kau kena jadikan temple tu sekarang 20 meter kiub, gitula citer senang dia.

so masa nak besarkan temple tu, puak2 paderi ni gi la jumpa Plato nak tanya pasal mimpi ni.

Plato tafsirkan mimpi tu as orang2 Greek ni jangan sibuk2 nak perang dan conquer tempat orang, sibuk-sibuk kan la diri belajar maths dan geometry.

sebab Plato ada gut feeling doubling the cube was not constructible.

Paderi2 tu pun leceh lecehkan Plato, ah pandai2 lak kau nak tafsirkan mimpi.

So what they did is besarkan temple tu.

Cara dia besarkan temple tu ialah, doublekan panjang, doublekan lebar dan doublekan tinggi temple tu.

So of course la kau dah doublekan semua side, mesti kau dapat doublekan volume tu kan?

Nope.

let say original length is s. being a cube, that means height and width is also s.

original volume = s3

sekarang what is the final volume.

Final volume is = 2s X 2s X 2s = 8 s3.

so kalau kau doublekan all sides, u get 8 times of the original volume.

Habis tu, berapa exact panjang, tinggi dan lebar yang kau nak panjangkan in order to get the cube EXACTLY twice the volume?

Tahun 1837, Pierre Wantzel je baru dapat jawapan tu. huhu.

http://en.wikipedia.org/wiki/Doubling_the_cube
http://en.wikipedia.org/wiki/Constructible_number

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The greeks said all calculation should be countable and commensurable.

In the sense that calculation mesti dapat exact number..atau tak dapat exact number pun..kau mesti dapat summarise dia dalam bentuk fraction.

“Past Greek conceptions dictated that they necessarily must be, for whole numbers represent discrete objects, and a commensurable ratio represents a relation between two collections of discrete objects”

http://en.wikipedia.org/wiki/Squaring_the_circle

Apa yang Greeks buat untuk determine Pi ialah dia try bahagikan square tu to many squares, so that diorang boleh jumpa EXACT RATIO untuk nilai Pi.

Antara orang2 yang tak setuju dengan pandangan nombor ni kena discrete dan boleh describes as fractions ialah encik Zeno.

Zeno ni basically he wants to criticise pandangan “determinism” dalam Greeks math.

So that’s why dia buat Zeno paradox.

Zeno Paradox tu…matlamatnya bukan pada paradox main2 atau apa..tapi nak cakap yang

length/distance atau time…tak semestinya boleh di-determine on fractions or absolute value.

Aku stress point “tak semestinya” tu.

http://en.wikipedia.org/wiki/Zeno_paradox

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Then the Greeks mula dapat the glimpse and idea when it comes to determining
square root of 2.

satu hari, Hippasus menggunakan theorem pythagoras untuk segitiga bersudut tegak yang ada equal sides.

then dia nak determine panjang hepotenus segitiga tu.

what happens is described here:-
http://en.wikipedia.org/wiki/Irrational_numbers

anyway sad story kat Hippasus, masa dia declare penemuan ni, puak2 Pythagoras tak puas hati lalu campak dia ke laut.

huhu.

anyway
from now onwards mathematicians dah jumpa 2 concept baru.
irrational numbers
and transcedental numbers.

pi tu is a transcedental number.

and that’s why Zeno paradox is taught dalam Maths syllabus as an introduction to irrational numbers.

http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational

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cerita Gauss dengan cikgu sekolah JG Buttner ni agak famous.

what is the sum of 1,2,3….99,100?

Apa dia nampak is 100+1 = 101
99+2 = 101
98+3 = 101 and so on.

ada 50 pairs of number between 1 to 100. and each pair the sum is 101.
so 50 x 101 = 5050.

mental arithmetic la ni.

things is easy if you can see the patterns.

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aku suka maths ni sbb maths ni seriously beautiful.

cuma nak menjumpai beauty of maths ni, all the mathematician ni mostly melalui

jalan yang penuh discipline.

dia ada study route dia semua.

ni bukan cerita mengolok-olokkan ke apa.

aih.

lotfi zadeh tu basically bila dia jumpa fuzzy theory,
he went at great lengths to study it, then he wrote a paper on it and publish it dalam peer review.

Sebab ilmu ni kan ada discipline dia.

bila ikut path dia tu..the world boleh dapat maximum benefit as the idea will be criticised and tested by many.

daripada paper set fuzzy theory, it was expressed at great lengths, ilmu dia berkembang sampai jadi teori fuzzy logic.

Mesin basuh sekarang yang boleh determine berat air, berapa cucian, berapa kali nak rinse semua tu based on fuzzy logic.

aku bukan child prodigy. as i wish i was.

Ini bukan mengolok-olok.

Tapi bila kau tengok non-muslims berlumba2 dalam ilmu dan benefit dia satu dunia dapat, tak kan kau tak rasa sedih?

sedih bila aku tengok satu child prodigy kata dia duk rumah main game x mahu pergi sekolah sebab speed CPU dia ferrari speed orang lain kancil.

sedih bila aku tengok satu prodigy gantung signboard kat diri dia sambil cakap dia adalah ketua saintis dunia.

sebab ke-prodigy-an diorang tu tak boleh di maximise dan dibenefit oleh semua.

kita perlu admit yang kita guna at this present time ialah hasil dari ilmu dan application by non muslims.

Lotfi zadeh tu pun masuk presbyterian school masa sekolah rendah weh.

Sedih, kita sibuk syarah sana sini, “Bacalah dengan Nama Tuhan mu.yang menciptakan.”..lepas tu citer pasal kepentingan ilmu, sejarah ilmu Islam tapi kita tak respect pun dengan ilmu dan discipline ilmu.

Tuhan tempelak orang yang tak buat apa dia cakap dalam Surah As-Saff ayat 1 sampai 4.

Macam mana nak jadi “tersusun rapat seolah2 bangunan kukuh? (bun ya nim mar sus) kalau cakap tak serupa bikin?

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masa aku 1st kerja dulu, senior aku sound.

“wei kau engineer, tolong respect engineering ni bleh?”

senior aku tu dia x de paper, tapi dia terror dalam kerja sebab dia memang dedicated dalam construction.

aku faham la, of course dia sakit hati bila tengok budak ada degree tapi buat kerja engineering ni main main.

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aku lahir 1982. so basically aku ambik Alam Dan Manusia during skolah rendah.

Alam Dan Manusia ni subject General Knowledge la kiranya.

anyway aku nk cakap pasal 3M.
Sekolah rendah dari darjah 1 sampai darjah 6, basically emphasise dan make sure the students empower 3M ni.
Membaca, Mengira, Menulis.
yes, there are people yg masuk form 1 tak ok lagi 3M ni.

so on what level of “Mengira” is needed before going to high school?

Time aku skolah rendah, syllabus maths is like this:-

standard 1 , the teachers objective is for kids to be able to handle arithmetics of plus and minus in terms of sa and puluh

standard 2 , be able to handle arithmetics of multiplication in terms of sa and puluh.
standard 3 , arithmetics in terms of hundreds.
standard 4 , arithmetics in terms of thousands.
standard 5 , arithmetics in terms of puluh thousands.
standard 6,  arithmetics in terms of hundred thousands to millions.

tapi perlu ke kita nak push bebudak ni buat speed mental arithmetic?

Adakah superior mental arithmetic needed for one to go to high school?

Aku focus kat point “push” tu.

Sebab, mental arithmetics will be an advantage to speeden things up. But merely an advantage.
Bila things speed up, basically easier for you to analysse the problem.
Sebab we need to teach the kids later to be able to “apply” that calculation.

but malaysia tak perlu seorang human-calculator.

Because by Form 1, the kids will taught Euclidean geometry (yang duk belajar sudut2 dan garis parallel tu la).
The emphasis is no longer on arithmetic.
Masa 12 and 24 jam kinda stuff is also taught.
The PMR (masa aku SRP da tukar kepada PMR)..the PMR syllabus..what is the emphasis of PMR syllabus?

Can I say that speed mental arithmetics is similar to teaching kids to write more beautiful?
Semua orang boleh menulis.
Tapi ada yang menulis cantik gila, ada yang menulis hodoh gila.
Tapi terlalu emphasize kan kepada “teknik menulis cantik”..ajar budak pegang pensil at 45 degrees..ada certain kedudukan jari…

The thing is, if the kid is “good enough” in writing and calculating in UPSR, he still have a good chance in PMR.
but the kid yang “excel” dalam writing and calculating in UPSR, not sure yet dia even boleh lulus PMR.

Kau kena faham concept “Matthew effect” to understand this.

http://en.wikipedia.org/wiki/Matthew_effect_%28education%29

http://en.wikipedia.org/wiki/Matthew_effect_%28sociology%29

It’s best to read the book “Outliers” by Malcolm Gladwell to understand this concept.

the thing is mental arithmetic skill is an advantage to increase your chances to understand further topics at later stage.
increase your chance.
bukan 100% probability.

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Something is wrong.
It’s the Malays perception of academic excellence.
Malays should focus on the kids to “think” rather than hafal.

dan academic excellence yang based on “thinking” lebih berguna dari academic excellence yang based on “hafal”.

syllabus tak salah.

all the syllabus in the Malaysian schools semua ada meaning, if you think in depth you will find it.

To be able to kaitkan certain dates with certain accounts, to prove your goddam point on certain issues ialah skill yang akan dapat kalau kau belajar Sejarah dengan betul.

Maths dan Add Maths in SPM is basically a “pengenalan” to the actual mathematical concept.

Same with Physics, Chems and Bio. kau kena pergi university degree level..and then look back baru kau akan faham.

syllabus tak salah.
how you perceive it might be wrong.

Having advanced mental arithmetic skill is like having good IQ.
but having these advantages only increase the chance for later understanding of more difficult complex.

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Lastly, ada story yang aku nak share.

Story ni melibatkan jiran Chinese aku.

Ada satu kali, aku dan ayah tiri jalan2 sekali ternampak jiran ni dengan anak dia.

Pak Tiri : Wah you punya anak tak sekolah ka? Saya selalu nampak waktu pagi?

Jiran : Dia tak berapa pandai baca buku. So I suruh dia tolong2 dekat kedai motor I..boleh tolong2 sikit. kalau dia sudah pandai repair motor, at least dia tahu macam mana mahu cari makan..

Aku rasa kena tahu apa absolutely reason hantar budak pergi sekolah dan belajar.
(ya, selain dari diwajibkan oleh kerajaan la)

Nak berjaya bukan satu cara sahaja.

Tapi lagi satu hal.

Budak melayu kalau tak pergi sekolah, best assured dia memonteng melepak.

Budak Chinese?

Aku antara yang suka baca blog kak di (comatosewithbraindamage)..dia pernah cerita pasal saudara dia di Singapore.

Masa time cuti sekolah, saudara dia (masa tu umur dia 12 tahun) buat kad2 yang kiut lepas tu jual kat orang ramai.

In all, dari case KSD, Adi Putra, anak jiran Cina, ngan kak di ni, aku rasa masyarakat kita masih keliru apa meaning of education and academic.

Entah. Just my point of view.
huhu.

But then,
mcm dalam Ayat 12, Surah Al Hujurat, tak payah lah aku nak sibuk2 sangat cari fault orang lain dan masyarakat.
At least aku nanti kalau da kawin dan ada anak (hahaha mimpi siang!)
Agak2 anak aku tu blur2 dalam belajar ni..aku suh dia wat business ke apa..huhu!