Very Big Numbers

In a word

A billion is certainly a large number but it used to be even bigger. In Britain, until the 1970s, a billion was considered to be a million million, that is one and twelve zeros (1 000 000 000 000). Since then, however, the American version of a billion has come into general usage worldwide and now it usually represents a thousand million, that is one and nine zeros (1 000 000 000). To keep in sequence then, a trillion is now reckoned to be a thousand billion, that is one and twelve zeros (1 000 000 000 000).

Mathematical shorthand

Mathematicians are shrewd operators and it did not take them long to realise that writing very large numbers with masses of zeros would be very tiresome. So they devised a type of mathematical shorthand where the number of zeros required could be represented by what became known as index numbers, or indices.

One hundred, for example, could be written as 10² as it is 10 x 10. Likewise 1000 would be 10³, meaning three tens multiplied together (10 x 10 x 10). 10000 is 104 and 100000 is 105. Further up the scale a million would be written as 106, a billion as 109 and a trillion as 1012.

There are also situations where the use of a decimal point can seriously reduce the number of zeros that have to be written. For example, a newspaper headline might give the transfer fee for an expensive footballer as £5.5 million instead of writing £5,500,000. A farmhouse with surrounding land might sell for £9.6 million (£9,600,000), or a large company could declare profits of £1.4 billion (£1,400,000,000).

Over budget

Construction companies are well known for going over budget as projects develop and get out of hand. What are already expensive undertakings at their conception seem to escalate out of control once steel and concrete begin to take shape.

Listed below are the approximate building costs of some of the world’s most expensive structures. Say the amounts out loud and then try to write them in full using the correct number of zeros each time.

An extra challenge

If $1 is worth about £0.67 ($1000 = £670), calculate what each of these construction projects would cost in pounds sterling.

Numerical Pounds Sterling
Tokamak Reactor, Cadarache, France, US$14 billion
Abraj Al Bait, Mecca, Saudi Arabia US$15 billion
Marina Bay Sands, Singapore US$5.5 billion
Resorts World Sentosa, Singapore US$5.38 billion
Emirates Palace, Abu Dhabi, UAE, US$4.46 billion
Cosmopolitan, Las Vegas, USA, US$4.16 billion
Marina Bay Sands Erwin Soo from Singapore, Singapore, CC BY 2.0 , via Wikimedia Commons
Marina Bay Sands, Singapore. Marina Bay Sands Erwin Soo from Singapore, Singapore, CC BY 2.0 , via Wikimedia Commons

Interplanetary travel

Distances within the solar system are so vast that it is difficult to comprehend them. These are the approximate average distances from each of the planets to the Sun. Rewrite them using index notation and the decimal equivalence method mentioned earlier.

Index Notation Decimal Equivalence
Mercury 58 000 000 km
Venus 110 000 000 km
Earth 150 000 000 km
Mars 30 000 000 km
Jupiter 80 000 000 km
Saturn  400 000 000 km
Saturn  400 000 000 km
Neptune  500 000 000 km
Solar System Tdadamemd, CC BY-SA 3.0 , via Wikimedia Commons

Now work conversely. Here are some facts about the Moon. This time write the approximate numbers in full.

Diameter = about 4 x 10³ km
Surface area = about 4 x 107 km²
Weight = about 8 x 1019 tonnes
Visible craters = about 3 x 104
Lunar Craters and seas What The Puck, CC BY-SA 4.0 , via Wikimedia Commons
Lunar Craters and seas. CC BY-SA 4.0 , via Wikimedia Commons

Big crowds

The current population of the world is reckoned to be about 7.2 billion. Below is a breakdown of the approximate population figures for each of the main continents. Write out the numbers in full and then arrange them in order of size, smallest to largest. Did the final order cause any surprises? Why is there such a vast difference between the smallest and the largest numbers?

Numerical Size (smallest to largest)
Africa 1.2 billion
Antarctica 5,000
Asia 4.5 billion
Europe 746.4 million
North America 579 million
Oceania/Australasia 46 million
South America 422.5 million
Central Intelligence Agency, Public domain, via Wikimedia Commons
Africa, population density. Central Intelligence Agency, Public domain, via Wikimedia Commons

If you enjoyed this interesting blog, then you will love the articles in our Megamaths issue. This article appeared in the Game Theory issue which you can access via our digital subscription!

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Words: Frances Durkin. Illustration: Yann Bastard