3.1.1 Powers of ten and scientific notation
In this study note you will look at how to write small and large numbers using scientific notation.
Study note: Powers of ten and scientific notation
Figure 11(a) shows the production of lead in tonnes (also known as metric tons) on a scale using different powers of ten (100, 102, etc.). When you see numbers written down, it is quite easy to read and understand them when they have few digits; for example, 0.01, 0.5, 4, 15 or 132. But when numbers have a lot of digits, for example, a small number such as 0.0000067, or a very large number such as 1 700 000 000, they are less easy to read, and consequently it is harder to understand what they are telling you. For example, if you are asked to say ‘75 kg’ you would probably respond immediately with ‘seventy-five kilograms’. But if you were asked to say the mass 330000000 tonnes, you would probably have to start counting the zeros.
To make large and small numbers easier to comprehend, there are two options. One is to use the prefixes for words illustrated in Table 2 below. The other is to use numbers as in the final column of Table 2 which is labelled ‘Power of ten’, where the power is the number of tens that are multiplied together. For example, 102, which you would say as ‘ten to the power of 2’, means that two tens are multiplied together (i.e. 10 × 10). So
102 = 100.
Similarly, ten to the power of three (i.e. 10 × 10 × 10) is
103 = 1000.
And so on. Clearly, 107 is easier to understand than 10 000 000. Note that 101 implies just one ten, that is, 101 = 10, so you do not add the power 1 in this case. When dealing with powers of 10 you could also just say that the power is the number of zeros after the 1, so 100 is just the number 1.
That covers numbers greater than 1, but what about numbers less than 1 such as 0.1? In powers of ten this would be written as 1 divided by 10, so
and this is written as 10–1. Similarly, 10–4 is 1 divided by 10 four times:
So how would you write the number 150 using powers of 10? The number 150 is 1.5 × 10 × 10, so would be written 1.5 × 102. This form of writing numbers is known as scientific notation. A number written in scientific notation always looks like this:
(number between 1 and 10) × 10some power.
This superscript notation can also be used to show powers of units. For example:
Square kilometres (for area):
Metres per second (for speed):
Square kilometres per year (e.g. for a change in area through time):
Table 2 Examples of different ways of writing large and small numbers
|Prefix||Prefix name||Meaning||Number or fraction||Decimal||Power of ten|
|G||giga||billion or thousand million||1 000 000 000||1 000 000 000||109|
|M||mega||million||1 000 000||1 000 000||106|
|µ||micro||millionth||1/1 000 000||0.000 001||10–6|
|n||nano||billionth||1/1 000 000 000||0.000 000 001||10–9|
Sounds, seismic waves and starlight all have something in common: they are measured in powers of ten. Each can vary by so much that logarithmic scales are needed to describe the whole range. For example, a sound level of 110 decibels (dB) is 10 times louder than one of 100 dB. An earthquake of magnitude 8.0 has seismic waves that are 10 times larger than in an earthquake of magnitude 7.0. The brightness (‘apparent magnitude’) of stars is also measured on a kind of logarithmic scale.
Activity 3 Powers of ten
Around 66 million years ago an asteroid or comet around 10 km wide hit the Earth, creating the 180 km wide Chicxulub crater in Mexico and causing a mass extinction including that of the dinosaurs. The impact has been estimated as causing a magnitude 13 earthquake. In recent times, the fifth largest earthquake ever measured (at the time of writing) was the 2011 Japanese Tōhoku earthquake, which had a magnitude of 9.
How many times larger would the seismic waves have been for the impact earthquake than the Tōhoku earthquake?