We have all been taught that a million is equal to 10^6, billion is 10^9, trillion is 10^12, quadrillion is 10^15, quintillion is 10^18 and the list continues. This type of system is what is taught in virtually the entirety of the English-speaking world. This system states that billion is a thousand million, trillion is a thousand billion, quadrillion is a thousand trillion, and so on is what is called the short system or the American system of numbers. This is opposed to the Old British system which states million is equal to 10^6, and billion is 10^12 or a million million (what is equivalent to trillion in the American system), and trillion is a million million million (what is equivalent to quintillion in the American system and so on. This old British system of billions and trillion and other such huge numbers is not used anymore, rather the American system is what is being used in Britain today. The American system is what all English-speaking mathematicians use regardless of nationality and this system is also what is taught in British schools. Prior to 1974, the old British system was used to denote billions and trillions, but in 1974, this system was officially replaced with the American system (also known as the short system.) The entire English-speaking world uses this system of denoting such exponentially huge quantities whilst all of continental Europe uses the old British system. Even though the American system is what is used in standard mathematics in the English-speaking world, the old British system or the long system is more mathematical consistent and makes more logical sense. Many of the older generations of English people will refer to the old British system as the correct number system and they are correct for that matter. Let us look at why.
The American system multiplies every upcoming number (million, billion, trillion) by a thousand every time these kinds of numbers increase. Here is an example of what I mean:
1000000 = million = thousand thousand = 1000^2
1000000000 = billion = thousand million = 1000^3
1000000000000 = trillion = thousand billion = 1000^4
1000000000000000 = quadrillion = thousand trillion = 1000^5
1000000000000000000 = quintillion = thousand quadrillion = 1000^6
I gave an example of how the most commonly used system, the American system assigns quantities to the numbers by multiplying by a thousand each time. After seeing how the old British system works, you’ll see why it makes more sense than the universally accepted American system. So let us take a look at the former British system.
The old British system multiplies every number after million by a million and assigns quantities to names of numbers:
1000000 = million = 1000000^1
1000000000000 = billion = 1000000^2
1000000000000000000 = trillion = 1000000^3
1000000000000000000000000 = quadrillion = 1000000^4
In this kind of system, the word matches with the quantity and is mathematically consistent. For example, the billion has the root “bi” meaning two and billion is a million raised to the power of two! So the name of the number has an etymological connection with the quantity which proves the word billion was originally meant to mean a million squared. Similarly, the word trillion has the root word “tri” meaning three and a trillion is a million raised to the power of three. The quadrillion has the root “quad” meaning four and quadrillion is a million raised to power of four. Quintillion has the root word “quin” meaning five and quintillion is a million raised to the power of five. This shows that the meaning of the word corresponds to the number itself whereas in the American system billion is said to be a thousand raised to power of three, which is rather odd because the root “bi” and the word three don’t match up. Like this, trillion is a thousand raised to the fourth power when the root “tri” and four don’t match up. So whence this kind of incompatibility between the word and the number in the American system? The short answer is that someone invented this as the new number system in the USA, but didn’t think about the actual mathematical meaning of the words like billion and trillion. This was blindly adopted as the official system of numbers in the UK. I would prefer the old British system but won’t use it due to the fact that this old system is now obsolete and the American system is now in use officially.
There is actually a word for a billion (in the American system) in the old British system. Since a billion is equal to 10^9 in the short system, the word for 10^9 in the long system is milliard, which was also known as a yard. The word for 10^15 (which would be quintillion in American system) would be known as a billiard in the long system.
Different cultures had their own system of defining numbers. The Mayans in fact didn’t even use the decimal system! They used the vigesimal system, or the base 20 system to count. So that means that they didn’t even use what we would call a million or a billion. Instead they had names for the following numbers: Hun for 1, which is 20^0, Kal for 20, which is 20^1, Bak for 400 which is 20^2, Pic for 8000 which 20^3, Cabal for 160000 which is 20^4, Kinchil for 3200000 which is 20^5 and so on.
The Greeks also had a rather interesting system. They used the word “myriad” to mean ten thousand. Myriad in English means huge or uncountably huge but in Greek it means ten thousand. The Greeks too used the long system but didn’t use words like million. Million was known as a hundred myriad in their system where myriad means ten thousand. Asia also has completely different systems of defining a million and billion out of which is the system of using words like lakh and crore which us Indians use.
That is it for today’s article about different systems of numbers and I hope you liked it. Thank you very much for reading this article and I hope you learned something new!
Update: I am writing a blog about the physics theory of quantum gravity which is coming out soon! Hope you will read it once it comes out and learn something new. Also, I would like to wish everyone a happy new year and I hope 2022 brings us something new!
Sources: Numberphile and Lumen Learning.
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