One million is a big number. To really comprehend numbers such as this we need some sort of scale. So, how big is a million? And what about even bigger numbers? Millions, billions, even trillions are thrown around in everyday conversation today without much thought about the differences in size of these huge numbers.

Typically there are about 2000 letters or other characters on each page of a paperback book. So a 500 page book contains around one million characters. A bookcase might contain 100 books of this size, so that would provide storage for about one hundred million characters. Ten good-sized bookcases could contain around one billion characters.

So there are a million characters in a large book and a billion characters in a collection of bookcases. But it would require a million large books to hold a trillion characters—one book for every character in a large book!

Manchester Central Library is the second largest lending library in the UK with 35 miles (56 kilometres) of shelf space, and it contains around one million books. So Manchester Central Library, housed in the ornate building shown below, contains somewhere in the region of a trillion characters in total. (However, this is probably an over-estimate as the average book length in the library is almost certainly under 500 pages.)

*St Peter’s Square, Manchester. Left to right: Midland Hotel, Central Library and Town Hall. Credit: Wikimedia – Gtosti.*

**About Time**

The relative sizes of these large numbers can also be gauged by considering some time intervals.

Puzzle: How long is one million seconds?

Answer: 11 days 13 hours 46 minutes and 40 seconds.

Puzzle: How long is one billion seconds?

Answer: About 31 years 251 days and 7 hours.

Puzzle: How long is one trillion seconds?

Answer: About 31,688 years.

**The Index**

In our usual base 10 number system one million is represented as 1,000,000 and one billion as 1,000,000,000. This notation becomes unwieldy as the numbers grow larger. It is often much simpler to use index notation. The index just counts the number of zeros, or equivalently the number of powers of ten, as in base 10 each power of ten adds another zero: 10 = 10^{1}, 100 = 10^{2}, 1000 = 10^{3}, so one million is 10^{6} and one billion is 10^{9}. More generally, a number such as 33,690 may be written as 3.369 × 10^{4}, because 33,690 = 3.369 × 10 × 10 × 10 × 10.

Index notation is useful when numbers are multiplied. For instance, the sum 100 × 1000 = 100,000 is equivalent to 10^{2} × 10^{3} = 10^{5}, so to multiply the powers of ten we simply add the indices. This simplification in the algorithm for multiplication offers great advantages in using index notation.

Similarly, division corresponds to subtraction of indices. For example, 100,000/1000 = 100 is equivalent to 10^{5}/10^{3} = 10^{2}.

In general, indices do not just refer to powers of ten, the index represents the number of times that any number is multiplied by itself, so that 2^{1} = 2, 2^{2} = 2 × 2 = 4, 2^{3} = 2 × 2 × 2 = 8 and so on.

Puzzle: How many seconds have there been since the Big Bang?

Answer: There are about 3.16 × 10^{7} seconds in a year and it is 13.8 × 10^{9} years since the Big Bang, so there have been something like 3.16 × 10^{7} x 1.38 x 10^{10} = 4.36 × 10^{17} seconds since the Big Bang.

**Roots**

Index notation was introduced to simplify expressions in which numbers are multiplied by themselves several times, with the index indicating the number of multiples. Now, by using the rules for multiplication and division, this notation can be consistently extended to include indices of zero, negative numbers and fractions, even though the meaning of such indices may not be immediately obvious. Using the division rule, 10^{2}/10^{3} = 10^{-1} and this is equivalent to 100/1000 = 1/10, so 10^{-1} = 1/10. Similarly, 10^{-2} = 1/10^{2} , 10^{-3} = 1/10^{3} and so on. In general, a number with a negative index is equal to the inverse of the same number with a positive index.

Again using the division rule, the fraction 1000/1000 = 1 is equivalent to 10^{3}/10^{3} = 10^{0} = 1. Similarly, the fraction 10,000/10,000 = 1 is equivalent to 10^{4}/10^{4} = 10^{0} = 1. In fact, any number with an index of zero is equal to 1. This might seem odd, but it follows automatically if we use the division rule for indices consistently. Any number divided by itself always equals one, and this is essentially what the index zero means.

So what does 10^{½} mean? By the addition rule 10^{½} x 10^{½} = 10^{1} = 10, so 10^{½} is a number which multiplied by itself equals 10. In other words 10^{½} must equal the square root of 10. Similarly, 10^{¼} is equal to the fourth root of 10.

**The Googol**

Index notation gives us a convenient and compact way to express very large numbers. It is powerful, but deceptively succinct. Take the number 1 followed by 100 zeros. This is an incredibly large number, but it can be represented as 10^{100}. Edward Kastner tells a charming story in the book *Mathematics and the Imagination* of when he introduced index notation to his young nephew Milton Sirotta in 1920. The nine-year old Milton was so impressed by the vastness of this huge number that he named it *one googol*.

In 1998 two students, Larry Page and Sergey Brin, studying for PhDs in computer science at Stanford University, California founded a new tech company. The company was based on an algorithm for ranking web pages that they had invented and incorporated into a new search engine. Less than two decades later the company is valued at well over $100 billion (or perhaps we should say $10^{11} ).

Larry Page and Sergey Brin were inspired by Kastner’s story when devising a name for their browser. *Google* was chosen to signify that its users would be able to mine vast quantities of information. The slight difference in spelling was apparently a mistake, but it may have proved useful for the purposes of trademarking the company.

Puzzle: How big would a library need to be to contain a total of one googol characters in its books?

Answer: Unfeasibly big. There are nowhere near enough atoms in the visible universe to form a googol characters. A rough calculation of the number of atoms in the universe will show that this must be the case. It will also illustrate just how powerful index notation is.

**Eureka!**

In the third century BC Archimedes wrote a short treatise known as *The Sand Reckoner* in which he attempted to calculate the number of grains of sand that would fill the entire universe.

We will aim even higher than Archimedes. Our universe is much larger than that imagined by Ancient Greek astronomers and our grains are atoms, which are much smaller than the sand grains considered by Archimedes.

The matter in the universe is three quarters hydrogen, the lightest atom, and one quarter helium, the next lightest, but as we are aiming for a rough estimate, we won’t need to take this into account.

The number of hydrogen atoms in one gram of hydrogen is known as Avogadro’s number N_{A}. It is about 6×10^{23}.

The mass of the Sun M* is 10^{30} kilograms or 10^{33} grams.

According to a recent estimate the mass of the galaxy M_{MW} is about 700 billion or 7 × 10^{11} solar masses. Much of this is dark matter, so it is not composed of atoms and, as yet, no-one knows what it consists of. But we will assume that it is all hydrogen atoms, so our final result will be an overestimate of the total number of atoms.

There are about N_{g} = 2 × 10^{11} galaxies in the visible universe.

Multiplying these numbers together we obtain a rough estimate of the number of atoms in the universe:

N_{A} × M* × M_{MW} × N_{g} = 6 × 10^{23} × 10^{33} × 7 × 10^{11} × 2 × 10^{11} = 84 × 10^{78} ~ 10^{80} ,

where the symbol ~ means approximately equal to.

So there are somewhere in the region of 10^{80} atoms in the visible universe, and this is likely to be a bit of an over-estimate.

Of course, 10^{80} is a vast vast number, one followed by 80 zeros. But it is far far less than a googol! Smaller by a factor of 10^{20}, which is itself an enormous number. So clearly, it would be impossible to form a library with sufficient books to hold a googol characters, even if the entire universe could be converted into ink. Nevertheless, we can easily write down a googol: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000

or simply 10^{100}.

Index notation is deceptively compact when applied to extremely large numbers. A number such as 10^{100} looks rather sweet and innocent, but it is almost impossible to comprehend a number of this magnitude. As we have seen, it is far greater than the number of atoms in the visible universe. This shows just how deceptive index notation can be.

**The Googolplex**

But young Milton Sirotta was not satisfied with the arithmetical peak that he had conquered and proposed the name *googolplex* for a number equal to ‘one, followed by writing zeroes until you get tired’. Kasner suggested to his nephew that they should adopt a more formal definition ‘because different people get tired at different times and it would never do to have the boxer Carnera be a better mathematician than Einstein, simply because he had more endurance and could write for longer.’ Kastner proposed that a googolplex should be equal to 10 to the power of a googol.

Of course, it is impossible to write down a googolplex in ordinary Arabic numerals, as it would be 1 followed by a googol zeros and that is more zeros than there are atoms in the universe. However, we can easily write down this behemoth using index notation: 10^{10100}.

The corporate headquarters of Google in Mountain View, California is known as the Googleplex.

**Further Information**

*Mathematics and the Imagination* by Edward Kasner and James R. Newman was published in 1940. It is still in print almost 80 years later.