is algebra a spoilsport in mathematics? up the wrong tree? little the guy knows that i have just...

Is Algebraa spoilsport in mathematics?

Lincoln, 17 January 2018

Evgeny Khukhro

Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 1 / 67

Heads and legsThe stage is set: a small railway station in the middle ofRussia, 35 years ago. We, I and my wife NadiaChuzhanova, are waiting for a train on the platform.

A somewhat disheveled, small man approaches:

GUY: “Here is a maths problem for you:”

A farmer keeps chickens and pigs.Altogether, there are 100 heads and 288 legs.How many of each kind?

GUY: “But can you solve it without X s and Y s”?

...That is, without algebra.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 2 / 67

Barking up the wrong tree?Little the GUY knows that I have just obtained by PhDin algebra.

Furthermore, at that time, I participated in organizingmaths competitions – maths olympiads – as well as hadbeen taking part in these when I was at school.

I knew this very problem since I was 12 – and we did notlearn using ‘X s and Y s’ until being 13.

So the GUY had no chance of embarrassing me.

Here is an ‘algebra-free’ solution that I produced to hisapparent disappointment.


Heads and legs: mental arithmeticA farmer keeps chickens and pigs. Altogether, there are100 heads and 288 legs. How many of each kind?

SOLUTION 1: Assume that every animal has 4 legs(“add two additional legs to each chicken”).

Then there will be 4× 100 = 400 legs.

Hence we added 400− 288 = 112 extra legs.

Each chicken got 2 extra legs, so there are 112 : 2 = 56chickens.

(And 100− 56 = 44 pigs.)


Heads and legs: ‘with X s and Y s’A farmer keeps chickens and pigs. Altogether, there are100 heads and 288 legs. How many of each kind?

SOLUTION 2:

Let X be the number of pigs, Y number of chickens.

Then X + Y = 100 and 4X + 2Y = 288.

From the first equation, Y = 100− X ;

substitute into the second: 4X + 2(100− X ) = 288.

Expand, collect terms, balance, solve:

4X + 200− 2X = 288; 2X = 288− 200 = 88;

Answer: X = 88 : 2 = 44.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 5 / 67

Heads and legs: algebra

Algebra in SOLUTION 2 means using letters, symbols forunknowns, solving equations almost mechanically.

With matrices, the simultaneous equations are writteneven more efficiently:[

1 14 2

]·[XY

]=

[100288

];

solve:[XY

]=

[1 14 2

]−1

·[

100288

]=

[−1 0.5

2 −0.5

]·[

100288

]=

[4456

].


Less fun?

Using X and Y made a difficult problem easy.


Algebra at school

When I was at school, we had “Arithmetic” textbookuntil year 6 (age 13). Somewhat artificially kept in thedark about using unknown variables.

Experiments with curriculum in 1970s showed thatalgebra, variables, cannot be introduced too early.


Maths textbooks in Russia in 1960s

“Arithmetic” for years 5–6(age 12–13)

“Algebra” for years 6–8(age 13–15)


Solving equations is a skill that most school childrenmaster successfully.

Example (in our diagnostic test)Alloy A contains 40% gold, alloy B contains 25% gold.How much of each of these alloys are to be meltedtogether to produce 600 grams of alloy containing 30%gold?

(Similar to “Heads and legs”...but not all students find it easy!)

Perhaps there should be more focus on translatingproblems into equations, when one has to choose,declare unknowns, variables.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 10 / 67

ExampleA motorboat on a river travels upstream from A to B in4 hours, and downstream from B to A in 3 hours. Howlong does it take for a raft to travel from B to A?

Seems to be too many unknowns!distance, speed of boat, speed of river current...

Introduce: d = distance from A to B ;

b = speed of the boat w.r.t. water;

w = speed of the river flow.

Translate the hypotheses:d

b − w= 4 and

d

b + w= 3.

Solve excluding b, find d/w .....Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 11 / 67

Mixing rocket fuelsWhen I was 12, I was thrilled to put algebra to work forcalculating the right mix of two or three types ofhome-made gunpowder (for home-made fireworks).

Of course, balancing chemical reactions also makes useof this type of algebra.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 12 / 67

Digression: more talking at the station...After initial disappointment of failing to impress orembarrass us with the heads–legs problem, otherscientific topics were discussed. The GUY turned out tobe a school teacher of physics and maths in a smallvillage near Khabarovsk, in the Far East of Russia, wherehe was going by train taking about 6 days.


Digression: more talking at the stationGUY: found nonsense in physics textbooks:“temperature is a measure of warmth”, while in fact,he ‘discovered’, it is measure of speed of molecules.

Other scientific ‘discoveries’ were written in a notebook,which was his only luggage, apart from galoshes in anet-bag – surprisingly little for such a long journey.


Digression: “Kvant” magazineThe GUY told us that he sent his discoveries to theAcademy of Sciences and to the physics-maths magazinefor school children “Kvant”. Apparently, he did notreceive proper recognition...


Digression: theory of everything

Finally he explained to me and my wife his

THEORY OF EVERYTHING: all things are governed bystruggle for existence, extending his version ofDarwinism to the Universe, nature, society, etc.

His final, incontrovertible argument in support of thistheory struck us speechless, in the form of hisrhetorical question: “You beat your wife, don’t you?!”


Geometry at schoolMy geometry textbooks, albeit already ‘closer topractice’ than previous ones, which were closer to Euclid,still contained quite a few of so-called ‘syntheticgeometry’ – as opposed to coordinate geometry.

Nowadays, synthetic geometry all but disappeared fromschool curriculum, often replaced with calculations incoordinates and applications of Pythagoras or cosinetheorems.

Remarkably, many students have no recollectionof a proof of the Pythagoras theorem!

(although a proof is part of Key Stage 3 curriculum)


One geometric example

TheoremThe bisector AL in a triangle ABC divides the side BCproportionally to the lengths of AC and AB :

BL

CL=

AB

AC.


‘Synthetic’ proof:

Construct a parallel line to AC through B .

Extend AL to intersection M . Then

∠MAB = ∠MAC = ∠AMB and ∠CBM = ∠BCA.

Hence, AB = BM and 4ACL is similar to 4MBL.

ThenBL

CL=

MB

AC=

AB

AC, as required.


‘Algebraic’ proof:Introducecoordinatesas on the picture,u, v ,w are ‘known’,while x is unknown,

aiming at x =w

w +√u2 + v 2

.

The condition that AL is a bisector translates as−→AB ·

−→AL

|AB | · |AL|=

−→AC ·

−→AL

|AC | · |AL|(equates cosines),

giving an equation on x .


Calculation is tedious but elementary and produces therequired result:

u(w + x(u − v)) + v · xv√u2 + v 2

=w(w + x(u − v)) + 0 · xv

w.

Actually straightforward, methods works, ‘problem solvesitself’.


Advent of algebra in geometry

Nicole Oresme, Tractatus de configurationibusqualitatum et motuum, manuscript of 14th century,abridged version published by end of 15th century.

Pierre de Fermat, Ad locos planos et solidos isagoge,manuscript of 1636.

Rene Descartes, La Geometrie, published in 1637,Latin transl. 1647.


Advent of algebra in geometry

...What we now know as Cartesian coordinates.

Descartes anecdotally boasted “I solved all problems!”

Change of paradigm:

before, geometry was used as basis for proofs;

now algebra became basis for proofs(both for calculus, and for geometry).

May be ‘boring’, but a successful method, became basisfor development of calculus by Newton and Leibnitz.


Constructions with compass andstraightedge

straight line through two known points;

circle centred at known pointof known radius (defined by two known points);

intersections of these.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 24 / 67

Algebraic interpretationChoose a segment ‘of length 1’:distances ↔ numbers.Compass and straightedge can construct perpendiculars,parallel lines, copy given segments. Then arithmeticoperations can be performed:


Reduction to algebraic equationsThus all possible construction problems can be solved:calculate requisite lengths by solving equations, thenperform those arithmetic operations by compass andstraightedge.No fun at all...E.g.: including difficult Apollonius circles...


ImpossibilityThree famous construction problems,by using only compass and straightedge:

– squaring a circle

– doubling a cube

– trisecting an angle.

Since Ancient Greece, geometers tried in vain to comeup with a solution.

Only in 19th century, algebra gave an answer: all threeconstructions are impossible in general.

Proof is based on coordinate interpretationand properties of algebraic field extensions.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 27 / 67

Only quadratic and linear equationsGiven coordinates of known points, equations arewell-known:

Straight line through (a, b) and (c , d)

(y − b)(c − a) = (x − a)(d − b).

Circle of radius r centred at (u, v)

(x − u)2 + (y − v)2 = r 2.

Intersection points are defined by simultaneous equations.

Solutions are always given by quadratic (or linear)equations in x and y , with coefficients composed ofknown coordinates.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 28 / 67

Field extensions by constructible pointsEvery new point constructed has coordinates in aquadratic extension of the field generated by previouspoints:

Q ⊂ Q(r1) ⊂ Q(r1, r2) ⊂ Q(r1, r2, r3) ⊂ · · · ⊂ F ,

where each extension has degree 2 (=dimension as avector space over preceding field).

Tower Theorem: the resulting dimension is the productof these dimensions.

Hence, whatever constructions we perform, we can onlybuild an extension F of dimension 2n over Q.


Impossibility of doubling a cubeLet given cube have side length 1, so volume is 1.

The side of a cube of volume 2 is 3√

2.

Adjoining 3√

2 to Q generates an extension of degree 3(since x3 − 2 is an irreducible polynomial over Q).

If 3√

2 was constructed in

Q ⊂ Q(r1) ⊂ Q(r1, r2) ⊂ Q(r1, r2, r3) ⊂ · · · ⊂ F ,

then by Tower Theorem applied to

Q ⊂ Q( 3√

2) ⊂ F

we would have dimension of F over Q divisible by 3.

But it is 2n, a contradiction.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 30 / 67

Ultimate spoilsport: killing the problem

Similarly for trisecting an angle: this would mean solvinga certain cubic equation (for cosine), again adjoining anelement of degree 3, impossible, as we saw.

For squaring a circle (of radius 1), one would need toconstruct

√π, which is not even a root of any

polynomial (a more difficult theorem).


Algebraization

Hard to believe that

vector spaces, matrices,

let alone other algebraic systems such as groups,

only appeared relatively recently, in 18–19 centuries.

Modern concept of a vector space

was definitively formulated by Peano as late as in 1888.


Theorem (Euler’s rotation theorem)Any displacement of a rigid body (sphere) in 3D with afixed point (centre) is a single rotation about some axis.

GEOMETRIC PROOF: first we use a reflection in aplane through O to superpose R1(X ) = X ′.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 33 / 67

Then both R1(Y ) and Y ′ must be on the great circleperpendicular to OX ′:

Therefore R1(Y ) and Y ′ are symmetric w.r.t. the planethrough O, X ′ and midpoint M .

Reflection R2 in this plane superposes R2(R1(Y )) = Y ′

(leaving R1(X ) = X ′ unchanged).


Then R2(R1(Z )) = Z ′ automatically:

(if orientation is preserved).


Thus, our displacement is achieved by two mirrorreflections (if orientation is preserved).

Finally, two consecutive reflections in the planes is thesame as a rotation about the intersection line of theplanes, through the angle double the angle between theplanes:


Algebraic proof of Euler’s rotation theoremIsometry of a sphere extends to a isometric(=orthogonal) linear transformation of the whole space.

Equivalent matrix form: ~v 7→ T~v , where ~v =

xyz

, and

T is a 3× 3 orthogonal matrix.

Need a fixed axis: ~v 6= ~0 such that T~v = ~v .

Rewrite as (T − I)~v = ~0, where I =

1 0 00 1 00 0 1

is the identity matrix.Nontrivial solution ⇔ det(T − I) = 0.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 37 / 67

Polynomial χ(λ) = det(T − λI); roots = eigenvalues.

Degree 3 with real coeff. ⇒ at least one real root, whilecomplex roots are in conjugate pairs β, β.

For any eigenvalue α there is an eigenvector ~u 6= ~0such that T ~u = α~u.

Isometry: |~u| = |T ~u| = |α| · |~u|; hence |α| = 1.

Preserves orientation when detT > 0.

detT = product of eigenvalues. Complex β · β = 1.

Then real eigenvalue is 1, as required.

(Complex eigenvalues cos θ ± i sin θ determine therotation angle θ.)

If all eigenvalues are real, then −1,−1, 1 or 1, 1, 1.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 38 / 67

Power of linear algebra

Algebraic calculation produces new coordinates where

the matrix is T =

1 0 00 cos θ − sin θ0 sin θ cos θ

.

Less reliance on ‘geometric imagination’.

Thrill of seeing how algebra works for ‘real-life’,‘physical’ problem.

Works just as well in dimension 4, or 5, etc.


Algebra for differential equations

differentiation as an operator:df

dx= D(f ).

Differential operators are manipulated by algebraic rules:

e.g. multiplied:d2f

dx2= D2(f ).

Example:d2f

dx2+ 3

df

dx+ 2f = sin x takes the form

D2(f ) + 3D(f ) + 2f = sin x or (D2 + 3D + 2)f = sin x .

Use D2 + 3D + 2 = (D + 1)(D + 2).....


Digression: George Boole (1815–1864)

A famous mathematicianborn and bred in Lincoln,and later a professor inCork, Ireland.

Self-educated, great man,local and global hero.


Boole’s work on differential equationsBoole extended algebraic ‘D-methods’; was the first toapply decompositions into algebraic partial fractions withoperators.

Boole’s paper on this subject, “On a general method inanalysis” was published in Philosophical Transactions ofthe Royal Society in 1844

(while he lived in Lincoln, at 29 years of age, running aboarding school).

For this paper he was awardedthe Royal Medal of the RoyalSociety.


Boole’s algebraic approach

Probably, algebraic approach to differential equationsprompted Boole’s ideas of algebraic manipulations withsymbols in Logic, for which Boole is most famous.


Algebra of logic

Logic is part of human mind, reasoning.

Aristotle (384–322 BC)

Early theoretical, formalapproach: Aristotle’s logic.

Syllogisms:

Every man is mortal;

Socrates is a man;

hence Socrates is mortal.


Other formal logic systems proposed by

G. W. Leibniz

(1646–1716)

Richard Whately

(1787–1863)

Augustus

De Morgan

(1787–1863)

But none of the forerunners went as far as George Boolein making logic algebraic.


Boole’s books on logic

Mathematical Analysis of Logic,

1847, written in Lincoln

Investigation of Laws of Thought,

1854, written in Cork


Boole’s algebra of logic

Universe of discourse is denoted by 1.

Logical statement = ‘elective symbol’, x , y , ...Convention: x elects elements X , etc.

Product xy = composition of x and y .

So xy elects elements satisfying both x and y .


Classes

Later Boole regarded x as a class (= (sub)set of allelements of the ‘universe’ for which x is true).

Then xy is the intersection of the classes, where both xand y are true.

Empty set is 0.

Negation: 1− x is the class of all non-X .

In particular, x(1− x) = 0, ‘principle of contradiction’.

Addition: x + y , union of x and y , where either xor y is true (by Boole only for disjoint classes).

x + (1− x) = 1 the law of excluded middle.


Inclusion (implication)To express that x is contained in y

(that is, x implies y),

in modern notation: x ⊆ y , or x ⇒ y .

But Boole prefers equations:

x(1− y) = 0, (of course the same as x ⊆ y).


Boole’s algebra for a syllogismAll Y s (men) are Z s (mortal);all Ss (Socrates) are Y s (men);hence all Ss (Socrates) are Z s (is mortal).

In terms of corresponding ‘elective symbols’, or classes:y ⊆ z and s ⊆ y ; need s ⊆ z . Translate a la Boole:

y(1− z) = 0; and s(1− y) = 0;

need to show that s(1− z) = 0.

We have s = s × 1 = s(y + (1− y)

)= sy + s(1− y)

Substitute into s(1− z) =(sy + s(1− y)

)(1− z)

= sy(1− z) + s(1− y)(1− z) = s× 0 + 0× (1− z) = 0,

as required: s(1− z) = 0, means exactly s ⊆ z .Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 50 / 67

Algebra of logic

Boole’s calculations of logical equations later becameobsolete, forgotten, no longer used.

But his algebraic approach really signified a paradigmalshift, was a true start to development of symbolic logic.

... which later became a basis for development of logicaloperations in computers, both in software and hardware(‘Boolen gates’).

One might say, algebra colludes with machines to replacehumans! :-)


Algebra and abstractionAt the beginning of algebra: variables:

2 · 3− 3 = 3(2− 1) a · b − b = b(a − 1)

Equations: ax2 + bx + c = 0; x1,2 =−b ±

√b2 − 4ac

2a.

Algebraic theories — abstraction of algebraic structuresin various parts of maths.

Algebraic objects hidden in other mathematical theories— skeletons reflecting essential formal rules (often quitesimple!).

mathematics

sciences∼ algebra

mathematicsEvgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 52 / 67

Abstract group theoryAll bijections (=transformations) of a given object,preserving certain properties, form a group.

Transformations of a square (as a rigid figure):

four rotations about centre O:through 0◦, 90◦, 180◦, and 270◦;

four reflections:with axes: vertical, horizontal,and two diagonals.

These eight elements form thegroup of isometries D8 of a square.Evgeny Khukhro (Univ. of Lincoln) Is Algebra a spoilsport in mathematics? 53 / 67

‘Multiplication’in a group of transformations(often called group operation, or group composition):

product gh of two transformations g , h:

performed one after another:


Group is closed under multiplication

Example (Transformations of a square)

Let a = rotation 90◦ anticlockwise,

and b = reflection in the vertical axis.

Then what is ab?

(look where A,B go!): Aa−→ D

b−→ A, so Aab−→ A;

Ba−→ A

b−→ D, so Bab−→ D.

Hence ab must be the reflection in AC .


Other products:

a2 = aa = rotation 180◦,

b2 = bb = Id is identity.

ba = reflection in BD.

So a group can be non-commutative: ab 6= ba.

Here, ba = a3b.


Abstract group D8

Group of isometries of a square

D8 = {e, a, a2, a3︸︷︷︸rotations

, b, ab, a2b, a3b︸︷︷︸reflections

}.

(here e = Id is the identity transformation – whennothing is moved).

b2 = e, a4 = e, ba = a3b .

Abstract group – when we forget the squareand work just with these eight elements using theserelations.


Definition of an abstract group

A group is any set G with an operation called‘multiplication’ such that

g , h ∈ G ⇒ gh ∈ G (closed under multiplication);

(ab)c = a(bc) (associative);

there is identity (“neutral”) element e such thatea = ae = a for all a ∈ G ;

for any g ∈ G there is inverse g−1 such thatgg−1 = g−1g = e.


Associativity

Main example of a group:the group of transformations G (A).

All axioms are satisfied “automatically”, e = Id.

Associativity:


Free application

Associativity holds for compositions of any mappings:

abc = (ab)c = a(bc).

Corollary: Associativity of matrix multiplication

(AB)C = A(BC ) for compatible matrices A,B ,C .

Because matrices ↔ linear mappings

(of multi-dimensional spaces),

and matrix multiplication ↔ composition of mappings.


This is what we actually use solving equations:

AX = B ;

A−1(AX ) = A−1B ;

associativity!

(A−1A)X = A−1B ;

IX = A−1B ;

X = A−1B .


Underlying algebraic structures

We saw even in this lecture how algebra helps in:

‘Heads–Legs’ problems

Differential equations

Orthogonal transformations(Euler’s rotation theorem)

Etc.

Linear algebra:

matrices,

linear systems ofequations,

A~X = ~B ,

eigenvalues(roots of det(A− λI)),

etc.



Permutations of roots ofpolynomials

Symmetries, rotations ingeometry

Rubik’s cube

Elementary particles inphysics

Etc.

Group Theory:

subgroups,

quotients,

multiplicators,

etc.



Integers

Invariant theory

Sets of functions

Matrices

Etc.

Ring Theory:

ideals,

generators,

radicals,

etc.


Thanks to Algebra

Many algebraic structures arise as abstractions fromvarious mathematical objects and theories.

Then payback benefits are enormous: algebra makesthose theories and objects more transparent, providesanswers, techniques, insights.

Algebra is generous: she often gives morethan is asked for.

Jean d’Alembert


Example of recent researchHigher abstraction, limits of finite groups,commutativity-type conditions in recent research:


is algebra a spoilsport in mathematics? up the wrong tree? little the guy knows that i have just...

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