- [Instructor] They're hidden pieces of information
in all of our numbers.
From the quantity you see on your money
to the age on a birthday cake,
but numbers haven't always been this way.
In fact, the numbers that we use today
are the result of thousands of years
of inventing and refining.
Many ancient civilizations independently
came up with forms of basic tallying
where counting was tracked
by lines drawn in the earth.
But once the number gets big enough
a basic tally becomes unmanageable.
Because every time you want to know
how many there are, you have to count them all.
Even when the tallying number is small
it can be hard to distinguish what it is without counting.
Like could you tell me what this number was
just from looking at it?
The natural solution is to separate
the tallies into groups,
but it's at this next stage
of the evolution of number systems
that different cultures started to diverge.
While the modern tally uses
one of the popular options groups of five,
reflecting the numbers of fingers on a hand
different cultures use groupings
that were as small as three,
and the Babylonians used groupings as large as 60.
The counting system became even more different over time
as different cultures try to improve their systems.
Generally the next step was to create
some kind of shorthand symbol,
like Roman numerals which wrote five tallies as a V
and 10 tallies as an X.
The Greek civilizations had a similar idea
and created separate symbols
to represent different numbers.
Creating a math alphabet was a major accomplishment
but while these ancient systems
required remembering more and more symbols,
as you wanted to use bigger and bigger numbers,
the modern system manages to write
every positive whole number with just 10 symbols.
So these old systems were really
just more sophisticated tally systems
because you still have to add together all the symbols
to figure out what number was being represented.
Take this number for example,
even if you were familiar with the Roman numerals
it would take you a bit of time
to add all the symbols together
to figure out what number this was.
But if I write the same number in our modern notation
you can tell at a glance.
So what is the modern number system
and where did it come from?
This system is called place notation
and the first instance
of a fully developed version of it,
was by the Mayan civilization around 2000 years ago.
Well, in the tally system there's a convention
of writing the ungrouped lines at the end.
The number is still the same
if you switch around where they're placed,
but this is not the case with our modern number system.
The assumption that we make when we see the number 42
is learn at such an early age
that it can be easy to forget that it's there at all.
As we go from right to left
each digit represents the multiple of a factor of 10.
So 42 is really four times 10 plus two times one
and 42,199 is really four times 10,000
plus two times 1000, plus one times 100,
plus nine times 10, plus nine times one.
Another way to say this
is that we have four in the 10 thousands place,
two in the thousands place,
one in the hundreds place and so on.
So not only do we look at a number
and assume that it uses place notation
which historically hasn't always been the case,
but we specifically assume
that each place in the number
is associated with a factor of 10.
Now, as you'll remember
from the earliest days of tallying
there's no base number that our system
inherently needs to be centered around
and historically different cultures
have used different numbers.
In fact, the original Mayan system
used 20 as its base number.
So if the two major assumptions with our modern numbers
are the use of positional notation,
and the use of base 10, is it possible
that there's a better number system out there?
Like our ancestors before us
might we one day discover a new and better one?
Well, there are actually already non base-10 systems
being used all the time in the modern world.
For example, the way we measure time with 60 minutes
in an hour and 60 seconds in a minute,
or the fact that there are 360 degrees in a circle.
These are carried over
from the ancient Babylonian base-60 system.
On a larger scale classical computers
are completely based on a base-two system binary.
This means that every number
can be written using only two symbols, zero and one.
Classical computers have a physical reason
for using binary because fundamentally
computers are just a bunch of on-off switches.
If the computer reads out these on off switches
that translates to the binary number below it,
but humans don't have a similar restriction.
So what makes a number system good or bad for humans?
We've already learnt a few lessons
from the historical number systems
that are no longer in use.
We stopped using tallies because we wanted a system
that lets us know what the number is with a quick glance.
This requires creating a mathematical alphabet,
the set of symbols with a widely agreed upon meaning
but it's also possible to have too many symbols.
So we're looking for a balance.
We want one that's easy to do operations
like multiplication and addition on.
So what about binary?
It's got a nice, simple mathematical alphabet
with just two symbols,
which means it's multiplication
and addition tables are very easy to memorize.
But the downfall of binary for human use
is that the numbers get long very quickly.
The biggest four digit number is one, one, one, one
which is only 15 in decimal.
Even just 500 decimal is one, one, one, one,
one, zero, one, zero, zero in binary.
So like the tally system
the binary system is hard to read at a glance.
Since the decimal system already works
pretty well in terms of our three rules,
let's look at a couple of bases close to 10.
In a couple of his popular math book,
Isaac Asimov discusses the benefits
of the octal system, which is based on eight
and the duo decimal system, which is based in 12.
While we can use the first eight symbols
of the decimal system to write numbers in base-eight,
the base-12 system requires two additional symbols
which will borrow from the English alphabet.
Take a look at the size of the multiplication table
and the largest four digit number for each system.
If you were going to remake our number system
would you trade having numbers
that tend to have some extra digits
in exchange for having an easier times table to memorize?
If you're thinking you'd rather just stick
with the decimal system consider this,
the easiest times tables to remember
are the ones that are a factor of the base number.
Decimal being base-10 has easy two
and five times tables
because when you divide by those numbers,
you get an integer.
In other words, two and five are factors of 10.
This means that their multiplication tables
always follow a regular pattern.
Despite having a smaller times table
an octal system base-eight
also has two factors, four and two.
But unlike with decimal, these numbers allow you
to do simple integer halving of the base.
Half of eight is four, half of four is two
and half of two is one, whereas half of 10 is five
but then half of five is an innate whole number.
What about the duodecimal system, base-12?
Well, it has the best of both worlds
way more factors and better simple halving
than the decimal system.
The factors of 12, are two, three, four, and six.
So each of these follow an easy to remember pattern
in the times table.
And if you're worried about running out of fingers
just get a little creative.
All of this is to say
that while the decimal system
gets the job done really well,
even minor changes like using a slightly larger base
has the potential to make basic arithmetic much faster.
That said, it seems unlikely
that we'd go through all the trouble
of completely changing our number system,
unless there was some kind of new discovery
that fundamentally changed the way we do math.
But just like negative numbers
weren't always a part of everyday life
maybe sometime in the future
some new technology is developed
which makes imaginary numbers part of everyday life.
This would call for a number system
that easily incorporates imaginary numbers.
Number systems like this actually already exist.
One version is called quarter-imaginary base
and uses the imaginary number 2i as it's base.
It can represent almost every complex number
using only these four digits.
Another very unique number system
represents numbers as tangled knots.
These allow you to do mathematical operations
in a more visual and geometric way.
Decimal is so useful
that it allows children to do calculations
that even the best mathematicians
would have had a hard time with before,
because they weren't using
sophisticated enough number systems.
And just like it would have been difficult
for them to imagine the number system that we use today,
it's possible that sometime in the future
humans might use a totally different number system
that'll allow them to do bigger and better things.
You know, all this talk of numbers
has got me wondering, different civilizations
came up with different ways of representing numbers
but they all still perform the same mathematics?
One plus one still equals two,
independent of what system you're working with.
So on one hand, it seems like numbers
were a creation of humans
as different civilizations came up with different numbers.
But on the other hand,
all of these different representations
represent the same ideas.
So is mathematics invented or discovered?
This is an age-old question
and there are good answers to both sides.
That's why I'm making a video about it
over on Nebula.
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Thank you for watching
and I'll see you in the next episode.
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