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When studying science, you will likely come across numbers that are either extremely large or very small. It often isn't convenient (or practical) to write out the numbers ‘longhand’. An example of this is in using the constant for the speed of light in vacuum. The value of this constant is approximately three hundred million metres per second (m s⁻¹), which can be written as 300 000 000 m s⁻¹. This method of writing numbers is sometimes called non‐exponential form. A much simpler way of expressing this number is in exponential form, so 300 000 000 m s⁻¹ is written as 3 × 10⁸ m s⁻¹. The number 10⁸ represents one hundred million or 100 000 000. When is it written as × 10⁸ this means there are eight zeroes after the 3. Some numbers are very small: for example, the Planck constant is 6.6 × 10⁻³⁴ J s. The number 10⁻³⁴ indicates there are 33 zeroes after the decimal point and before the number itself. Clearly, it would be very tedious to write the number out in full!

Take care!

3 × 10² = 300 (i.e. two zeroes after the number before the decimal point) 3 × 10⁻² = 0.03 (i.e. just one zero after the decimal point before the number)

When a number is written in exponential form, it is called scientific notation. The standard format is that the number is written with one digit before the decimal point and the multiplier 10ⁿ is added after the number. The following examples show how numbers can be rewritten using scientific notation:

 4 is written as 4 × 100.Any number raised to the power of zero is equal to 1, so 100 = 1.

 42 is written as 4.2 × 101.It isn't usual to use the factors 100 and 101 unless specifically required to give an answer in scientific notation.

 242 is written as 2.42 × 102.

 2424 is written as 2.424 × 103.

To express numbers smaller than 1 using scientific notation, i.e. a decimal number, the number is converted to a number between 1 and 10 and the exponential factor 10^–n used to indicate the position of the decimal point, as in these examples:

 0.4 is written as 4 × 10−1.

 0.042 is written as 4.2 × 10−2.

 0.00242 is written as 2.42 × 10−3.