# The story of Algebra and its development

once upon a time there was a king he had
a mathematician in his court this clever mathematician invented the game of chess
and presented it to the king the king was delighted and asked the
mathematician to name a reward any reward nothing was too big
the mathematician looked at the chessboard consisting of 64 squares and
asked for some rice the King looked perplexed rice what kind of gift is that
the matter of addition then added that the rice needed to be given to him
according as this condition one grain of rice on the first square two grains of
rice in the second square four grains in the third square eight in the default and
so on until the last square the King was a bit offended at the simple request but
he told the mathematician if this is your wish I will grant that will the
King be able to grant this request welcome to another episode of the Max
Factor where we’re going to explore the intricacies of algebra this is no
textbook journey we’ll tell you stories riddles and real-life problems to bring
the subject life keep watching to see algebra like never
before back to a king and his mathematician he starts doling out the
rice according to his condition and so in the first day the mathematician got
one grain he got two and the second foreign the third and the king look
amused the stupidity of the seemingly brilliant mathematician if we progress
along the same lines in the sixth day the mathematician will get 32 grains of
rice on the 8th day he will get 128 grains on the 16th day he will get
32,768 cranes which is about 1 kg of rice but soon the numbers started
increasing and the King started getting restless soon the king realized that he
would need to resigned and as promised he just would not have enough rice to
algebra he clearly would not have been in the situation he was in
let’s see what he should have been able to work out the first square is 1 the
second to the third 2 into 2 which is 4 the fourth square 2 into 2 into 2 or 2
cube or 8 so the number of grains in the 64th square is 2 to the power of 63
which is wow that’s really large isn’t it well
the story is a sad ending not only was the King unable to keep his promise he
was so annoyed with her mathematician that he had him be had it much like the
King most of us don’t get the point of algebra do we it’s something we plowed
through in school and discard at the first point that we can but we could
shift perspective and look at algebra as a mystery as a search for the unknown X let’s put this in a more real context
here is chthonic carrying a bunch of balloons
he accidentally lets go of 3 and is left with 5 balloons so how many balloons did
done I have to start off with let’s call that number
the mysterious Hanks we know that when three balloons were taken away we’re
left with 5 balloons now if we work the equation out we get X is equal to 5 plus
3 which is equal to 8 so we know that Donna has 8 balloons besides helping Taniya figure out how
many balloons he has algebra powers much of our lives starting from working out
quantities in the kitchen to converting measurements from one system to another
many fundamental principles and physics can only be understood using the tools
in algebra modern algebra is applied in and propels almost every branch of
mathematics it is – mathematics what mathematics is – science now a lot of algebra originated from the
Middle East from Persia to be precise and one mathematician is central to this
story his name is Mahmoud Eden Musa al-khwarizmi he was born in 780 ad in
quarrelsome a small town in present-day as Pakistan he and his family then moved
to badonk at that time Bogdan was under the patronage of the
opposite khalif’s the khalif’s for three generations had put together a
magnificent scientific academy called the house of wisdom at bergdahl much
learning from all over the world was centred here through this time
al-khwarizmi flourished in this atmosphere he was a scholar the House of
Wisdom and was later pointed that khalif’s
librarian he wrote a famous book called Hossam al-jabbar
while macabre in 825 ad the word algebra comes from algebra
noir yzma’s books came to form the foundation of algebra and actually
established it as a separate discipline in fact they constituted university
textbooks till the 16th century since the books were not footnoted it is tough
to trace the sources of his information though he was clearly influenced by both
Indian and Creek thought now the title of his book teaches us algebra right
away let’s look at that a little more closely al-khwarizmi book was called
hisab algebra well McCallum in Arabic jabber is an algebraic operation of
restoration which is the act of subtracting terms on either side of the
equation take the equation X minus 2 is equal to 12 now we take the 2 from the
left hand side and restore it to the right hand side by making X is equal to
12 plus 2 is equal to 14 the other word in the title Macabre means balancing and
is the act of cancelling like the terms on opposite sides of the equation let’s
take this one X plus y is equal to y plus 7 now we get X is equal to 7 by
canceling a balancing the two sides of the equation
eventually the Macabre was left behind and this type of math came to be known
as algebra in many languages in fact the entire title has been translated to mean
the book of restoration and balancing on a journey through algebra we were
exploring al-khwarizmi his famous book that translates to the book of
restoration and balancing now the very same idea of restoration and canceling
comes up and we was Carroll classic Alice in Wonderland and this is not as
curious as it first appears syn Lewis Carroll who was in reality Charles
Dodgson was actually a mathematician at Christchurch College Oxford let’s enter
Wonderland and see what actually happens to Alice and more importantly what that
had to do with mathematics Alice plunges down a rabbit hole and
this kick starts her adventure she meets an advice-giving caterpillar who is
beside a large mushroom and is smoking a long hookah at this point of the story
Alice is already experimented with proportionality and is eaten a cake that
made her grow bigger and has drunk the liquid that made her grow smaller the caterpillar then gives her some
additional information one side will make you grow taller the other side will
make you grow shorter one side have walked the other side of what thought
Alice to herself of the mushrooms that the caterpillar just as if she’d asked
it aloud and another moment it was out of sight
Alice clearly a risk-taking young woman samples the mushroom however when she
eats to one side she discovers that growing taller translates to a neck
being stretched and growing shorter translates to a torso shrink she
concludes that in order to regain her proper size and proportions she must
ensure that she eats exactly the right balance of each side of the mushroom so
what does the sequence psychedelic as it seems have to do with algebra now if we
go back to our discussion of the title of al-khwarizmi s book it translates to
restoration and reduction cattle experiments with alice’s
reduction which is reducing a size to get it through the door and restoration
as she seeks to find something to restore her to her original size
now many mathematicians believe that Carroll’s writing was an exploration of
new modern mathematics that were emerging in the mid 19th century now
enough of Alice and are absurd adventures let’s head back to the
evolution of algebra in the Middle East one of the areas where algebra came in
very useful was Islamic inheritance law let’s try and see why consider this
problem here is a man who had two sons and two daughters then he dies and
leaves all his money to his two sons and two daughters his wife is already dead
the root of the law is that each son gets twice as much as the daughter but
the sons have to get the same amount as each other now if he has 24 thousand
gold coins how much will each of them get now let’s use algebra to try and
solve the problem if each georgia inherits X coins then each son inherits
2x with two sons and two daughters our equation will read 2x plus 2x plus X
plus X is equal to 24,000 6x is equal to 24,000 implies that X is equal to 4000
which means that each daughter gets 4,000 and each son 8,000 a bit Jen
devised but it does show us how useful algebra can be we have traveled to the Middle East to
see the contributions of al-khwarizmi now let’s travel back in time to India
in the seventh century to explore the mathematics of Brahma Gupta an Indian
scholar he was from the state of Rajasthan of Northwest India since he is
from Polamalu he is often referred to as Paloma Acharya now Brahma Gupta wrote
many important works focusing on mathematics in astronomy in fact he
became the head of the astronomical observatory at Aegean in central India Brahmagupta introduced extremely
influential concepts to basic mathematics including the use of zero in
mathematical calculations and the use of mathematics in describing in predicting
astronomical events much of what he worked on was not even thought about in
the West until a thousand years later mattias work was composed in elliptic
verse he once said as the Sun eclipses the Stars by his brilliance so the man
of knowledge will eclipse the fame of others in assemblies of people if he
proposes algebraic problems and still more if he solves them a lot of this
knowledge travelled from India to Rabia and influenced the thinking of scholars
like Al charisma on our journey through algebra we’ve
explored the work of Brahma Gupta and al-khwarizmi we next move to look at da
Fantas a famous mathematician from the third century now we know pretty little
about Diophantus his life except his age which is couched in the form of a riddle
sans intriguing let’s check out the riddle quite the riddle let’s take it a
line at a time and try and decipher this to do that let’s presume diophantus as
age is X now God gave him as boyhood 1/6 of his life means that his youth lasted
1/6 of his life 1/12 more is youth while whiskers grew ripe means he grew a beard
after 1/12 more and then yet 1/7 ere marriage began tells us that after 1/7
more of his life he married in five years there came a bouncing new Sun
which is quite clear in itself alas the dear child a master and sage after
attaining half the measure of his father’s life chill fate took him
sorrowful he tells us that the son lived only half as long as his father after
consoling his faith but the signs of numbers for 4 years
he ended his life and we realize the Diophantus died four years after his son
now we can use this information to craft this equation and if we keep simplifying this like
this we reach the conclusion the Diophantus lived to the age of 84 while
his son only lived to be 42 now let’s move to an area of algebra that sounds
is very simple one where we have a square of the unknown which means we’re
working with an x square not just an X the Latin word quadratus means square so
a general quadratic equation is ax square plus BX plus C is equal to 0
where a is not equal to 0 now the solution with this is pretty complicated
this is the answer let’s take a simple example and try and work out from a
comprehensible way of looking at it take the equation X square plus 6x is equal
to 27 let’s try and solve it if X is equal to
1 then 1 plus 6 is not equal to 27 so 1 is not the solution if X is 2 then 4
plus 12 which is not equal to 27 so 2 is not the solution we could keep going
like this but let’s look at it differently think of the first term as the area of a
square with side X now the 6x is the area of a rectangle with the sides of 6
+ X or we can split it up into two rectangles where one side is X and the
other side is 3 now if we attach the rectangles onto the square will produce
an area which is equivalent to X square plus 6 X now according to open equation
this is equal to 27 quite right 27 now look at the missing corner aren’t
watching to fill it with another square go ahead and do that
this square is a side of 3 and so our square has an area of 9 we need to add
this to both sides now the right hand side is a square with a side measuring X
plus 3 so we can write the equation as X plus 3 whole square is equal to 27 plus
9 which is equal to 36 now the whole complexity of the quadratic has been
ironed out hasn’t it now very simply X plus 3 is equal to 6 which means X is
equal to 3 this solution was actually suggested by al-khwarizmi
and what completes this elegance is that it can be used as a solution for any
quadratic equation I need to add in just one point that al-khwarizmi missed when
we look at the square root of 36 both plus 6 and -6 are possibilities during
aleca resumes time knowledge of negative numbers was not so prevalent so the
other solution to a problem is X plus 3 is equal to minus 6 so X is equal to
minus 9 and that ties in beautifully with a starting formula and explains why
there is a plus and minus option before the square root now what happens when the equations get
higher say a cubic equation it’s of the form X cube plus BX square plus CX Plus
D is equal to zero where a is not equal to zero now let’s look at a concrete
example 2 X cube plus 3 X square plus X plus 1 is equal to 0 how can we solve
this if X is 1 then 2 plus 3 plus 1 plus 1 is not equal to 0 so 1 is not the
solution if X is 2 then 16 plus 12 plus 2 plus 1 which is not equal to 0 so 2 is
not the solution in Europe at this time mathematical
contests and public forums was common mathematicians tried to solve equations
such as these and the winner earned valuable money for these let explore one
such contest in the republic of venice during the sixteenth century between
nicola Fontana Tartaglia and antonia maria Fiore Fiore had learned one way of
solving cubics equations from his teacher curiously enough a few days
before the contest Tartaglia managed to find the general method of solving
another kind of cubic at the contest the problems Tartaglia posed were all of the
second type and so he merged the winner later however both methods were
published together in a book called ARS Magna by Gerolamo Cardano another famous
mathematician let’s fast-forward another 13 years
Cardno had a student called la DeVito Ferrari Ferrari managed to solve the
quartet which equations where X goes up ^ then in a repeat of history Ferrari
Challenge Tartaglia to another public context Tartaglia was extremely
reluctant to dispute with Ferrari still a relatively unknown mathematician but
finally agreed on 10th August 15:48 the contest finally took place by the end of
the first day it was a clear that things were not going to Italia’s way Ferrari
clearly understood the cubic and quartic equations most heartily and won the
contest after all this the solution to cubic
equations is usually known as cardano’s formula Porte Italia died penniless and
unknown clearly algebra needs development have
more than its fair share of trauma I hope you have got to see a little of its
magic as we journey through Arabia India and Greece solving riddles and problems
for more fun mathematics don’t forget to keep watching

### 3 thoughts on “The story of Algebra and its development”

• June 4, 2019 at 11:28 pm

Amazing presentation.