How big is that number?

As an antidote to the current national fixation and frustration related to Brexit can I offer a brief antidote.  In doing so I hope it can expose how poor we are as humans at assessing certain facets of the world around us and that our intuitive response may not always be best relied on and that we need time for more consideration or an expert view….  I think that’s very relevant given the rise of “easy answer” rhetorical politics.

I’ll use numbers as an example…. Try and put these numbers in order of magnitude, so what is the largest to the smallest….

• The number of seconds since the start of the Universe
• The number of ways you can order a pack of 52 cards
• The number of stars in the visible universe
• The number of atoms of Carbon in a piece of Coal the size of your fist (say 12g)

Some of the above we can be more certain of (or even precise) than others…. but as shown below these numbers are so different as to illicit a reaction of disbelief in most people.  Forgive me, this exercise has been done I guess many times (albeit using different examples), so I am certainly not claiming a new or unique insight, nor do I profess to be an expert in any of the relevant scientific disciplines! It is also quite possible that I could have made an error of the order of “10* ” given all the zeros below! (I am sure you will let me know)

Lets start – and I will round up and down a little to better compare the numbers in tens, hundreds, thousands, etc. 

How many seconds since the big-bang?

How many seconds since the big-bang?  Well this is a case of applying simple arithmetic against the current received scientific consensus that Universe is approximately 13.8 billion years old, that’s 13,800,000,000 years old!

So let’s do this…

60 seconds in a minute * 60 minutes in an hour * 24 Hours is   86,400 seconds per day

For a year of 365 days this is then 31,536,000 seconds

Given 13,8000,000,000 years this is 4.2*10¹⁷

Or   420,000,000,000,000,000 seconds!

For the purposed of our comparison let’s just call it 10¹⁷ -it’s a big number!

How many stars in the visible Universe?

So how many stars in the visible universe…. well this is a number that seems to increase over time as we see further into the universe and with higher resolution.  The current consensus is that a typical galaxy has approximately 400 billion (that’s 400*10⁹ or 400,000,000,000) stars and that there are approximately 200 billion galaxies.

This then suggest the number of starts in the observable universe as

200 billion * 400 billion

That’s 8*10²² or rounded up, 10²³

Or 100,000,000,000,000,000,000,000

That’s approximately one million times bigger than the figure for the number of seconds since the big bang!   So for every second since the big bang there are approximately 1,000,0000 stars.   That’s’ a staggeringly large number!

How many atoms in a piece of coal?

Let look at that piece of coal now.   Atoms and all the stuff inside an atom (electrons and protons & neutrons and their component quarks) are pretty small.   The size of an atom of Carbon (using atomic radius) is about 70pm (picometres). 

A picometre is 1*10^{-12 M}, that’s 0.000,000,000,001 M

That’s very small!

In Physics and Chemistry there also is a concept of molar mass which is used to better understand and analyse chemical reactions.  For many years 12g of Carbon-12 was used to define a concept called the “mole” and the number of atoms in a mole.  Carbon-12 also has an atomic mass number of 12 (containing as it does 6 electrons and 6 protons).

Without trawling through the history and the background of Chemistry and Physics, scientists have used a concept known as “Avogadro’s number” which defines the number of atoms in a mole; and a mole of every element has the same number of atoms  – so a mole of carbon has the same number of atoms as a mole of lead, or gold or silver, etc. 

Avogadro’s number is 6.02214076×10²³

or 6.022   * 100,000,000,000,000,000,000,000

For one mole of Carbon-12, which weighs 12g (about the size of piece of coal you can hold in your hand) we therefore have 6.02214076×10^{23 atoms}.

Let’s round it up for our question and call that 10²⁴.  That’s a lot…about 10 times more than the current estimate for the number of stars in the visible universe!

So you can hold in your hand more atoms than there are stars in the visible universe!

How many ways can you sequence the 52 cards in a deck of cards?

Let’s look at a pack of 52 cards – how many ways are there of ordering those cards?

Let’s start with 3 cards as an example.  If you lay them out and move them around you will find there are 6 ways of ordering those 3 cards.

With 4 cards the number increases to 24, with 5 it is 120.

Try it – it is slightly counter intuitive and produces bigger numbers than most of us intuitively think.

The maths is based on “factorial”, expressed as n!  which is defined as follow

n! = n*(n-1) *(n-2) *(n-3) ….3*2*1

So 5! = 5*4*3*2*1 which equals 120

Then 6! = 6*5*4*3*2*1 which is 720

10!  = 3,628,800

You can see how quickly that number gets big as the number of cards increases.

If you do that for 52 cards the answer is a little mind blowing if you haven’t done this before

52! = 8.0658 * 10⁶⁷

Or

80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000

Let’s call it 10⁶⁸ for our comparison…it’s a huge number, just from a pack of 52 cards!

In fact if you efficiently shuffle a pack of cards you will end up with a sequence never before manifest in all of human card playing history; the same the next time, and the next. In fact there are so many possible sequences, each shuffle is effectively unique. And you can hold a deck of cards in the palm of your hand.

So overall, we have:

	Rounded up/down figure to nearest 101
Numbers of seconds since big bang	1017	100,000,000,000,000,000
Numbers of stars in the visible Universe	1023	100,000,000,000,000,000,000,000
Numbers of atoms in a 12g piece of coal	1024	1,000,000,000,000,000,000,000,000
Number of ways of ordering the 52 cards in a deck	1068	100,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000

So from the smallest to the largest the answer to the question is:

• The number of seconds since the start of the Universe
• The number of stars in the visible universe
• The number of atoms of Carbon in a piece of Coal the size of your fist (say 12g)
• The number of ways you can order a pack of 52 cards

I guess the lesson, for all of us, is that we can’t always rely on our intuition.  Especially in these anti-expert brexity times, we actually need expertise and objective analysis more than ever.