Hello. I’m Professor Von Schmohawk and welcome to Why U.
This series of lectures is an introduction to Algebra.
But before we discuss Algebra
we should start by taking a closer look at the things we call numbers.
In even the earliest cultures
understanding and communicating quantities has been essential to everyday life.
Anthropologists tell us that even the most primitive stone-age cultures
had some concept of “number”.
However, early number systems were much more limited than today's base-10 number system.
Life was simple back then.
We didn't need names for exact quantities.
If you knew that there was a herd of gazelles nearby
it didn't matter exactly how many gazelles were in the herd
or exactly how many miles away they were.
What was important was that there were a lot of gazelles and they were just over the hill.
Even recently, certain Australian aboriginal tribes counted only to two
with any number larger than two called "much" or "many".
South American Indians along the Amazon had names for numbers up to six
although three was called "two-one", four was "two-two" and so on.
Bushmen of South Africa had a similar way of naming quantities
but stopped at ten because the names became too long.
This tribe would not do "financial" transactions involving numbers greater than two.
For example, they would not trade two cows for four pigs.
Instead, they would trade one cow for two pigs
and then in a second transaction trade another cow for another two pigs.
If you had never heard of numbers and you wanted to describe to someone
exactly how many gazelles you had seen just over the hill
you might use your fingers to represent how many gazelles there were.
Of course this would become difficult if there were more than ten gazelles.
You could cut marks on a stick
or maybe gather a group of pebbles to represent the number of gazelles
but then, this would also become cumbersome if there were a lot of gazelles.
Another option would be to invent different names and symbols for each possible quantity.
This seems like a simple solution but there is still a problem.
Remember that you know nothing about our modern base-10 number system
which uses only ten symbols in different combinations
so you will have to invent a new word and symbol for every possible quantity.
For instance, in my primitive tribe on Cocoloco island
here are the first thirty numbers we use.
zoop
floop
dop
trom
mim
zap
weeny
glumby
bimpy
neechy
frump
wackit
trimble
walaki
kravitz
jolo
ponzo
kolob
krub
wallop
zoomy
mombozo
balleemi
toramoo
fallazit
smip
bazooloo
eekeena
eechiwa
and ZORTAN.
After Zortan we just say "a whole bunch".
There were several problems with this number system.
First of all, since we had over thirty symbols to memorize, math was really difficult.
For instance, if you have “walaki” coconuts
and then you eat “mim” coconuts
how many are left?
The answer is obviously “bimpy” coconuts
but it takes many years of school to memorize all the combinations.
Also, what happens if you have “bazooloo” coconuts
and then someone gives you “trom” more?
Well, then you’d have “a whole bunch of” coconuts.
In the past, some of the mathematicians on Cocoloco Island
actually invented an "advanced" number system
with several thousand names and symbols to handle problems like this.
But as soon as they did
someone always came up with a problem which they did not have a number for
by adding “zoop” to the biggest number.
Then one especially brilliant Cocoloconian mathematician
came up with the idea of combining symbols.
After zortan, the next number would be zortan-zoop
then zortan-floop, and so on.
When you got to zortan-zortan
you then would go to zortan-zortan-zoop
and zortan-zortan-floop.
This worked well but numbers could get quite long.
For instance, our number "one-thousand" in Cocoloconian would be
zortan-zortan-zortan
zortan-zortan-zortan-zortan-zortan-zortan
zortan-zortan-zortan-zortan-zortan
zortan-zortan-zortan-zortan-zortan-zortan
zortan-zortan-zortan-zortan-zortan
zortan-zortan-zortan
zortan-zortan-zortan-zortan
neechy
which we would write as ...
After many years people realized that it wasn’t necessary to write zortan over and over.
The first symbol could just represent the number of zortans.
For instance, the number zortan-zortan-zortan-zoop would be dop-zoop
three zortans plus a zoop.
Zortan-zortan-zortan-zortzan-zoop would be trom-zoop
four zortans plus a zoop.
The first symbol could represent up to zortan zortans, or 900.
This two-symbol system still couldn’t quite get up to 1000
but nobody had that many coconuts anyway.
Then one day a boat arrived from the distant island of Bongopongo.
We were all amazed.
The Bongoponganians had invented a much better system
which could represent really big numbers with much fewer symbols.
Apparently, in early pre-history of Bongopongo they must have counted on their fingers
because they only used ten different symbols
one
two
three
four
five
six
seven
eight
nine
and zero
which they called “fingers” or “digits”.
They counted up to bimpy, which they called “nine”, using a single digit.
After “nine”, they used two digits.
The first digit represented some number of tens
and the second digit would add from nothing to nine to that.
For instance, the number 32 would represent three tens and two ones.
The number 47 would be four tens and seven ones.
Likewise, the number 80 would be eight tens and zero ones.
But here is where their number system was really amazing.
After getting up to 99, you would think that they would have to stop.
But no!
They would just tack on another symbol in front
which would now be used to represent hundreds.
This could take you all the way to 999
which meant nine hundreds
plus nine tens
plus nine ones.
Every time they got to the biggest number they could represent
they would just tack on another digit which represented the next bigger number
and kept going!
This system had some big advantages.
First of all there were only ten different symbols to remember.
Secondly, every time they added a digit
they could represent a number ten times bigger.
So with six digits they could represent almost a million.
That’s a lot of coconuts!
Looking back, it is easy to see the problem with the first Cocoloconian number system.
Each quantity required a new symbol
so when you ran out of symbols, you ran out of numbers.
With thirty symbols, the largest quantity that could be represented was thirty.
The second system allowed these symbols to be combined in a single number
and their values added.
But since each symbol could add no more than thirty to a number
numbers got big really quick.
The third Cocoloconian system could combine two symbols
one of which was multiplied by a factor of thirty.
This allowed larger quantities to be created with only two digits.
However, the largest number was still less than 1000.
On the other hand, the Bongoponganian system used only ten symbols.
However, with just these ten symbols
very large quantities could be represented with just a few digits
since each additional digit in a number
represented a quantity ten times bigger than the previous digit.
This system was simple and efficient
and the quantities that could be represented were unlimited.
That is why this is the number system that the world uses today.