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.