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Astronomers, biologists, engineers, physicists and many others encounter
quantities whose measures involve very small or very large numbers.
For example, the distance of the Earth from the Sun is approximately
144,000,000,000 metres and the distance that light will travel in 1 year
is 5,870,000,000,000 metres.
It is sometimes tedious to write or work with such numbers. This
difficulty is overcome by writing such numbers in standard form.
E.g. 144,000,000,000 = 1.44 × 10¹¹
5,870,000,000,000 = 5.87 × 10¹²
If a quantity is written as the product of a power of 10 and a number
that is greater than or equal to 1 and less than 10, then the quantity is
said to be expressed in standard form (or scientific notation).
For example, 658 = 6.58 × 10²
Note:
- We have expressed 658 as a product of 6.58 and a power of 10.
Clearly, 6.58 is between 1 and 10. So the standard form of 658
is 6.58 × 10².
- The decimal point is shifted to the left by 2 places, and 2 appears
as the positive index in the power of 10.
In general:
In converting a number to standard form, if the decimal point is
shifted to the left p places, then p
appears as a positive index in the power of
10.
Note:
The decimal point is shifted to the right by 2 places, and the negative
of 2 appears as the index in the power of 10.
In general:
In converting a number to standard form, if the decimal point is
shifted to the right p places, then
the negative of p appears as a negative
index in the power of 10.
Example 15
Solution:
Key Terms
standard form, scientific notation
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