6 Scientific notation

It is often necessary to use very large or very small numbers such as 78000000 and 0.00000034. Scientific notation can be used to express such numbers in a more concise form. Each number is written in the form

$a\times 10^n$

where $a$ is a number between 1 and 10. We can make use of the following facts:

$10=10^1,100=10^2,1000=10^3\text{and so on}$

and

$0.1=10^{-1},0.01=10^{-2},0.001=10^{-3}\text{and so on.}$

For example,

$\bullet$ the number 5000 can be written $5\times 1000=5\times 10^3$

$\bullet$ the number 403 can be written $4.03\times 100=4.03\times 10^2$

$\bullet$ the number 0.009 can be written $9\times 0.001=9\times 10^{-3}$

Furthermore, to multiply a number by $10^n$ the decimal point is moved $n$ places to the right if $n$ is a positive integer, and $n$ places to the left if $n$ is a negative integer. (If necessary additional zeros are inserted to make up the required number of digits before the decimal point.)

Task!

Write the numbers 0.00678 and 123456.7 in scientific notation.

Answer

6.1 Engineering constants

Many constants appearing in engineering calculations are expressed in scientific notation. For example the charge on an electron equals $1.6\times 10^{-19}$ coulomb and the speed of light is $3\times 10^8{\text{m s}}^{-1}$ . Avogadro’s constant is equal to $6.023\times 10^{26}$ and is the number of atoms in one kilomole of an element. Clearly the use of scientific notation avoids writing lengthy strings of zeros.

Your scientific calculator will be able to accept numbers in scientific notation. Often the $E$ button is used and a number like $4.2\times 10^7$ will be entered as $4.2E7$ . Note that $10E4$ means $10\times 10^4$ , that is $10^5$ . To enter the number $10^3$ say, you would key in $1E3$ . Entering powers of 10 incorrectly is a common cause of error. You must check how your particular calculator accepts numbers in scientific notation.

The following Task is designed to check that you can enter numbers given in scientific notation into your calculator.

Task!

Use your calculator to find $4.2\times 10^{-3}\times 3.6\times 10^{-4}$ .

Answer

Exercises

1. Express each of the following numbers in scientific notation:
   1. 45,
   2. 456,
   3. 2079,
   4. 7000000,
   5. $0.1$ ,
   6. $0.034$ ,
   7. 0.09856
2. Simplify $6\times 10^{24}\times 1.3\times 10^{-16}$

Answer