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Hey guys, I recently learned
that the number zero was forbidden for 1,500 years.
At first, I was like, what, how could a number be forbidden?
And then I was like, "Why would a number be forbidden?"
Also, hasn't zero just always been around?
We're so familiar with it today, but it turns out
there was a time when the concept of zero didn't exist.
Humans had to invent it, and when we did
it caused a lot of drama.
How could a number that literally means nothing
cause so much controversy?
This is the story of zero's rise from rags to riches
or you could say from zero to hero.
(bell dings)
The time before zero was before the beginning
of recorded history.
So paleontologists have had to piece together
the birth of mathematics from bits of stone and bone.
They discovered that many ancient civilizations
independently came up with counting in the form of tallying,
lines drawn in the earth or carved into animal bones.
But none of these civilizations had a symbol for zero.
The concept didn't exist, and this kind of makes sense
if you think about it.
Stone Age mathematics stemmed from a need
to keep track of things.
Now we don't know exactly what early humans
were keeping track of,
maybe how many rocks they'd collected
or how many children they'd had,
but we do know they were keeping track of something.
They didn't need to keep track of zero rocks
or zero children.
You either had children or you didn't.
For the average Stone Age human
the concept of zero was unnecessary
so nobody thought to invent a symbol for it.
It also requires the idea that the absence of something
is a thing in itself
which was a pretty advanced idea for our caveman ancestors.
But if not in the Stone Age, when did humans invent zero?
(bell dings)
Fast forward about 2,000 years to the Bronze Age.
It was in the flourishing state of Babylon
or modern day Iraq that zero first entered the world,
but it was slightly different than how we use it today.
To see why
we need to understand the Babylonian counting system.
The Babylonians only used two symbols to represent numbers.
This wedge-looking thing which represented a one
and this arrow-looking thing which represented a 10.
To write different numbers
they would just combine these two symbols.
2 was just two 1 symbols together.
20 was just two 10 symbols together
and 17 was one 10 symbol and seven 1 symbols.
They used a sexagesimal system,
meaning rather than using base 10 like we do today,
they used base 60.
Once they got to 59, they just used the same symbol as 1
to represent 60.
They used positional notation as well,
meaning the position of each symbol
had its own specific meaning.
We also use positional notation.
When we write the number 111,
we know the value of what each different 1 means,
because of its position.
There's a 1 in the ones column, a 1 in the tens column
and a 1 in the hundreds column.
Adding them together gives 111.
Rather than multiples of 10,
the Babylonians used multiples of 60.
So this number would be
3,600 plus 60 plus 1, 3,661.
There was a problem though.
Imagine we have the number 3,601.
So a 1 in the 3,600s column,
nothing in the 60 column and a 1 in the ones column.
This is fine if we always have columns to guide us,
but when it came to writing down numbers
the Babylonians just wrote freehand on clay tablets.
This made it very difficult to tell numbers apart.
Was this 3,601,
3,660 or 61?
Imagine writing 101 like this,
110 like this and 11 like this.
That's basically what the Babylonians were working with
and just like we use a zero to distinguish between numbers
so too did they.
By around 300 BC,
the Babylonians invented this symbol
to represent the empty column.
It stood as a placeholder
which made it easy to tell the difference between numbers.
zero was born into the world,
but it wasn't quite like we used it today.
It was just a symbol that meant there is nothing there.
It wasn't a number.
It had no place on the number line,
and it wasn't used in any mathematical operations,
like addition or subtraction.
It wasn't that the Babylonians never thought
of making zero into a number, but rather it confused them.
See, zero doesn't behave like other numbers.
Add any number to any other number and it gets bigger.
Subtract a number from any other number
and it gets smaller, but not zero.
zero never makes any number bigger or smaller.
Furthermore, the number line can be viewed as a rubber band
and multiplication as stretching the rubber band.
Multiplying by 2 stretches the rubber band
by a factor of 2,
multiplying by 3 stretches the rubber band by a factor of 3,
but when you multiply by zero, you get zero.
There's no stretch.
The rubber band vanishes into thin air
and don't even get me started
with what happens when you divide by zero.
We still don't know how to handle that situation
so imagine how scary it would've been for the Babylonians.
No, they weren't ready for zero as a number.
Still it had a place in the world, but not for long.
(bell dings)
This is my favorite part of the story,
because it's so dramatic.
The ancient Greeks heard about zero
from the Babylonians and they hated it.
Not only did they reject it as a number,
they didn't like it as a placeholder
and avoided it wherever they could.
They didn't even have a name or symbol for it.
How could anyone feel so strongly about a number?
Well, it all started with a guy named Pythagoras.
You might remember him from such theorems
as Pythagoras' theorem.
But back in ancient Greece
he was a charismatic scholar and teacher.
He was also pretty eccentric.
He was convinced he was the reincarnated soul
of the Trojan Hero Euphorbos
and that all souls transmigrated to other bodies after death
including animals, which was why he was vegetarian.
Beans though were taboo
as he thought they were like genitalia.
Anyway, he developed
a cult-like following around mathematics.
His teachings revolved around the idea
that reality is mathematical in nature.
He believed that there was no distinction
between shapes and numbers
and a number couldn't exist without a shape.
The number 16 was a square number
as it could be arranged into 4 rows of 4.
The number 10 was a triangular number
as it could be arranged into, well, a triangle.
They derived relationships
between numbers in a geometric way.
For example, square numbers can be divided
up into two triangular numbers.
Zero didn't really fit with this view.
After all, what shape could zero be?
It's easy to think of a square
with width and height of 3 or a line of length 5
but was a square with a width and height of zero,
still a square?
Was a line of zero length, still a line?
To treat zero as a number
the Greeks would have to revamp their entire way
of doing mathematics and well, they didn't want to
but there was an even deeper reason
the Greeks rejected zero.
They were very mystical people
and thought that math the universe and God
were all intertwined.
Numbers represented concepts like earth, fire and water,
justice, reason and family.
Zero was associated with non-existence,
oblivion and the denial of God.
According to the Aristotelian doctrine
there was no void and to contemplate zero
was to question the existence of God.
This doctrine dominated the West
and zero was forbidden for over 1,500 years.
Now, the ancient Greeks did a lot of great things
but their rejection of zero
was probably one of their biggest blunders.
It's thought to be the only thing which stops them
from discovering calculus.
To show you how, let me tell you
about a paradox you might have heard of before.
Zeno's Paradox, where the athlete Achilles
challenges the tortoise to a race.
Achilles is faster than the tortoise,
he runs it 1 meter per second
while the tortoise runs at 1/2 meter per second.
To make it fair
Achilles gives the tortoise a 1 meter head start.
At what point will Achilles pass the tortoise?
To overtake the tortoise,
Achilles first needs to catch up to the place
where the tortoise was.
At 1 second, Achilles is at the 1 meter mark,
but by this time the tortoise has moved ahead
by 1/2 a meter meter.
At 1 1/2 seconds, Achilles is at the 1 1/2 meter mark,
but by this time, the tortoise has moved ahead
by a 1/4 of a meter.
At 1 3/4 of a second, Achilles is at the 1 3/4 mark,
but the tortoise has moved ahead by 1/8 of a meter.
Logically, this would continue indefinitely.
Zeno argued that because the distance
can be divided an infinite number of times
Achilles has an infinite number of distances to travel
to overtake the tortoise, therefore, he never will.
Now, Zeno didn't believe this is what would actually happen
in the real world.
It was an argument based purely on logic,
but the ancient Greeks couldn't find the flaw
in his argument and Zeno's paradox went unsolved
for over two millennia.
As we'll see, we would need to wait for zero to solve it.
(bell dings)
While zero was an outcast in the West,
it was welcomed with open arms in the East.
Like the Greeks, the Indians had learnt about zero
from the Babylonians,
but unlike the Greeks
they didn't have this geometric philosophy
or fear of nothingness.
In fact, the idea of the void was at the core
of the Hindu religion.
Emptiness was associated with spiritual peace
and creative potential.
Indian mathematicians embraced the idea of zero
and were the first in recorded history to transform it
from a placeholder into its very own number.
The exact details of how and when this happened
are lost to time,
but their quick acceptance of zero was probably due to
how they viewed mathematics.
They saw numbers as separate from the physical world,
entities to be manipulated without any physical meaning.
This meant numbers didn't have to make geometric sense.
You can't remove 4 acres of land from a 2-acre plot
as negative area doesn't make sense,
but you can subtract 4 from 2 to get -2.
This led to the idea of negative numbers
and the foundations of modern day algebra.
The ancient Indians were content to have zero
in their number system
since it fit perfectly between the positives and negatives.
Now, zero had a place on the number line.
Zero as we know it today had arrived.
(bell dings)
After a bumpy start to life,
zero would have the last laugh.
Over time, zero spread to Africa
where an Italian mathematician named Fibonacci was studying.
You might know his name from the Fibonacci sequence
but it turns out he also introduced
both zero and Arabic numerals to Europe.
Italian tradesmen couldn't resist how easy
they made calculations.
Despite the church still insisting that zero was Satanic,
its usefulness was irresistible
and zero spread throughout the West.
Finally, 1,500 years after its banishment
the West had accepted zero as a number
and it's a good thing they did,
because it forms the core
of a lot of modern day mathematics.
Let's go back to Zeno's paradox.
It's true that the distance Achilles must cover
to overtake the tortoise
can be divided an infinite number of times,
but that doesn't mean he must travel an infinite distance.
I'm sure you've noticed that the distances he must travel
get smaller and smaller.
Today, we would say that the terms are approaching zero.
We can see that as the terms approach zero.
The sum approaches 2.
This is called a limit.
The sum will never go beyond 2.
It's heading straight there.
As the terms approach zero, the limit of the sum is 2.
So Achilles will overtake the tortoise at the 2-meter mark.
The limit is the backbone of calculus.
It's the reason we can land rockets on the moon
and build the world's tallest skyscrapers.
The Greeks couldn't solve Zeno's Paradox
because they didn't have zero
so they couldn't really let the distance approach zero.
If they had,
they would most likely have discovered calculus.
I've made a video called
Three Paradoxes that gave us Calculus.
If you'd like to know more about limits
and the two other key ideas that make up calculus.
We've come a long way since the days
that zero was feared and rejected.
Today, zero has countless uses.
It's thanks to calculus that we can analyze
the progression of cancer,
set the minimum amount for credit card payments
and predict the orbit of the planets.
The computer or phone you're watching this video on
runs on strings of zeros and ones.
Without zero, there would be no computers, no social media
and no online games.
Modern day geometry and navigation rely on the idea of zero.
So next time you're in this situation
just think, maybe zero isn't so bad after all.
You know what I find most fascinating about this story
is just how much your beliefs can influence everything.
Like the ancient Greeks and Indians
had different beliefs about what the void meant
which influenced their mathematics,
which influenced what they could and couldn't do
with their mathematics.
Also, the fact that different cultures
invented different ways of doing mathematics,
but ultimately in the end, all the same ideas
ended up bubbling to the surface.
It raises the question, is math invented or discovered?
If the ancient Indians hadn't made zero into a number
would someone else have?
Did they really invent zero or did they discover it?
This question fascinated me so much
I made a feature documentary about it on Nebula.
What's Nebula?
Some creative friends and I teamed up
with our own streaming platform
where we don't have to worry about
the pressure of the YouTube algorithm
or click-bait titles to keep creating.
It's called Nebula and we're partnering
with our friends at Curiosity Stream.
Nebula is a place where we can focus on passion projects,
experiment with original content
or while staying completely ad-free.
Nebula features a lot of YouTube's top educational creators
like Real Engineering, Joe Scott, Isaac Arthur,
Medlife Crisis, and many more.
If you're a deep thinker
and like to contemplate the boundary between
math and philosophy, I think you'll like my documentary.
There are entire series
of exclusive content on there as well,
like Real Science's amazing new documentary series
about the key moments in our human evolution.
So what does all of this have to do with Curiosity Stream?
Well, Curiosity Stream make award-winning documentaries
ranging from science to history to nature.
So they obviously value education.
They want to support the creation
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So we've worked out a deal
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Curiosity Stream is offering 26% off their annual plan.
That's less than $15 a year
for both Curiosity Stream and Nebula.
I loved this four-part documentary series
on Curiosity Stream called The Story of Maths
which goes way more in depth
about the development of mathematics throughout the ages.
If you liked today's video, you will love this documentary.
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Bye.
(upbeat electro-pop music)