Today, every elementary school student learns how to do basic arithmetic.
What's so hard? Ask any of the math geniuses of ancient Greece like Pythagoras, Euclid, Aristotle, Plato, or Archimedes. Ask the Romans.
How do you figure this?
To make things worse, how about?
What are we taking for granted, exactly? Why didn't the ancients figure this out? How close did they come? What were the tools of ancient times that were begging for this leap? How would the world be different today if they had this clarity? Here we will explore these questions and dip our toes in some answers.
Confusion of the Day
The lack of clarity about numbers was pervasive even way after the new numbers were introduced to Europe. Take a look at this:
Man against Machine: Modern math against the abacus!
Calculating-Table by Gregor Reisch: Margarita Philosophica, 1503. The woodcut shows Arithmetica instructing an algorist and an abacist (inaccurately represented as Boethius and Pythagoras). There was keen competition between the two from the introduction of the Algebra into Europe in the 12th century until its triumph in the 16th.
I'm no expert in antiquities but the first thing I notice is that the perspective of the sums are wrong. The slant of the sums should align with the slant of the table edge. To me, that says the math was added to the print by someone other than the artist. It tells me that the artist didn't know numbers and most likely the subjects in the print didn't either. The second thing is that I don't understand what is being added and what is the result. Is it me or is it totally wrong? Regardless, it doesn't have the clarity to earn you a passing grade in elementary school today.
What was missing from the early number systems?
The main missing feature was the concept of constant base. What does that mean? For example, 1, 10, 100, 1000 all leverage the power of 10. The second key missing feature was a "place" holder, which holds a place without a value. The final abstraction is zero as a stand alone number (known as the additive identity) to make math manipulations complete.Here is the first known number using the modern number system with a zero place holder digit:
The number 605 in Khmer numerals, copied from the Sambor inscriptions in 683 AD. The earliest material use of zero.What is fascinating was that many early number systems had some, but not all, of these features, and often the rules were not always applied consistently.
What does a place value number system give us?
It gives us a dense way to represent numbers to very large and very small scales and high precision using decimal places. Somewhat miraculously, there are simple rules to add, subtract, multiply and divide any number. This applies to super large numbers, super small numbers, negative numbers and more complex numbers. At a deeper level, it allows a linear combination of digits by place. In other words you can add or subtract digits and keep the remainder value in place. Any value higher than the base will add to the place above and any value less than zero will borrow from the place above. This is simply not possible with IV + VI = X.From a math perspective an understanding of place value numbers with constant base sets a solid foundation and presents paths to quick advancement. For example, the simple and consistent rules begs us to imagine short cuts for large numbers.
and small numbers:
and then move on to non integer base and figure out compound interest for loans:
and then with some more imagination you are off to algebra:
and then you get comfortable with abstractions:
Roman Numerals
Let's look at Roman numerals and do a comparison with modern decimal numbers. What do we notice so far? Things start pretty much the same with I and 1 but then they start walking down different paths. The II uses two symbols whereas 2 uses a single, new symbol. We start to see a ray of sunshine. Let's move on. III is just using the same rule as with II and I. Just add them up: I + I + I = 3. Now IV is something new. First, we understand V to mean 5. Next, we are introduced to a new rule: take away I from V or take away 1 from 5, which equals 4. This rule was not in the original Roman numbers. It was introduced to shorten the numbers, since IIII is harder to read and write than IV. This is another sign that things are not going so well. When adding or subtracting or any other operations it is best to convert to the longer original format. Next.More of the same. Add all of the symbol values up. As we mentioned earlier IX is really just VIIII. In fact 9 is really just IIIIIIIII. There is nothing new here except for the X which is the star of this show:
The new number system introduces a completely new concept that makes math much easier:
The placement of a digit determines its scale. The scale is a constant, like 10 for decimal.
The other key ingredient is what you don't see: zero. Early concepts of zero were invented by the Chinese and Indians but the use of zero as a number and place holder was either forgotten or not fully baked until after the Roman empire was in decay thousands of years later. A real example of the absence of 0: there is no Year 0 AD. The Common era started with 1 because there was no 0 in the Roman numerals. Want to find out more? See: 0 Year
To really understand what is going on here let's continue our analysis.
Tally Numbers
The Greek and Roman number system is just a Tally system with fancy symbols. It's just a bunch of sticks that can be added up.Tally numbers are simple but cumbersome. In order to deal with large numbers we need a new concept. What is the problem with Roman numerals? A major problem is that the numbers count up too slowly. In our modern number system when adding a digit to the right of an integer makes the number ten times as big. This is a big deal. As an example, writing one million in the modern number system takes a couple of seconds: 1,000,000. In pure tally form it would take days to write out one million marks. In order to make numbers manageable the ancients came up with new symbols as shorthand. Unfortunately, this turned out to lead them down a dark path in terms of ease of use and understanding.
A path to the new number system
So, what was the breakthrough concept that shortened the number representation and at the same time made basic operations easy? Well, let's rewind to the tally system and take a step in a different direction.Imagine a scale - A perfectly balanced bar on a pivot. Imagine that the bar doesn't have any weight to clutter up your imagination. Now, let's say you only have pennies to count with. We set some rules. On the right of the pivot you can only stack pennies. This represents pure tallies. On the left you cannot stack pennies. The bar has to balance evenly on the pivot to satisfy us that the numbers mean the same thing.
Now, let's imagine some new ways to count. Let's start counting!
The left side shows somewhat compressed numbers compared with the stacks of pennies on the right hand side of the pivot. This system works but it has some problems. One of the problems is starting to appear with 3 and is pretty clear with number 4. One problem is that 3 can be represented in two ways: 3 or 2+1. The number 4 can be also be represented in 2 ways: 4 or 3+1. Moving to bigger numbers the duplication increases. Why is this a problem? If there is more than one way to represent a number then all sorts of problems arise in math. Also, it makes us think that there must be a denser way to represent the number. So we explore other representations.
Let's say that the 3rd place cannot have a penny. That solves the duplication for numbers 3 and 4. It turns out that when we notice duplication then we exclude that place then we see the rest of the numbers on the right side of the drawing and get the binary numbers.
What are binary numbers? They have digits that only have two values: either 1 or 0; on or off. Notice how the representations use space. At first glance, the binary numbers seem to use more space since they appear more spread out. But, the contrary is true since we never have to fill the slots that are not used. This makes binary numbers more efficient than the two other representations discussed here.
What do we gain from this? Lots. With just 2 digits (1 and 0) we can represent any integer. 10 = 8 + 2. Really, we're saying 10 = 2*2*2 + 2. Or just 10 = 2^3 + 2^1. This is (number of allowed digit values)^(place). Why is this so much more powerful than the tally system. Since the number doubles with every new digit the number grows quickly with very few bits. For example, 32 bits can hold a number up to 4 billion. Try writing that in tally.
Wow, we've come a long way! But what's missing here? We need a way to tell which slot the penny is in. How to do that? We need something that doesn't add any weight. So we create zero. It took thousands of years to imagine nothing.
Many things in life can be expressed as 10 or less. And now, we can easily grow the countable things by 10 times for every new digit added on the left! With this new power there is no need to add any new digits! The Roman number system needs a new symbol as numbers grow in order to write down large numbers efficiently.
Does anyone know the Roman numeral for 10 billion? The other awkwardness is demonstrated with one less than a big number. For example: 99 = XCIX and 999 = CMXCIX. Notice that the Roman numeral system has rules of placement but they are not simple and not powerful.
In the new number system, placement is more important to the final value than the digits themselves. Every movement in place multiplies or divides by 10. Off by one place and the number is either 10x bigger or 10x smaller. The benefits, however, are huge in terms of expression, comprehension, and math operations.
What about showing the number of atoms in the universe, you ask? Oh, about 10^80. All we need to do is add one new operator: the exponent. This is just another short hand to add a bunch of zeros to the right of the number 1. 10^0 = 1, 10^1 = 10, 10^2 = 100 and so forth.
There is a sense of place in the Romain numerals but it's different than what we know as place. IX = 9, XI = 11. How about IC = 99? No, this is not allowed. 99 = XCIX.
Now try 1990 - 11 = 1979:
Now imagine doing multiplication or division. That's hard.
And that's not all, there's more. In the year 1492, the decimal point was first used. It turned out that this new decimal placeholder number system could represent partial numbers as well. To represent them you can add a decimal point and add digits to the right. Imagine that you can do this and still have all the math operations like addition, subtraction, multiplication, division continue to work properly without any changes. Was this luck or what?
The borrowing of digits for subtraction and carrying for addition is much cleaner with the modern decimal numbers. You only have to borrow or carry one digit instead of a bunch. The rules are simple and easy to explain and understand.
The one invention the Greeks and Romans employed was base 10 numbers. The Babylonians used base 60 numbers which can be evenly divided by 2,3,4,5,6,10,12,15, and 30 making it useful for many calculations. That's where we get 60 minutes per hour, 360 degrees in a circle, and 360 days in the year (just kidding -- it may be that 360 was adopted at least partially due to it's closeness to the days in a year).
But wait there's more... How close were the ancient societies to adopting the modern constant base numbers? How close was Archimedes? So close, it turns out! Stay tuned for more.
A Time-line for the History of Mathematics
History of mathematics
zerorigindia.org - will research ancient writings in India to attempt to find the true origin of zero as a digit.
What do we gain from this? Lots. With just 2 digits (1 and 0) we can represent any integer. 10 = 8 + 2. Really, we're saying 10 = 2*2*2 + 2. Or just 10 = 2^3 + 2^1. This is (number of allowed digit values)^(place). Why is this so much more powerful than the tally system. Since the number doubles with every new digit the number grows quickly with very few bits. For example, 32 bits can hold a number up to 4 billion. Try writing that in tally.
Wow, we've come a long way! But what's missing here? We need a way to tell which slot the penny is in. How to do that? We need something that doesn't add any weight. So we create zero. It took thousands of years to imagine nothing.
Modern Decimal Numbers
This binary number system is great but it can be made even more practical. How to make this more human? How about if we have 9 different coins with weight 1 through 9. This gives us 0 through 9 power or a total of 10 times the weight per slot.Many things in life can be expressed as 10 or less. And now, we can easily grow the countable things by 10 times for every new digit added on the left! With this new power there is no need to add any new digits! The Roman number system needs a new symbol as numbers grow in order to write down large numbers efficiently.
Does anyone know the Roman numeral for 10 billion? The other awkwardness is demonstrated with one less than a big number. For example: 99 = XCIX and 999 = CMXCIX. Notice that the Roman numeral system has rules of placement but they are not simple and not powerful.
In the new number system, placement is more important to the final value than the digits themselves. Every movement in place multiplies or divides by 10. Off by one place and the number is either 10x bigger or 10x smaller. The benefits, however, are huge in terms of expression, comprehension, and math operations.
What about showing the number of atoms in the universe, you ask? Oh, about 10^80. All we need to do is add one new operator: the exponent. This is just another short hand to add a bunch of zeros to the right of the number 1. 10^0 = 1, 10^1 = 10, 10^2 = 100 and so forth.
Math Operations
When did you first realize that there aren't any zeros in Roman numerals? M = 1000, C = 100, X = 10. Without a zero it's hard to imagine place holders that work with arithmetic or other operations.There is a sense of place in the Romain numerals but it's different than what we know as place. IX = 9, XI = 11. How about IC = 99? No, this is not allowed. 99 = XCIX.
Now try 1990 - 11 = 1979:
Now imagine doing multiplication or division. That's hard.
Other Inventions making arithmetic easier
The + and - signs were invented in 1489. This was 2000 years after Pythagoras and his gang first developed geometry and arithmetic.And that's not all, there's more. In the year 1492, the decimal point was first used. It turned out that this new decimal placeholder number system could represent partial numbers as well. To represent them you can add a decimal point and add digits to the right. Imagine that you can do this and still have all the math operations like addition, subtraction, multiplication, division continue to work properly without any changes. Was this luck or what?
The borrowing of digits for subtraction and carrying for addition is much cleaner with the modern decimal numbers. You only have to borrow or carry one digit instead of a bunch. The rules are simple and easy to explain and understand.
The one invention the Greeks and Romans employed was base 10 numbers. The Babylonians used base 60 numbers which can be evenly divided by 2,3,4,5,6,10,12,15, and 30 making it useful for many calculations. That's where we get 60 minutes per hour, 360 degrees in a circle, and 360 days in the year (just kidding -- it may be that 360 was adopted at least partially due to it's closeness to the days in a year).
The Blindspot
So, the question remains. How is it possible that the ancient Greek math geniuses didn't figure out what every child today learns in elementary school? These are the people who discovered Pi, proved that square root of 2 cannot be represented by dividing two integers, and accurately measured the circumference of the Earth (see: ). Imagine how the world would be different today if math was more accessible to people over a two thousand year period. It must be one of the biggest blindspots in math history so far.But wait there's more... How close were the ancient societies to adopting the modern constant base numbers? How close was Archimedes? So close, it turns out! Stay tuned for more.
Some links
There are so many great writings on this subject. To explore more see the links below.A Time-line for the History of Mathematics
History of mathematics
zerorigindia.org - will research ancient writings in India to attempt to find the true origin of zero as a digit.
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